## Dr. Kourosh Parand

Professor of Scientific Computing

Department of Computer Science

Shahid Beheshti University

## About Me

##### Spectral Methods, ODEs, PDEs And Scientific Computing

### I am Professor @ Department of Computer Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University

My main research field is Scientific Computing, Spectral Methods, Meshless methods, Ordinary Differential Equations(ODEs), Partial Differential Equations(PDEs) and Computational Neuroscience Modeling.

- Age46
- ResidenceIran
- AddressFaculty of Mathematical Sciences, Shahid Beheshti University,Shahriyari Sq. Evin, Tehran
- e-mailk_parand@sbu.ac.ir
- Phone+9821 29903003

### Interested in

#### Scientific Computing

is a rapidly growing multidisciplinary field that uses advanced computing capabilities to understand and solve complex problems.

#### Spectral Methods

are a class of techniques used in applly mathemathic and scientific computing to numerically solve certain differential equations, potentially involving the use of the Fast Fourier Transform.

#### Meshless methods

are uniquely simple, yet provide solution accuracies for certain classes of equations that rival those of finite elements and boundary elements, without requiring the need for mesh connectivity.

#### Neuroscience Modeling

is a branch of neuroscience which employs mathematical models, theoretical analysis and abstractions of the brain to understand the principles of the nervous system.

### My Activities

#### Artices

163#### Conferances

14#### Publications

14#### Phd&Msc

25## Students/People

##### Spectral Methods, ODEs, PDEs And Scientific Computing

### I am Professor @ Department of Computer Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University

My main research field is Scientific Computing, Spectral Methods, Meshless methods, Ordinary Differential Equations(ODEs), Partial Differential Equations(PDEs) and Computational Neuroscience Modeling.

- Age46
- ResidenceIran
- AddressFaculty of Mathematical Sciences, Shahid Beheshti University,Shahriyari Sq. Evin, Tehran
- e-mailk_parand@sbu.ac.ir
- Phone+9821 29903003

### Postdoc

##### 2010

#### Dr First Name Last Name

Phd Degree, Shahid Beheshti UniversityName of project or reaserches.

##### 2012

#### Dr Fist Name Last Name

Phd degree , shahid beheshti universityName of project and etc.

##### 2008

#### Specialization Course

University of StudiesMauris magna sapien, pharetra consectetur fringilla vitae, interdum sed tortor.

### Phd

##### 2010

#### First Name Last Name

Phd Degree, Shahid Beheshti UniversityName of project or reaserches.

##### 2012

#### Fist Name Last Name

Phd degree , shahid beheshti universityName of project and etc.

##### 2008

#### Specialization Course

University of StudiesMauris magna sapien, pharetra consectetur fringilla vitae, interdum sed tortor.

### MSc

##### 2010

#### First Name Last Name

Phd Degree, Shahid Beheshti UniversityName of project or reaserches.

##### 2012

#### Fist Name Last Name

Phd degree , shahid beheshti universityName of project and etc.

##### 2008

#### Specialization Course

University of StudiesMauris magna sapien, pharetra consectetur fringilla vitae, interdum sed tortor.

## Resume

##### 6 Years of Experience

### Article

##### 2018

#### Using modified generalized Laguerre functions, QLM and collocation method for solving an Eyring–Powell problem

K Parand, Z Hajimohammadi Journal of the Brazilian Society of Mechanical Sciences and Engineering 2018(4) 250-254Abstract:The purpose of this paper is to introduce an effective strategy for solving boundary layer flow of an Eyring–Powell fluid over a stretching sheet in unbounded domain. This paper introduces a combination of Modified Generalized Laguerre, the

#### The pseudospectral Legendre method for solving the HIV infection model of CD4+T cells

Kourosh Parand, Fatemeh Mirahmadian, Mehdi Delkhosh Nonlinear Studies 2018(25) 241-250Abstract:: In this paper, the Pseudospectral Legendre (PL) method is presented to approximate the solution of the HIV infection model of CD4+T cells. The method is based on the Lagrangian interpolation and the zeros of Legendre polynomial. Using the proposed method, the given system of three differential equations is converted into a system of algebraic equations. By solving the system of the algebraic equations, the unknown coefficients are computed and then the approximate solutions are obtained. The main aim of this paper is to demonstrate the PL method, and a comparison is made with the existing methods in the literature.

#### An efficient analytic approach for solving Hiemenz flow through a porous medium of a non-Newtonian Rivlin-Ericksen fluid with heat transfer

Kourosh Parand, Yasaman Lotfi, Jamal Amani Rad Nonlinear Engineering 2018 in pressAbstract:In the present work, the problem of Hiemenz flow through a porous medium of a incompressible non-Newtonian Rivlin-Ericksen fluid with heat transfer is presented and newly developed analytic method, namely the homotopy analysis method (HAM) is employed to compute an approximation to the solution of the system of nonlinear differential equations governing the problem. This flow impinges normal to a plane wall with heat transfer. It has been attempted to show capabilities and wide-range applications of the homotopy analysis method in comparison with the numerical method in solving this problem. Also the convergence of the obtained HAM solution is discussed explicitly. Our reports consist of the effect of the porosity of the medium and the characteristics of the Non-Newtonian fluid on both the flow and heat.

#### Solving the Boundary Layer Flow of Eyring–Powell Fluid Problem via Quasilinearization–Collocation Method Based on Hermite Functions

Kourosh Parand, Zahra Kalantari, Mehdi Delkhosh INAE Letters 2018 in pressAbstract:In this paper, the combination of quasilinearization and collocation methods is used for solving the problem of the boundary layer flow of Eyring–Powell fluid over a stretching sheet. The proposed approach is based on Hermite function collocation method. The quasilinearization method is used for converting the non-linear Eyring–Powell problem to a sequence of linear equations and the Hermite collocation method is applied for solving linear equations at each iteration. In the end, the obtained result of the present work is compared with the obtained results in other papers.

#### Generalized Lagrangian Jacobi Gauss collocation method for solving unsteady isothermal gas through a micro-nano porous medium

Kourosh Parand, Sobhan Latifi, Mehdi Delkhosh, Mohammad M. Moayeri The European Physical Journal Plus 2018 in pressAbstract:In the present paper, a new method based on the Generalized Lagrangian Jacobi Gauss (GLJG) collocation method is proposed. The nonlinear Kidder equation, which explains unsteady isothermal gas through a micro-nano porous medium, is a second-order two-point boundary value ordinary differential equation on the unbounded interval [0,∞) . Firstly, using the quasilinearization method, the equation is converted to a sequence of linear ordinary differential equations. Then, by using the GLJG collocation method, the problem is reduced to solving a system of algebraic equations. It must be mentioned that this equation is solved without domain truncation and variable changing. A comparison with some numerical solutions made and the obtained results indicate that the presented solution is highly accurate. The important value of the initial slope, y′(0) , is obtained as −1.191790649719421734122828603800159364 for η=0.5 . Comparing to the best result obtained so far, it is accurate up to 36 decimal places.

#### An iterative kernel based method for fourth order nonlinear equation with nonlinear boundary condition

Babak Azarnavid, Kourosh Parand, Saeid Abbasbandy Communications in Nonlinear Science and Numerical Simulation 2018(59) 544-552Abstract:This article discusses an iterative reproducing kernel method with respect to its effectiveness and capability of solving a fourth-order boundary value problem with nonlinear boundary conditions modeling beams on elastic foundations. Since there is no method of obtaining reproducing kernel which satisfies nonlinear boundary conditions, the standard reproducing kernel methods cannot be used directly to solve boundary value problems with nonlinear boundary conditions as there is no knowledge about the existence and uniqueness of the solution. The aim of this paper is, therefore, to construct an iterative method by the use of a combination of reproducing kernel Hilbert space method and a shooting-like technique to solve the mentioned problems. Error estimation for reproducing kernel Hilbert space methods for nonlinear boundary value problems have yet to be discussed in the literature. In this paper, we present error estimation for the reproducing kernel method to solve nonlinear boundary value problems probably for the first time. Some numerical results are given out to demonstrate the applicability of the method.

#### An efficient numerical method for solving nonlinear foam drainage equation

Kourosh Parand, Mehdi Delkhosh Indian Journal of Physics 2018(92) 231-243Abstract:In this paper, the nonlinear foam drainage equation, which is a famous nonlinear partial differential equation, is solved by using a hybrid numerical method based on the quasilinearization method and the bivariate generalized fractional order of the Chebyshev functions (B-GFCF) collocation method. First, using the quasilinearization method, the equation is converted into a sequence of linear partial differential equations (LPD), and then these LPDs are solved using the B-GFCF collocation method. A very good approximation of solutions is obtained, and comparisons show that the obtained results are more accurate than the results of other researchers.

##### 2017

#### The meshfree strong form methods for solving one dimensional inverse Cauchy-Stefan problem

J.A. Rad, K. Rashidi, K. Parand, H. Adibi Engineering with Computers 2017(33) 547–571Abstract:In this paper, we extend the application of meshfree node based schemes for solving one-dimensional inverse Cauchy-Stefan problem. The aim is devoted to recover the initial and boundary conditions from some Cauchy data lying on the admissible curve s(t) as the extra overspecifications. To keep matters simple, the problem has been considered in one dimensional, however the physical domain of the problem is supposed as an irregular bounded domain in R^2. The methods provide the space-time approximations for the heat temperature derived by expanding the required approximate solutions using collocation scheme based on radial point interpolation method (RPIM). The proposed method makes appropriate shape functions which possess the important Delta function property to satisfy the essential conditions automatically. In addition, to conquer the ill-posedness of the problem, particular optimization technique has been applied for solving the system of equations Ax=b in which A is a nonsymmetric stiffness matrix. As the consequences, reliable approximate solutions are obtained which continuously depend on input data.

#### Pricing American options under jump-diffusion models using local weak form meshless techniques

Jamal Amani Rad, Kourosh Parand International Journal of Computer Mathematics 2017(10) 1-25Abstract:Recently, several numerical methods have been proposed for pricing options under jump-diffusion models but very few studies have been conducted using meshless methods [12, 40]. Indeed, only a strong form of meshless methods have been employed in these lectures. We propose the local weak form meshless methods for option pricing under Merton and Kou jump-diffusion models. Predominantly in this work we will focus on meshless local Petrov-Galerkin (MLPG), local boundary integral equation (LBIE) methods based on moving least square approximation (MLS) and local radial point interpolation (LRPI) based on Wendland’s compactly supported radial basis functions (WCS-RBFs). The key feature of this paper is applying a Richardson extrapolation technique on American option which is a free boundary problem to obtain a fixed boundary problem. Also the implicit-explicit (IMEX) time stepping scheme is employed for the time derivative which allows us to obtain a spars and banded linear system of equations. Numerical experiments are presented showing that the presented approaches are extremely accurate and fast.

#### Numerical Study of Astrophysics Equations by Meshless Collocation Method Based on Compactly Supported Radial Basis Function

Kourosh Parand, Mohamad Hemami International Journal of Applied and Computational Mathematics 2017 1-23Abstract:In this paper, we propose compactly supported radial basis functions for solving some wellknown classes of astrophysics problems categorized as non-linear singular initial ordinary differential equations on a semi-infinite domain. To increase the convergence rate and to decrease the collocation points, we use the compactly supported radial basis function through the integral operations. Afterwards, some special cases of the equation are presented as test examples to show the reliability of the method. Then we compare the results of this work with some results and show that the new method is e_cient and applicable.

#### A numerical investigation of the boundary layer flow of an Eyring-Powell fluid over a stretching sheet via rational Chebyshev functions

Kourosh Parand, Mohammad Mahdi Moayeri, Sobhan Latifi, Mehdi Delkhosh The European Physical Journal Plus 2017(132) 325Abstract:In this paper, a spectral method based on the four kinds of rational Chebyshev functions is proposed to approximate the solution of the boundary layer flow of an Eyring-Powell fluid over a stretching sheet. First, by using the quasilinearization method (QLM), the model which is a nonlinear ordinary differential equation is converted to a sequence of linear ordinary differential equations (ODEs). By applying the proposed method on the ODEs in each iteration, the equations are converted to a system of linear algebraic equations. The results indicate the high accuracy and convergence of our method. Moreover, the effects of the Eyring-Powell fluid material parameters are discussed.

#### The generalized fractional order of the Chebyshev functions on nonlinear boundary value problems in the semi-infinite domain

Kourosh Parand, Mehdi Delkhosh : The European Physical Journal Plus 2017(6)Abstract:A new collocation method, namely the generalized fractional order of the Chebyshev orthogonal functions (GFCFs) collocation method, is given for solving some nonlinear boundary value problems in the semi-infinite domain, such as equations of the unsteady isothermal flow of a gas, the third grade fluid, the Blasius, and the field equation determining the vortex profile. The method reduces the solution of the problem to the solution of a nonlinear system of algebraic equations. To illustrate the reliability of the method, the numerical results of the present method are compared with several numerical results.

#### Novel orthogonal functions for solving differential equations of arbitrary order

Kourosh Parand, Mehdi Delkhosh, Mehran Nikarya Tbilisi Mathematical Journal 2017(10)Abstract:Fractional calculus and the fractional differential equations have appeared in many physical and engineering processes. Therefore, an efficient and suitable method to solve them is very important. In this paper, novel numerical methods are introduced based on the fractional order of the Chebyshev orthogonal functions (FCF) with Tau and collocation methods to solve differential equations of the arbitrary (integer or fractional) order. The FCFs are obtained from the classical Chebyshev polynomials of the first kind. Also, the operational matrices of the fractional derivative and the product for the FCFs have been constructed. To show the efficiency and capability of these methods we have solved some well-known problems: the momentum, the Bagley-Torvik, and the Lane-Emden differential equations, then have compared our results with the famous methods in other papers.

#### Fractional order of rational Jacobi functions for solving the non-linear singular Thomas-Fermi equation

Kourosh Parand, Pooria Mazaheri, Hossein Yousefi, Mehdi Delkhosh The European Physical Journal Plus 2017(10)Abstract:In this paper, a new method based on Fractional order of Rational Jacobi (FRJ) functions is proposed that utilizes quasilinearization method to solve non-linear singular Thomas-Fermi equation on unbounded interval [0,∞) . The equation is solved without domain truncation and variable changing. First, the quasilinearization method is used to convert the equation to the sequence of linear ordinary differential equations. Then, by using the FRJs collocation method the equations are solved. For the evaluation, comparison with some numerical solutions shows that the proposed solution is highly accurate.

#### Numerical pricing of American options under two stochastic factor models with jumps using a meshless local Petrov-Galerkin method

J.A. Rad, K. Parand Applied Numerical Mathematics 2017(4)Abstract:The most recent update of financial option models is American options under stochastic volatility models with jumps in returns (SVJ) and stochastic volatility models with jumps in returns and volatility (SVCJ). To evaluate these options, mesh-based methods are applied in a number of papers but it is well-known that these methods depend strongly on the mesh properties which is the major disadvantage of them. Therefore, we propose the use of the meshless methods to solve the aforementioned options models, especially in this work we select and analyze one scheme of them, named local radial point interpolation (LRPI) based on Wendland's compactly supported radial basis functions (WCS-RBFs) with C6, C4 and C2 smoothness degrees. The LRPI method which is a special type of meshless local Petrov-Galerkin method (MLPG), offers several advantages over the mesh-based methods, nevertheless it has never been applied to option pricing, at least to the very best of our knowledge. These schemes are the truly meshless methods, because, a traditional non-overlapping continuous mesh is not required, neither for the construction of the shape functions, nor for the integration of the local sub-domains. In this work, the American option which is a free boundary problem, is reduced to a problem with fixed boundary using a Richardson extrapolation technique. Then the implicit-explicit (IMEX) time stepping scheme is employed for the time derivative which allows us to smooth the discontinuities of the options' payoffs. Stability analysis of the method is analyzed and performed. In fact, according to an analysis carried out in the present paper, the proposed method is unconditionally stable. Numerical experiments are presented showing that the proposed approaches are extremely accurate and fast.

#### A Numerical Approach to Solve Lane-Emden Type Equations by the Fractional Order of Rational Bernoulli Functions

K Parand, H Yousefi, M Delkhosh Romanian Journal of Physics 2017(62)Abstract:In this paper, a numerical method based on the hybrid of the quasilinearization method (QLM) and the collocation method is suggested for solving wellknown nonlinear Lane-Emden-type equations as singular initial value problems, which model many phenomena in mathematical physics and astrophysics. First, by using the QLM method, the nonlinear ordinary differential equation is converted into a sequence of linear differential equations, and then the linear equations by the fractional order of rational Bernoulli collocation (FRBC) method on the semi-infinite interval [0;1) are solved. This method reduces the solution of these problems to the solution of a system of algebraic equations. Computational results of several problems are presented to demonstrate the viability and powerfulness of the method. Further, the fractional order of rational Bernoulli functions has also been used for the first time. The first zeros of standard Lane-Emden equation and the approximations of y(x) for Lane-Emden-type equations are given with unprecedented accuracy.

#### New numerical solutions for solving Kidder equation by using the rational Jacobi functions

K Parand, P Mazaheri, M Delkhosh, A Ghaderi SeMA Journal 2017(74) 569–583Abstract:In this paper, a new method based on rational Jacobi functions (RJ) is proposed that utilizes quasilinearization method to solve non-linear singular Kidder equation on unbounded interval. The Kidder equation is a second order non-linear two-point boundary value ordinary differential equation on unbounded interval [0,∞) The equation is solved without domain truncation and variable changing. First, the quasilinearization method is used to convert the equation to sequence of linear ordinary differential equations. Then, by using RJ collocation method equations are solved. For the evaluation, comparison with some numerical solutions shows that the proposed solution is highly accurate. Using 200 collocation points, the value of initial slope that is important is calculated as −1.1917906497194217341228284 −1.1917906497194217341228284 for κ=0.5

#### An effective numerical method for solving the nonlinear singular Lane-Emden type equations of various orders

K Parand, M Delkhosh Jurnal Teknologi 2017(79) 25-36Abstract:The Lane-Emden type equations are employed in the modeling of several phenomena in the areas of mathematical physics and astrophysics. These equations are categorized as non-linear singular ordinary differential equations on the semi-infinite domain. In this paper, the generalized fractional order of the Chebyshev orthogonal functions (GFCFs) of the first kind have been introduced as a new basis for Spectral methods, and also presented an effective numerical method based on the GFCFs and the collocation method for solving the nonlinear singular Lane-Emden type equations of various orders. Obtained results have compared with other results to verify the accuracy and efficiency of the presented method.

#### Accurate solution of the Thomas–Fermi equation using the fractional order of rational Chebyshev functions

Kourosh Parand, Mehdi Delkhosh Journal of Computational and Applied Mathematics 2017(317) 624-642Abstract:In this paper, the nonlinear singular Thomas–Fermi differential equation for neutral atoms is solved using the fractional order of rational Chebyshev orthogonal functions (FRCs) of the first kind, FT n α (t, L), on a semi-infinite domain, where L is an arbitrary numerical parameter. First, using the quasilinearization method, the equation be converted into a sequence of linear ordinary differential equations (LDEs), and then these LDEs are solved using the FRCs collocation method. Using 300 collocation points, we have obtained a very good approximation solution and the value of the initial slope y′(0)=− 1.58807102261137531271868450942395…, highly accurate to 37 decimal places.

#### Solving the nonlinear Schlomilch’s integral equation arising in ionospheric problems

Kourosh Parand, Mehdi Delkhosh Afrika Matematika 2017(28) 459–480Abstract:In this paper, the linear and nonlinear Schlomilch’s integral equations and their generalized forms are studied. The Schlomilch’s integral equations are used for many ionospheric problems, atmospheric and terrestrial physics. The generalized fractional order of the Chebyshev orthogonal functions (GFCF) collocation method is used to handle many forms of Schlomilch’s integral equations. The GFCF method can be used in the applied physics, applied mathematics, and engineering applications. The reliability of the GFCF method is justified through illustrative examples.

#### Quasilinearization-Lagrangian method to solve the HIV infection model of CD4 T cells

Kourosh Parand, Zahra Kalantari, Mehdi Delkhosh SeMA Journal 2017 113Abstract:Abstract In this paper, the quasilinearization and Lagrangian methods are used for solving a model of the HIV infection of CD4\(^+\) T cells. This approach is based on the Lagrangian method by using the collocation points of transformed Hermite polynomials. The quasilinearization method is used for converting the non-linear problem to a sequence of linear equations and the Hermite Lagrangian method is applied for solving linear equations at each iteration. In the end, the obtained results have been compared with some other well-known results and show that the present method is efficient.

#### Solving magneto-hydrodynamic squeezing flow between two parallel disks with suction or injection using three classes of polynomials

Kourosh Parand, Amin Ghaderi, Hossein Yousefi, Mehdi Delkhosh Palestine Journal of Mathematics 2017(6) 333-347Abstract:Abstract In this paper, three numerical methods based on the Spectral methods to consider magneto-hydrodynamic (MHD) squeezing flow of a viscous incompressible fluid between two parallel disks with suction or injection are introduced. It is assumed that upper disk is movable in upward and downward directions while the lower disk is fixed but permeable. First, the governing partial differential equations by using viable similarity transforms convert to a system of nonlinear ordinary differential equations. Then, the system is solved by collocation method by using polynomials of shifted Chebyshev, Euler, and Bessel. Influence of flow parameters is discussed and to show the efficiency and capability of these methods, our results are compared with each other and with other researchers. Numerical solutions are obtained by using few numbers of collocation points.

#### New Numerical Solution For Solving Nonlinear Singular Thomas-Fermi Differential Equation

Kourosh Parand, Mehdi Delkhosh Bulletin of the Belgian Mathematical Society 2017(24)Abstract:Abstract In this paper, the nonlinear singular Thomas-Fermi differential equation on a semi- infinite domain for neutral atoms is solved by using the generalized fractional order of the Chebyshev orthogonal functions (GFCFs) of the first kind. First, this collocation method reduces the solution of this problem to the solution of a system of nonlinear algebraic equations. Second, using solve a system of nonlinear equations, the initial value for the unknown parameter $ L $ is calculated, and finally, the value of $ L $ to increase the accuracy of the initial slope is improved and the value of $ y'(0)=-1.588071022611375312718684509$ is calculated. The comparison with some numerical solutions shows that the present solution is highly accurate.

#### An Efficient Numerical Solution of Nonlinear Hunter–Saxton Equation

Kourosh Parand, Mehdi Delkhosh Communications in Theoretical Physics 2017(67) 483-492Abstract:Abstract In this paper, the nonlinear Hunter–Saxton equation, which is a famous partial differential equation, is solved by using a hybrid numerical method based on the quasilinearization method and the bivariate generalized fractional order of the Chebyshev functions (B-GFCF) collocation method. First, using the quasilinearization method, the equation is converted into a sequence of linear partial differential equations (LPD), and then these LPDs are solved using the B-GFCF collocation method. A very good approximation of solutions is obtained, and comparisons show that the obtained results are more accurate than the results of other researchers.

#### Shifted Lagrangian Jacobi collocation scheme for numerical solution of a model of HIV infection

K Parand, S Latifi, MM Moayeri SeMA Journal 2017 1-20Abstract:In this paper, a system of nonlinear ordinary differential equations (NODEs), namely the equation model of the human immunodeficiency virus (HIV) infection of CD4 + T cells, is studied. Our approach is implemented by using the Shifted-Lagrangian Jacobi (SLJ) polynomials formed by Shifted-Jacobi-Gauss-Radau (SJ-GR) points. In a new insight, by applying Quasilinearization method (QLM) the system of NODE’s is simplified and changed into a system of Linear ordinary differential equations (LODE’s) and instead of working on a system of NODE’s, all processes and works are done on a system of LODE’s. Therefore, unlike the most of the current studies working on nonlinear algebraic equations, the problem is reduced to a system of linear algebraic equations. Then, to solve the problem and find the unknown approximation coefficients, a system of Ax=b has been solved. At the end, the accuracy and reliability of this method are shown and comparisons with the other current work’s results are made.

#### Collocation Method using Compactly Supported Radial Basis Function for Solving Volterra's Population Model

Kourosh Parand, Mohammad Hemami Caspian Journal of Mathematical Sciences 2017(6) 77-86Abstract:In this paper, indirect collocation approach based on compactly supported radial basis function is applied for solving Volterras population model. The method reduces the solution of this problem to the solution of a system of algebraic equations. Volterras model is a non-linear integro-differential equation where the integral term represents the effect of toxin. To solve the problem, we use the well-known CSRBF: Wendland3, 5. Numerical results and residual norm 2 show good accuracy and rate of convergence.

##### 2016

#### Rational and Exponential Legendre Tau Method on Steady Flow of a Third Grade Fluid in a Porous Half Space

F Baharifard, Saeed Kazem, K Parand International Journal of Applied and Computational Mathematics 2016(2) 679-698Abstract:In this paper, we decide to compare rational and exponential Legendre functions Tau approach to solve the governing equations for the flow of a third grade fluid in a porous half space. Firstly, we estimate an upper bound for function approximation based on mentioned functions in semi-infinite domain, and discuss that the analytical functions have a superlinear convergence for these basis. Also the operational matrices of derivative and product of these functions are presented to reduce the solution of this problem to the solution of a system of nonlinear algebraic equations. The comparison of the results of rational and exponential Legendre Tau methods with numerical solution shows the efficiency and accuracy of these methods. We also make a comparison between these two methods themselves and show that using exponential functions, leads to more accurate results and faster convergence in this problem.

#### NUMERICAL SOLUTION OF AN INTEGRO-DIFFERENTIAL EQUATION ARISING IN OSCILLATING MAGNETIC FIELDS

K. Parand, M. Delkhosh The Journal of the Korean Society for Industrial and Applied Mathematics 2016(20) 15Abstract:In this paper, an integro-differential equation which arises in oscillating magnetic fields is studied. The generalized fractional order Chebyshev orthogonal functions (GFCF) collocation method used for solving this integral equation. The GFCF collocation method can be used in applied physics, applied mathematics, and engineering applications. The results of applying this procedure to the integro-differential equation with time-periodic coefficients show the high accuracy, simplicity, and efficiency of this method. The present method is converging and the error decreases with increasing collocation points.

#### Operational Matrices to Solve Nonlinear Volterra-Fredholm Integro-Differential Equations of Multi-Arbitrary Order

Kourosh Parand, Mehdi Delkhosh Gazi University Journal of Science 2016(29) 895-907Abstract:Fractional calculus has been used for modelling many of physical and engineering processes, that many of them are described by linear and nonlinear Volterra- Fredholm integro- differential equations of multi-arbitrary order. Therefore, an efficient and suitable method for the solution of them is very important. In this paper, the generalized fractional order of the Chebyshev functions (GFCFs) based on the classical Chebyshev polynomials of the first kind is used to obtain the solution of the linear and nonlinear multi-order Volterra-Fredholm integro-differential equations. Also, the operational matrices of the fractional derivative, the product, and the fractional integration to transform the equations to a system of algebraic equations are introduced. Some examples are included to demonstrate the validity and applicability of the technique.

#### A new approach for solving nonlinear Thomas-Fermi equation based on fractional order of rational Bessel functions

K. Parand, A. Ghaderi, M. Delkhosh, H. Yousefi Electronic Journal of Differential Equations 2016(2016) 1-18Abstract:In this paper, the fractional order of rational Bessel functions collocation method (FRBC) to solve Thomas-Fermi equation which is defined in the semi-infinite domain and has singularity at x = 0 and its boundary condition occurs at infinity, have been introduced. We solve the problem on semi-infinite domain without any domain truncation or transformation of the domain of the problem to a finite domain. This approach at first, obtains a sequence of linear differential equations by using the quasilinearization method (QLM), then at each iteration solves it by FRBC method. To illustrate the reliability of this work, we compare the numerical results of the present method with some wellknown result

#### The rational Chebyshev of second kind collocation method for solving a class of astrophysics problems

Kourosh Parand, Sajjad Khaleqi The European Physical Journal Plus 2016(131) --Abstract:The Lane-Emden equation has been used to model several phenomena in theoretical physics, mathematical physics and astrophysics such as the theory of stellar structure. This study is an attempt to utilize the collocation method with the rational Chebyshev function of Second kind (RCS) to solve the Lane-Emden equation over the semi-infinite interval\([0,+\ infty [\). According to well-known results and comparing with previous methods, it can be said that this method is efficient and applicable.

#### Solving Volterra’s population growth model of arbitrary order using the generalized fractional order of the Chebyshev functions

Kourosh Parand, Mehdi Delkhosh Ricerche di Matematica 2016(65)Abstract:Volterra’s model for population growth in a closed system includes an integral term to indicate accumulated toxicity in addition to the usual terms of the logistic equation, that occurs in ecology. In this paper, a new numerical approximation is introduced for solving this model of arbitrary (integer or fractional) order. The proposed numerical approach is based on the generalized fractional order Chebyshev orthogonal functions of the first kind and the collocation method. Accordingly, we employ a collocation approach, by computing through Volterra’s population model in the integro-differential form. This method reduces the solution of a problem to the solution of a nonlinear system of algebraic equations. To illustrate the reliability of this method, we compare the numerical results of the present method with some well-known results in order to show that the new method is efficient and applicable.

#### An operation matrix method based on Bernstein polynomials for Riccati differential equation and Volterra population model

K Parand, Sayyed A Hossayni, JA Rad Applied Mathematical Modelling 2016(40) 21Abstract:In this study, we present a modified configuration, including an exact formulation, for the operational matrix form of the integration, differentiation, and product operators applied in the Galerkin method. Previously, many studies have investigated the methods for obtaining operational matrices (derivative, integral, and product) for Fourier, Chebyshev, Legendre, and Jacobi polynomials, and some have considered the non-orthogonal bases that almost all of them operate on approximately. However, in this study, we aim to obtain the exact operational matrices (EOMs), which can be used for many classes of orthogonal and non-orthogonal polynomials. Similar to previous approaches, this method transforms the original problem into a system of nonlinear algebraic equations. To retain the simplicity of the procedure, the samples are considered in one-dimensional contexts, although the proposed technique can also be employed for two- and three-dimensional problems. Two examples are presented to verify the accy of the proposed new approach and to demonstrate the superior performance of EOMs compared with ordinary operational matrices. The corresponding results demonstrate the increased accuracy of the new method. In addition, the convergence of the EOM method is studied numerically and analytically to prove the efficiency of the method.

#### A novel numerical technique to obtain an accurate solution to the Thomas-Fermi equation

Kourosh Parand, Hossein Yousefi, Mehdi Delkhosh, Amin Ghaderi The European Physical Journal Plus 2016(131) --Abstract:In this paper, a new algorithm based on the fractional order of rational Euler functions (FRE) is introduced to study the Thomas-Fermi (TF) model which is a nonlinear singular ordinary differential equation on a semi-infinite interval. This problem, using the quasilinearization method (QLM), converts to the sequence of linear ordinary differential equations to obtain the solution. For the first time, the rational Euler (RE) and the FRE have been made based on Euler polynomials. In addition, the equation will be solved on a semi-infinite domain without truncating it to a finite domain by taking FRE as basic functions for the collocation method. This method reduces the solution of this problem to the solution of a system of algebraic equations. We demonstrated that the new proposed algorithm is efficient for obtaining the value of y'(0)y'(0) , y(x)y(x) and y'(x)y'(x) . Comparison with some numerical and analytical solutions shows that the present solution is highly accurate.

#### A comparison of numerical and semi-analytical methods for the case of heat transfer equations arising in porous medium

K Parand, JA Rad, M Ahmadi The European Physical Journal Plus 2016(131) --Abstract:Natural convective heat transfer in porous media which is of importance in the design of canisters for nuclear waste disposal has received considerable attention during the past few decades. This paper presents a comparison between two different analytical and numerical methods, i.e. pseudospectral and Adomian decomposition methods. The pseudospectral approach makes use of the orthogonal rational Jacobi functions; this method reduces the solution of the problem to a solution of a system of algebraic equations. Numerical results are compared with each other, showing that the pseudospectral method leads to more accurate results and is applicable on similar problems.Keywords

##### 2015

#### Pricing European and American options by radial basis point interpolation

Jamal Amani Rad, Kourosh Parand, Luca Vincenzo Ballestra Applied Mathematics and Computation 2015(251) 15Abstract:We propose the use of the meshfree radial basis point interpolation (RBPI) to solve the Black–Scholes model for European and American options. The RBPI meshfree method offers several advantages over the more conventional radial basis function approximation, nevertheless it has never been applied to option pricing, at least to the very best of our knowledge. In this paper the RBPI is combined with several numerical techniques, namely

#### Local weak form meshless techniques based on the radial point interpolation (RPI) method and local boundary integral equation (LBIE) method to evaluate European and American options

Jamal Amani Rad, Kourosh Parand, Saeid Abbasbandy Communications in Nonlinear Science and Numerical Simulation 2015(22) 23Abstract:For the first time in mathematical finance field, we propose the local weak form meshless methods for option pricing; especially in this paper we select and analysis two schemes of them named local boundary integral equation method (LBIE) based on moving least squares approximation (MLS) and local radial point interpolation (LRPI) based on Wu’s compactly supported radial basis functions (WCS-RBFs). LBIE and LRPI schemes are the truly meshless methods, because, a traditional non-overlapping, continuous mesh is not required, either for the construction of the shape functions, or for the integration of the local sub-domains. In this work, the American option which is a free boundary problem, is reduced to a problem with fixed boundary using a Richardson extrapolation technique. Then the ?-weighted scheme is employed for the time derivative. Stability analysis of the methods is analyzed and performed by the matrix method. In fact, based on an analysis carried out in the present paper, the methods are unconditionally stable for implicit Euler (?=0) and Crank–Nicolson (?=0.5) schemes. It should be noted that LBIE and LRPI schemes lead to banded and sparse system matrices. Therefore, we use a powerful iterative algorithm named the Bi-conjugate gradient stabilized method (BCGSTAB) to get rid of this system. Numerical experiments are presented showing that the LBIE and LRPI approaches are extremely accurate and fast.

#### Pricing European and American options using a very fast and accurate scheme

Jamal Amani Rad, Kourosh Parand, Saeid Abbasbandy Proceedings of the National Academy of Sciences, India Section A 2015(85) 15Abstract:In this paper, a method for the numerical pricing of American and European options under the Black–Scholes model is introduced. This approach is meshless local Petrov–Galerkin (MLPG) based on local weak form and the moving least squares approximations. The MLPG offers advantages over conventional and strong meshless methods of radial basis function approximations. In this paper, the American option which is a free boundary problem, is reduced to a fixed boundary problem using a Richardson extrapolation technique. Then a time stepping method is employed for the time derivative. Finally numerical results are presented in two test cases. These experiments show that MLPG approach is accurate and fast, and performs significantly better compared to the conventional radial basis functions collocation methods.

#### Application of the exact operational matrices for solving the Emden-Fowler equations, arising in Astrophysics

Sayyed A Hossayni, JA Rad, K Parand, S Abbasbandy International Journal of Industrial Mathematics 2015(7) 24Abstract:The objective of this paper is applying the well-known exact operational matrices (EOMs) idea for solving the Emden-Fowler equations, illustrating the superiority of EOMs over ordinary operational matrices (OOMs). Up to now, a few studies have been conducted on EOMs ; but the solved differential equations did not have high-degree nonlinearity and the reported results could not strongly show the excellence of this new method. So, we chose Emden-Fowler type differential equations and solved them utilizing this method. To confirm the accuracy of the new method and to show the preeminence of EOMs over OOMs, the norm 1 of the residual and error function for both methods are evaluated for multiple $m$ values, where $m$ is the degree of the Bernstein polynomials. We report the results by some plots to illustrate the error convergence of both methods to zero and also to show the primacy of the new method versus OOMs. The obtained results demonstrate the increased accuracy of the new method

##### 2014

#### Application of Bessel functions for solving differential and integro-differential equations of the fractional order

K Parand, M Nikarya Applied Mathematical Modelling 2014(38) 4137-4147Abstract:In this paper, a new numerical algorithm to solve the linear and nonlinear fractional differential equations (FDE) is introduced. Fractional calculus and fractional differential equations have many applications in physics, chemistry, engineering, finance, and other sciences. The proposed approach is based on the first kind of Bessel functions collocation method. The first kind of Bessel function is an infinite series, which is convergent for any x in R . In this method, we reduce the solution of a nonlinear fractional problem to the solution of a system of the nonlinear algebraic equations. To illustrate the reliability of this method, we solve some important equations of fractional order, and present numerical results of the present method to show convergence rate, applicability and reliability of this method.

#### Optimal control of a parabolic distributed parameter system via radial basis functions

JA Rad, S Kazem, K Parand Communications in Nonlinear Science and Numerical Simulation 2014(19) 2559-2567Abstract:This paper attempts to present a meshless method to find the optimal control of a parabolic distributed parameter system with a quadratic cost functional. The method is based on radial basis functions to approximate the solution of the optimal control problem using collocation method. In this regard, different applications of RBFs are used. To this end, the numerical solutions are obtained without any mesh generation into the domain of the problems. The proposed technique is easy to implement, efficient and yields accurate results. Numerical examples are included and a comparison is made with an existing result.

#### A new numerical algorithm based on the first kind of modified Bessel function to solve population growth in a closed system

K Parand, JA Rad, M Nikarya International Journal of Computer Mathematics 2014(91) 1239-1254Abstract:Volterra's model for population growth in a closed system includes an integral term to indicate accumulated toxicity in addition to the usual terms of the logistic equation. In this research, a new numerical algorithm is introduced for solving this model. The proposed numerical approach is based on the modified Bessel function of the first kind and the collocation method. In this method, we aim to solve the problems on the semi-infinite domain without any domain truncation, variable transformation in basis functions and shifting the problem to a finite domain. Accordingly, we employ two different collocation approaches, one by computing through Volterra's population model in the integro-differential form and the other by computing by converting this model to an ordinary differential form. These methods reduce the solution of a problem to the solution of a nonlinear system of algebraic equations. To illustrate the reliability of these methods, we compare the numerical results of the present methods with some well-known results in other to show that the new methods are efficient and applicable.

#### Solving Steady Flow of a Third-Grade Fluid in a Porous Half Space via Normal and Modified Rational Christov Functions Collocation Method

K. Parand, E. Hajizadeh Zeitschrift für Naturforschung A 2014(69) 7Abstract:The present study is an attempt to find a solution for steady flow of a third-grade fluid by utilizing spectral methods based on rational Christov functions. This problem is described as a nonlinear twopoint boundary value problem. The following method tries to solve the problem on the infinite domain without truncating it to a finite domain and transforms the domain of the problem to a finite domain. Researchers in this try to solve the problem by using anew modified rational Christov functions and normal rational Christov function. Finally, the findings of the current study, i. e., proposal methods, numerical out cames and other methods were compared with each other.

#### Application of Meshfree methods for solving the inverse one-dimensional Stefan problem

K. Rashidi, H. Adibi, J. A. Rad, K. Parand Engineering Analysis with Boundary Elements 2014(40) 22Abstract:This work is motivated by studies of numerical simulation for solving the inverse one and two-phase Stefan problem. The aim is devoted to employ two special interpolation techniques to obtain space-time approximate solution for temperature distribution on irregular domains, as well as for the reconstruction of the functions describing the temperature and the heat flux on the fixed boundary x=0 when the position of the moving interface is given as extra specification. The advantage of applying the methods is producing the shape functions which provide the important delta function property to ensure that the essential conditions are fulfilled. Due to ill-posedness of the problem, the process is intractable numerically, so special optimization technique is used to obtain the regularized solution. Numerical results for the typical benchmark test examples, which have the input measured data perturbed by increasing amounts of noise and continuity to the input data in the presence of additive noise, are obtained, which present the efficiency of the proposed method.

#### Numerical solution of fractional differential equations with a Tau method based on Legendre and Bernstein polynomials

JA Rad, S Kazem, M Shaban, K Parand, A Yildirim Mathematical Methods in the Applied Sciences 2014(37) 15Abstract:In this paper, we state and prove a new formula expressing explicitly the integratives of Bernstein polynomials (or B-polynomials) of any degree and for any fractional-order in terms of B-polynomials themselves. We derive the transformation matrices that map the Bernstein and Legendre forms of a degree-n polynomial on [0,1] into each other. By using their transformation matrices, we derive the operational matrices of integration and product of the Bernstein polynomials. These matrices together with the Tau method are then utilized to reduce the solution of this problem to the solution of a system of algebraic equations. The method is applied to solve linear and nonlinear fractional differential equations.

#### Solving the Unsteady Isothermal Gas Through a Micro-Nano Porous Medium via Bessel Function Collocation Method

K Parand, M Nikarya Journal of Computational and Theoretical Nanoscience 2014(11)Abstract:In this study, a new numerical algorithm is introduced to solve the unsteady isothermal flow of a gas through a semi-infinite micro-nano porous medium. The unsteady gas equation is a second-order non-linear two-point boundary value ordinary differential equation (ODE) on the semi-infinite domain, with a boundary condition in the infinite. The proposed approach is based on the first kind of Bessel functions collocation method. The first kind of Bessel function is an infinite series, that is convergent for any x ? R. In this work, we aim to solve the problem on the semi-infinite domain without any domain truncation, variable transformation in basis functions or transformation of the domain of the problem to a finite domain. In this work, we reduce the solution of a nonlinear problem to the solution of a system of the nonlinear algebraic equations. To illustrate the reliability of this method, we compare the numerical results of the present method with some well-known results in order to show the applicability and efficiency of our method, and also.

#### Numerical solution of Maxwell equations using local weak form meshless techniques

S. Sarabadan, M. Shahrezaee, J.A. Rad, K. Parand Journal of Mathematics and Computer Science 2014(13) 8Abstract:In this paper we propose a method for solving some well-known classes of Lane-Emden type equations which are nonlinear ordinary differential equations on the semi-infinite domain. The proposed approach is based on an Unsupervised Combined Artificial Neural Networks (UCANN) method. Firstly, The trial solutions of the differential equations are written in the form of feed-forward neural networks containing adjustable parameters (the weights and biases); results are then optimized with the combined neural network. The proposed method is tested on series of Lane-Emden differential equations and the results are reported. Afterward, these results are compared with the solution of other methods demonstrating the efficiency and applicability of the proposed method.

#### Using Hermite function for solving Thomas-Fermi equation

F.B. Babolghani, K. Parand International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering 2014(8) 123-126Abstract:In this paper, we propose Hermite collocation method for solving Thomas-Fermi equation that is nonlinear ordinary differential equation on semi-infinite interval. This method reduces the solution of this problem to the solution of a system of algebraic equations. We also present the comparison of this work with solution of other methods that shows the present solution is more accurate and faster convergence in this problem.

##### 2013

#### Analytical solution of the transpiration on the boundary layer flow and heat transfer over a vertical slender cylinder

S Abbasbandy, D Modarrespoor, K Parand, JA Rad Quaestiones Mathematicae 2013(36) 28Abstract:An analysis is carried out to study the effects that blowing/injection and suction on the steady mixed convection or combined forced and free convection boundary layer flows over a vertical slender cylinder with a mainstream velocity and a wall surface temperature proportional to the axial distance along the surface of the cylinder. Both cases of buoyancy forces aid and oppose the development of the boundary layer are considered. For the investigated problem, the governing non-linear partial differential equations and their associated boundary conditions are transformed into coupled non-linear ordinary differential equations by using similarity transformations. This equation is solved both by a newly developed analytic technique, namely homotopy analysis method (HAM) and by a numerical method employing the shooting method. The convergence of the obtained series solutions is carefully checked. The physical significance of interesting parameters on the velocity profile and the temperature profile are shown through graphs and discussed in detail. The values of the skin friction coefficient, the local Nusselt number, curvature parameter, buoyancy or mixed convection parameter and Prandtl number are tabulated. Comparison is also made with the corresponding results of viscous fluid with no mixed convection and an excellent agreement is noted.

#### Numerical approach of flow and mass transfer on nonlinear stretching sheet with chemically reactive species using rational Jacobi collocation method

K Parand, L Hosseini International Journal of Numerical Methods for Heat & Fluid Flow 2013(23) 17Abstract:The aim is to present in this paper an effective strategy in dealing with a semi-infinite interval by using a suitable mapping that transforms a semi-infinite interval to a finite interval. The authors introduce a new orthogonal system of rational functions induced by general Jacobi polynomials with the parameters alpha and beta. It is more flexible in applications. In particular, alpha and beta could be regulated, so that the systems are mutually orthogonal in certain weighted Hilbert spaces. This approach is applied for solving a non-linear system two-point boundary value problem (BVP) on semi-infinite interval, describing the flow and diffusion of chemically reactive species over a nonlinearly stretching sheet immersed in a porous medium. The new approach reduces the solution of a problem to the solution of a system of algebraic equations. The paper presents an effective strategy in dealing with a semi-infinite interval by using a suitable mapping that transforms a semi-infinite interval to a finite interval.

#### Solving non-linear Lane–Emden type equations using Bessel orthogonal functions collocation method

Kourosh Parand, Mehran Nikarya, Jamal Amani Rad Celestial Mechanics and Dynamical Astronomy 2013(116) 11Abstract:The Lane–Emden type equations are employed in the modeling of several phenomena in the areas of mathematical physics and astrophysics. These equations are categorized as non-linear singular ordinary differential equations on the semi-infinite domain [0,?) . In this research we introduce the Bessel orthogonal functions as new basis for spectral methods and also, present an efficient numerical algorithm based on them and collocation method for solving these well-known equations. We compare the obtained results with other results to verify the accuracy and efficiency of the presented scheme. To obtain the orthogonal Bessel functions we need their roots. We use the algorithm presented by Glaser et al. (SIAM J SciComput 29:1420–1438, 2007) to obtain the N roots of Bessel functions.

#### Radial basis functions approach on optimal control problems

Jamal Amani Rad, Saeed Kazem, Kourosh Parand Journal of Vibration and Control 2013(20) 1394-1416Abstract:A numerical method for solving optimal control problems is presented in this work. The method is based on radial basis functions (RBFs) to approximate the solution of the optimal control problems by using collocation method. We applied Legendre–Gauss–Lobatto points for RBFs center nodes to use numerical integration method more easily, then the method of Lagrange multipliers is used to obtain the optimum of the problems. For this purpose different applications of RBFs are used. The differential and integral expressions which arise in the system dynamics, the performance index and the boundary conditions are converted into some algebraic equations which can be solved for the unknown coefficients. Illustrative examples are included to demonstrate the validity and applicability of the technique.

#### Kansa method for the solution of a parabolic equation with an unknown spacewise-dependent coefficient subject to an extra measurement

K Parand, JA Rad Computer Physics Communications 2013(184) 14Abstract:Parabolic partial differential equations with an unknown spacewise-dependent coefficient serve as models in many branches of physics and engineering. Recently, much attention has been expended in studying these equations and there has been a considerable mathematical interest in them. In this work, the solution of the one-dimensional parabolic equation is presented by the method proposed by Kansa. The present numerical procedure is based on the product model of the space–time radial basis function (RBF), which was introduced by Myers et al. Using this method, a rapid convergent solution is produced which tends to the exact solution of the problem. The convergence of this scheme is accelerated when we use the Cartesian nodes as center nodes. The accuracy of the method is tested in terms of Error and RMS errors. Also, the stability of the technique is investigated by perturbing the additional specification data by increasing the amounts of random noise. The numerical results obtained show that the proposed method produces a convergent and stable solution.

#### Solving a laminar boundary layer equation with the rational Gegenbauer functions

K Parand, Mehdi Dehghan, F Baharifard Applied Mathematical Modelling 2013(37) 13Abstract:In this paper, a collocation method using a new weighted orthogonal system on the half-line, namely the rational Gegenbauer functions, is introduced to solve numerically the third-order nonlinear differential equation, af?+ff?=0af?+ff?=0, where a is a constant parameter. This method solves the problems on semi-infinite domain without truncating it to a finite domain and transforming the domain of the problems to a finite domain. For a=2a=2, the equation is the well-known Blasius equation, which is a laminar viscous flow over a semi-infinite flat plate. We solve this equation by considering 1?a?21?a?2 and compare the new results with the established results to show the efficiency and accuracy of the new method.

#### A numerical approach on Hiemenz flow problem using radial basis functions

Saeid Abbasbandy, K Parand, S Kazaem, AR Sanaei Kia International Journal of Industrial Mathematics 2013(5) 9Abstract:Abstract In this paper, we propose radial basis functions (RBF) to solve the two dimensional flow of fluid near a stagnation point named Hiemenz flow. The Navier-Stokes equations governing the flow can be reduced to an ordinary differential equation of third order using similarity transformation. Because of its wide applications the flow near a stagnation point has attracted many investigations during the past several decades. We satisfy boundary conditions such as infinity condition, by using Gaussian radial basis function through the both differential and integral operations. By choosing center points of RBF with shift on one point in uniform grid, we increase the convergence rate and decrease the collocation points.

#### Application of the exact operational matrices based on the Bernstein polynomials

K Parand, Sayyed A Kaviani Journal of mathematics and computer Science, 2013(6) 24Abstract:Abstract This paper aims to develop a new category of operational matrices. Exact operational matrices (EOMs) are matrices which integrate, differentiate and product the vector (s) of basis functions without any error. Some suggestions are offered to overcome the difficulties of this idea (including being forced to change the basis size and having more equations than unknown variables in the final system of algebraic equations). The proposed idea is implemented on the Bernstein basis functions. By both of the newly extracted Bernstein EOMs and ordinary operational matrices (OOMs) of the Bernstein functions, one linear and one nonlinear ODE is solved. Special attention is given to the comparison of numerical results obtained by the new algorithm with those found by OOMs.

#### The Sinc-collocation method for solving the Thomas–Fermi equation

K Parand, Mehdi Dehghan, A Pirkhedri Journal of Computational and Applied Mathematics 2013(273) 9Abstract:A numerical technique for solving nonlinear ordinary differential equations on a semi-infinite interval is presented. We solve the Thomas–Fermi equation by the Sinc-Collocation method that converges to the solution at an exponential rate. This method is utilized to reduce the nonlinear ordinary differential equation to some algebraic equations. This method is easy to implement and yields very accurate results.

##### 2012

#### A new Reliable Numerical Algorithm Based on the First Kind of Bessel Functions to Solve Prandtl–Blasius Laminar Viscous Flow over a Semi-Infinite Flat Plate

K Parand, M Nikarya, JA Rad, F Baharifard Zeitschrift für Naturforschung A 2012(67)Abstract:In this paper, a new numerical algorithm is introduced to solve the Blasius equation, which is a third-order nonlinear ordinary differential equation arising in the problem of two-dimensional steady state laminar viscous flow over a semi-infinite flat plate. The proposed approach is based on the first kind of Bessel functions collocation method. The first kind of Bessel function is an infinite series, defined on R and is convergent for any x?R. In this work, we solve the problem on semi-infinite domain without any domain truncation, variable transformation basis functions or transformation of the domain of the problem to a finite domain. This method reduces the solution of a nonlinear problem to the solution of a system of nonlinear algebraic equations. To illustrate the reliability of this method, we compare the numerical results of the present method with some well-known results in order to show the applicability and efficiency of our method.

#### The numerical study on the unsteady flow of gas in a semi-infinite porous medium using an RBF collocation method

S Kazem, JA Rad, K Parand, M Shaban, H Saberi International Journal of Computer Mathematics 2012(89)Abstract:In this paper, we study a nonlinear two-point boundary value problem on semi-infinite interval that describes the unsteady gas equation. The solution of the mentioned ordinary differential equation (ODE) is investigated by means of the radial basis function (RBF) collocation method. The RBF reduces the solution of the above-mentioned problem to the solution of a system of algebraic equations and finds its numerical solution. To examine the accuracy and stability of the approach, we transform the mentioned problem into another nonlinear ODE which simplifies the original problem. The comparisons are made between the results of the present work and the numerical method by shooting method combined with the Runge–Kutta technique. It is found that our results agree well with those by the numerical method, which verifies the validity of the present work.

#### A numerical solution of the nonlinear controlled Duffing oscillator by radial basis functions

JA Rad, S Kazem, K Parand Computers & Mathematics with Applications 2012( 64) 17Abstract:In this research, a new numerical method is applied to investigate the nonlinear controlled Duffing oscillator. This method is based on the radial basis functions (RBFs) to approximate the solution of the optimal control problem by using the collocation method. We apply Legendre–Gauss–Lobatto points for RBFs center nodes in order to use the numerical integration method more easily; then the method of Lagrange multipliers is used to obtain the optimum of the problems. For this purpose different applications of RBFs are used. The differential and integral expressions which arise in the dynamic systems, the performance index and the boundary conditions are converted into some algebraic equations which can be solved for the unknown coefficients. Illustrative examples are included to demonstrate the validity and applicability of the technique.

#### A meshless method on non-Fickian flows with mixing length growth in porous media based on radial basis functions

S Kazem, JA Rad, K Parand Computers & Mathematics with Applications 2012(64)Abstract:The present study aims to introduce a solution for parabolic integro-differential equations arising in heat conduction in materials with memory, which naturally occur in many applications. Two Radial basis functions (RBFs) collocation schemes are employed for solving this equation. The first method tested is an unsymmetric method, and the second one, which appears to be more efficient, is a symmetric one. The convergence of these two schemes is accelerated, as we use the cartesian nodes as the center nodes.

#### Numerical investigation on nano boundary layer equation with Navier boundary condition

AR Rezaei, M Shaban, K Parand Mathematical Methods in the Applied Sciences 2012(35)Abstract:In this paper, by applying rational Legendre collocation technique and relaxation method, the classical laminar boundary layer equations with the nonlinear Navier boundary conditions are investigated. The features of the flow characteristics for different values of n are discussed. Numerical approaches are used to find solutions for the cases n?>?1?/?2 corresponding to the flow past a wedge and n?=?1?/?2 corresponding to the flow in a convergent channel. During the comparison, the effectively and stability of the applied methods are demonstrated. The effects of the varying slip length, index parameter, components of velocity, and tangential stress are analyzed.

#### Comparison between Rational Gegenbauer and Modified Generalized Laguerre Functions Collocation Methods for Solving the Case of Heat Transfer Equations Arising in Porous Medium

K Parand, F Baharifard, F Bayat Babolghani International Journal of Industrial Mathematics 2012(4)Abstract:In this paper, we provide the collocation method for natural convection heat transfer equations embedded in porous medium which are of great importance in the design of canisters for nuclear waste disposal. This problem is a non-linear, three-point boundary value problem on semi-infinite interval. We use two orthogonal functions namely rational Gegenbauer and modified generalized Laguerre functions which are defined as basis functions in this approach and compare them together. We also present the comparison of these works with Runge-Kutta solution, moreover, in the graph of the ?Res? 2 , we show that the present solutionsare accurate and applicable.

#### Radial basis functions methods for solving Fokker–Planck equation

S Kazem, JA Rad, K Parand Engineering Analysis with Boundary Elements 2012(36) 9Abstract:In this paper two numerical meshless methods for solving the Fokker–Planck equation are considered. Two methods based on radial basis functions to approximate the solution of Fokker–Planck equation by using collocation method are applied. The first is based on the Kansa's approach and the other one is based on the Hermite interpolation. In addition, to conquer the ill-conditioning of the problem for big number of collocation nodes, two time domain Discretizing schemes are applied. Numerical examples are included to demonstrate the reliability and efficiency of these methods. Also root mean square and Ne errors are obtained to show the convergence of the methods. The errors show that the proposed Hermite collocation approach results obtained by the new time-Discretizing scheme are more accurate than the Kansa's approach.

#### The use of Sinc?collocation method for solving Falkner–Skan boundary?layer equation

K Parand, M Dehghan, A Pirkhedri International Journal for Numerical Methods in Fluids 2012(68)Abstract:The MHD Falkner–Skan equation arises in the study of laminar boundary layers exhibiting similarity on the semi-infinite domain. The proposed approach is equipped by the orthogonal Sinc functions that have perfect properties. This method solves the problem on the semi-infinite domain without truncating it to a finite domain and transforming domain of the problem to a finite domain. In addition, the governing partial differential equations are transformed into a system of ordinary differential equations using similarity variables, and then they are solved numerically by the Sinc-collocation method. It is shown that the Sinc-collocation method converges to the solution at an exponential rate.

#### Applying the Modified Generalized Laguerre Functions for Solving Steady Flow of a Third Grade Fluid in a Porous Half Space

K Parand, F Bayat Babolghani World Applied Sciences 2012(17)Abstract:In this paper we provide a collocation method for the problem of steady flow of third fluid in a porous half space. This problem is a non-linear, two point boundary value problem (BVP) on semi-infinite interval. This approach is based on a modified generalized Laguerre which is an orthogonal function. We also present the comparison of this work with solution of other methods; moreover, in the graph of the ||Res||2, we show that the present solution is more accurate and faster convergence in this problem.

#### Modified generalized laguerre functions for a numerical investigation of flow and diffusion of chemically reactive species over a nonlinearly stretching sheet

K Parand, FB Babolghani World Applied Sciences Journal 2012(17)Abstract:In this paper we provide a collocation method for the problem of flow and diffusion of chemically reactive species over a nonlinearly stretching sheet. This approach is based on a modified generalized Laguerre that it is an orthogonal function. Collocation method reduces the solution of these problems to the solution of systems of algebraic equations. We also compare this work with Homotopy Analysis Method (HAM). Moreover, in the graph of the ||Res||2, we show that the present solution is more accurate and faster convergence in this problem

#### Exp-function method for some nonlinear PDE’s and a nonlinear ODE’s

K Parand, JA Rad Journal of King Saud University-Science 2012(24)Abstract:In this paper, we apply the Exp-function method to find some exact solutions for two nonlinear partial differential equations (NPDE) and a nonlinear ordinary differential equation (NODE), namely, Cahn-Hilliard equation, Allen-Cahn equation and Steady-State equation, respectively. It has been shown that the Exp-function method, with the help of symbolic computation, provides a very effective and powerful mathematical tool for solving NPDE’s and NODE’s. Mainly we try to present an application of Exp-function method taking to consideration rectifying a commonly occurring errors during some of recent works. The results of the other methods clearly indicate the reliability and efficiency of the used method.

#### Numerical solution of nonlinear Volterra–Fredholm–Hammerstein integral equations via collocation method based on radial basis functions

K Parand, JA Rad Applied Mathematics and Computation 2012(218)Abstract:A numerical technique based on the spectral method is presented for the solution of nonlinear Volterra–Fredholm–Hammerstein integral equations. This method is a combination of collocation method and radial basis functions (RBFs) with the differentiation process (DRBF), using zeros of the shifted Legendre polynomial as the collocation points. Different applications of RBFs are used for this purpose. The integral involved in the formulation of the problems are approximated based on Legendre–Gauss–Lobatto integration rule. The results of numerical experiments are compared with the analytical solution in illustrative examples to confirm the accuracy and efficiency of the presented scheme.

#### Application of Exp-function method for a class of nonlinear PDE's arising in mathematical physics

K. Parand1 , Z. Delafkar , J. A. Rad , S. Kazem International Journal of Nonlinear Science 2012(13) 39-50Abstract:In this paper, a new powerful approach, called rational Legendre collocation method (RLC) is used to obtain the solution for nonlinear ordinary deferential equations that often appear in boundary layers problems arising in heat transfer. These kinds of the equations contain infinity boundary condition. The main objective is to reduce the solution of the problem to a solution of a system of algebraic equations, which do not require linearization and imposing the asymptotic condition transforming and physically unrealistic assumptions. Numerical results are compared with those of other methods, showing that the collocation method leads to more accurate results.

##### 2011

#### Rational Chebyshev Tau method for solving natural convection of Darcian fluid about a vertical full cone embedded in porous media whit a prescribed wall temperature

K. Parand1, Z. Delafkar2 , F. Baharifard3 Journal of Applied Mathematics and Informatics 2011(29) 763-779Abstract:In this paper we apply the Exp-function method to obtain traveling wave solutions of three nonlinear partial differential equations, namely, generalized sinh-Gordon equation, generalized form of the famous sinh-Gordon equation, and double combined sinh-cosh-Gordon equation. These equations play a very important role in mathematical physics and engineering sciences. The Exp-Function method changes the problem from solving nonlinear partial differential equations to solving a ordinary differential equation. Mainly we try to present an application of Exp-function method taking to consideration rectifying a commonly occurring errors during some of recent works.

#### Quasilinearization–Barycentric approach for numerical investigation of the boundary value Fin problem

A. Rezaei, F. Baharifard , K. Parand International Journal of Computer and Information Engineering 2011(5)Abstract:The problem of natural convection about a cone embedded in a porous medium at local Rayleigh numbers based on the boundary layer approximation and the Darcy’s law have been studied before. Similarity solutions for a full cone with the prescribed wall temperature or surface heat flux boundary conditions which is the power function of distance from the vertex of the inverted cone give us a third-order nonlinear differential equation. In this paper, an approximate method for solving higher-order ordinary differential equations is proposed. The approach is based on a rational Chebyshev Tau (RCT) method. The operational matrices of the derivative and product of rational Chebyshev (RC) functions are presented. These matrices together with the Tau method are utilized to reduce the solution of the higher-order ordinary differential equations to the solution of a system of algebraic equations. We also present the comparison of this work with others and show that the present method is applicable.

#### Solving MHD Falkner-Skan Boundary-Layer Equation Using Collocation Method

A Pirkhedri, HHS Javadi, K Parand, N Fatahi, S Lotfi World Applied Sciences Journal 2011(5)Abstract:In this paper we improve the quasilinearization method by barycentric Lagrange interpolation because of its numerical stability and computation speed to achieve a stable semi analytical solution. Then we applied the improved method for solving the Fin problem which is a nonlinear equation that occurs in the heat transferring. In the quasilinearization approach the nonlinear differential equation is treated by approximating the nonlinear terms by a sequence of linear expressions. The modified QLM is iterative but not perturbative and gives stable semi analytical solutions to nonlinear problems without depending on the existence of a smallness parameter. Comparison with some numerical solutions shows that the present solution is applicable.

#### Numerical solution of the Falkner-Skan equation with stretching boundary by collocation method

Kourosh Parand, Nasrollah Pakniat, Zahra Delafkar 2011(16)Abstract:Abstract Based on a new approximation method, namely pseudospectral method, a solution for the three order nonlinear ordinary differential laminar boundary layer Falkner–Skan equation has been obtained on the semi-infinite domain. The proposed approach is equipped by the orthogonal Hermite functions that have perfect properties to achieve this goal. This method solves the problem on the semi-infinite domain without truncating it to a finite domain and transforming domain of the problem to a finite domain. In addition, this method reduces solution of the problem to solution of a system of algebraic equations. We also present the comparison of this work with numerical results and show that the present method is applicable.

#### A numerical study on reaction–diffusion problem using radial basis functions

K Paranda, S Kazemb, AR Rezaei International Journal of Nonlinear Science 2011(11) 275--283Abstract: In the present paper, a solution for the boundary value problem over a semi-infinite interval has been obtained by transforming the two-dimensional laminar boundary equations into a nonlinear ordinary equation using similarity variables. Moreover, a collocation method is proposed to solve the Falkner-Skan equation. This method is based on rational Legendre functions and convert the Falkner-Skan equation to a system of nonlinear algebraic equations. The results are tabulated and compared with some other methods

#### An efficient computational algorithm for solving the nonlinear Lane-Emden type equations

Gholamreza Hojjati, Kourosh Parand Communications in Numerical Analysis 2011(2011)Abstract:Abstract In this paper, the collocation approach, based on the indirect radial basis functions on boundary value problems (IRBFB), is used to obtain a solution for the problem of a non- linear model of reaction–diffusion in porous catalysis pellets for the case of nth–order reaction. One of the boundaries of porous slab is impermeable and the other one is held at constant concentration. We applied this method through the integration process on the boundary value reaction–diffusion problem. The Thiele modulus thus measures the relative importance of the diffusion and reaction phenomena. Interestingly, for the large Thiele modulus the IRBFB offer a reasonable solution. Numerical results and findings obtained by the comparison with finite difference method, show a good accuracy and appropriate convergence rate of IRBFB process.

#### An approximate solution of the MHD Falkner–Skan flow by Hermite functions pseudospectral method

K Parand, AR Rezaei, SM Ghaderi World Academy of Science, Engineering and Technology 2011(5)Abstract:In this paper we propose a class of second derivative multistep methods for solving some well-known classes of Lane-Emden type equations which are nonlinear ordinary differential equations on the semi-infinite domain. These methods, which have good stability and accuracy properties, are useful in deal with stiff ODEs. We show superiority of these methods by applying them on the some famous Lane-Emden type equations.

#### a new method for solving steady flow of a third-grade fluid in a porous half space based on radial basis functions

Saeed Kazem, Jamal Amani Rad, Kourosh Parand, Saied Abbasbandy Communications in Nonlinear Science and Numerical Simulation 2011(16) 274-283Abstract:Based on a new approximation method, namely pseudospectral method, a solution for the three order nonlinear ordinary differential laminar boundary layer Falkner–Skan equation has been obtained on the semi-infinite domain. The proposed approach is equipped by the orthogonal Hermite functions that have perfect properties to achieve this goal. This method solves the problem on the semi-infinite domain without truncating it to a finite domain and transforming domain of the problem to a finite domain. In addition, this method reduces solution of the problem to solution of a system of algebraic equations. We also present the comparison of this work with numerical results and show that the present method is applicable.

#### A novel application of radial basis functions for solving a model of first-order integro-ordinary differential equation

K Parand, S Abbasbandy, S Kazem, JA Rad Zeitschrift für Naturforschung A 2011(66) 591-598Abstract: In this study, flow of a third-grade non-Newtonian fluid in a porous half space has been considered. This problem is a nonlinear two-point boundary value problem (BVP) on semi-infinite interval. We find the simple solutions by using collocation points over the almost whole domain [0;?). Our method based on radial basis functions (RBFs) which are positive definite functions. We applied this method through the integration process on the infinity boundary value and simply satisfy this condition by Gaussian, inverse quadric, and secant hyperbolic RBFs.We compare the results with solution of other methods.

#### Numerical and analytical solution of gas flow through a micro-nano porous media

JA Rad, SM Ghaderi, K Parand Communications in Nonlinear Science and Numerical Simulation 2011(16 ) 4250-4258Abstract: In this paper two common collocation approaches based on radial basis functions (RBFs) have been considered; one is computed through the differentiation process (DRBF) and the other one is computed through the integration process (IRBF). We investigate these two approaches on the Volterra's Population Model which is an integro-differential equation without converting it to an ordinary differential equation. To solve the problem, we use four well-known radial basis functions

#### Collocation method using sinc and Rational Legendre functions for solving Volterra’s population model

K Parand, Z Delafkar, N Pakniat, A Pirkhedri, M Kazemnasab Haji Journal of Computational and Theoretical Nanoscience 2011(8) 2033-2041Abstract: In this paper, we study a non-linear two-point boundary value problem (BVP) on semi-infinite interval that describes the unsteady gas equation. The solution of the mentioned ordinary differential equation (ODE) is investigated by means of the Hermite functions collocation method and the Homotopy analysis method (HAM). The Hermite functions collocation method reduces the solution of above-mentioned problem to the solution of a system of algebraic equations and finds its the numerical solution. The homotopy analysis method is also one of the most effective methods in obtaining series solutions for these types of problems and finds their analytic solution. Through the convergence of these methods we determine the accurate initial slope y'(0) with good capturing the essential behavior of y (x). Numerical and analytical evaluations and comparisons with the results obtained are also discussed at the last part of the paper.

#### Comparison between two common collocation approaches based on radial basis functions for the case of heat transfer equations arising in porous medium

Kourosh Parand, Saeid Abbasbandy, Saeed Kazem, AR Rezaei Communications in Nonlinear Science and Numerical Simulation 2011(16) 1811-1819Abstract: This paper proposes two approximate methods to solve Volterra's population model for population growth of a species in a closed system. Volterra's model is a nonlinear integro-differential equation on a semi-infinite interval, where the integral term represents the effect of toxin. The proposed methods have been established based on collocation approach using Sinc functions and Rational Legendre functions. They are utilized to reduce the computation of this problem to some algebraic equations. These solutions are also compared with some well-known results which show that they are accurate.

#### Numerical study on gas flow through a micro-nano porous media based on special functions

Alireza Rezaei, Kourosh Parand, Ali Pirkhedri Communications in Nonlinear Science and Numerical Simulation 2011(16) 1396-1407Abstract: In this paper two common collocation approaches based on radial basis functions have been considered; one be computed through the integration process (IRBF) and one be computed through the differentiation process (DRBF). We investigated the two approaches on natural convection heat transfer equations embedded in porous medium which are of great importance in the design of canisters for nuclear wastes disposal. Numerical results show that the IRBF be performed much better than the common DRBF, and show good accuracy and high rate of convergence of IRBF process.

#### An improved numerical method for a class of astrophysics problem based on radial basis functions

K Parand, S Abbasbandy, S Kazem, AR Rezaei Journal of Computational and Theoretical Nanoscience 2011(8) 282-288Abstract: In this paper unsteady isothermal flow of a gas through a semi-infinite Micro-Nano porous medium which is a non-linear two-point boundary value problem (BVP) on semi-infinite interval has been considered. We solve this problem by two different pseudospectral approaches and compare their results with solution of other methods. The proposed approaches are equipped by the orthogonal rational Legendre and Sinc functions. These methods reduce solution of the problem to solution of a system of algebraic equations. Also through the convergence of these methods we determine the accurate initial slope y'(0) with good capturing the essential behavior of y (x).

#### Spectral Method for Solving Differential Equation of Gas Flow Through a Micro-Nano Porous Media

Taghavi, Amir; Parand, Koroush; Shams, Alireza; Sofloo, Hadi Ghezel Physica Scripta 2011(83) 015011Abstract: In this paper, we propose radial basis functions for solving some well-known classes of astrophysics problems categorized as nonlinear singular initial ordinary differential equations on a semi-infinite domain. To increase the convergence rate and to decrease the collocation points, we use the even radial basis functions through the integral operations. Afterwards, some special cases of the equation are presented as test examples to show the reliability of the method. Then we compare the results of this work with some recent results and show that the new method is efficient and applicable.

##### 2010

#### Numerical approximations for population growth model by Rational Chebyshev and Hermite Functions collocation approach

K. Parand , A. R. Rezaei, A. Taghavi Mathematical Methods in the Applied Sciences 2010(2) 542-546Abstract:In this paper we propose, Spectral method to solve unsteady gas equation which is a nonlinear ordinary differential equation on semi-infinite interval. These approaches are based on Modified generalized Laguerre functions. These methods reduces the solution of this problem to the solution of systems of algebraic equations. We also compare this works with some other numerical results.

#### Lagrangian method for solving Lane-Emden type equation arising in astrophysics on semi-infinite domains

K Parand, A R Rezaei, A Taghavi* Acta Astronautica 2010(33) 2076-2086Abstract:This paper aims to compare rational Chebyshev (RC) and Hermite functions (HF) collocation approach to solve the Volterra’s model for population growth of a species within a closed system. This model is a nonlinear integro-differential equation where the integral term represents the effect of toxin. This approach is based on orthogonal functions which will be defined. The collocation method reduces the solution of this problem to the solution of a system of algebraic equations. We also compare these methods with some other numerical results and show that the present approach is applicable for solving nonlinear integro-differential equations.

#### Modified generalized Laguerre function Tau method for solving laminar viscous flow

K. Parand, M. Dehghan, A. Taghavi International Journal of Numerical Methods for Heat & Fluid Flow 2010(67) 673-680Abstract:In this paper we propose a Lagrangian method for solving Lane-Emden equation which is a nonlinear ordinary differential equation on semi-infinite interval. This approach is based on a Modified generalized Laguerre functions Lagrangian method. The method reduces the solution of this problem to the solution of a system of algebraic equations. We also present the comparison of this work with some well-known results and show that the present solution is acceptable.

#### Sinc-Collocation method for solving astrophysics equations

K. Parand ,A. Pirkhedri New Astronomy 2010(20) 728-743Abstract:The purpose of this paper is to propose a Tau method for solving nonlinear Blasius equation which is a partial differential equation on a flat plate. Design/methodology/ approach–The operational matrices of derivative and product of modified generalized Laguerre functions are presented. These matrices together with the Tau method are then utilized to reduce the solution of the Blasius equation to the solution of a system of nonlinear equations. Findings–The paper presents the comparison of this work with some well-known results and shows that the present solution is highly accurate. Originality/value–This paper demonstrates solving of the nonlinear Blasius equation with an efficient method.

#### Some solitary wave solutions of generalized Pochhammer-Chree equation via Exp-function method

Kourosh Parand, Jamal Amani Rad 2010(15) 533–537Abstract:In this paper we propose Sinc-Collocation method for solving Lane–Emden equation which is a nonlinear ordinary differential equation on a semi-infinite interval. It is found that Sinc procedure converges with the solution at an exponential rate. This method is utilized to reduce the computation of this problem to some algebraic equations. We also compare this solution with some well-known results and show that it is accurate.

#### An approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method

K. Parand, Mehdi Dehghan , A.R. Rezaeia , S.M. Ghaderia Computer Physics Communications 2010(4) 991-996Abstract:In this paper, Exp-function method is used for some exact solitary solutions of the generalized Pochhammer-Chree equation. It has been shown that the Exp-function method, with the help of symbolic computation, provides a very effective and powerful mathematical tool for solving nonlinear partial differential equations. As a result, some exact solitary solutions are obtained. It is shown that the Exp-function method is direct, effective, succinct and can be used for many other nonlinear partial differential equations.

#### Solution of a laminar boundary layer flow via a numerical method

K. Parand , M. Shahini , Mehdi Dehghan Commun Nonlinear Sci Numer Simulat 2010(181) 1096-1108Abstract:In this paper we propose a collocation method for solving some well-known classes of LaneEmden type equations which are nonlinear ordinary differential equations on the semi-infinite domain. They are categorized as singular initial value problems. The proposed approach is based on a Hermite function collocation (HFC) method. To illustrate the reliability of the method, some special cases of the equations are solved as test examples. The new method reduces the solution of a problem to the solution of a system of algebraic equations. Hermite functions have prefect properties that make them useful to achieve this goal. We compare the present work with some well-known results and show that the new method is efficient and applicable.

#### RATIONAL CHEBYSHEV COLLOCATION METHOD FOR SOLVING NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS OF LANE-EMDEN TYPE

KOUROSH PARAND AND MEHDI SHAHINI International Journal of Information and Systems Sciences 2010(15) 360-367Abstract:In this paper, the numerical solution of the Blasius problem is obtained using the collocation method based on rational Chebyshev functions. The Blasius equation is a nonlinear ordinary differential equation which arises in the boundary layer flow. The method reduces solving the equation to solving a system of nonlinear algebraic equations. The results presented here demonstrate reliability and efficiency of the method.

#### Analytical solution of Gas Flow Through a Micro-Nano Porous Media by Homotopy Perturbation method

Jamal Amani Rad, Kourosh Parand International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering 2010(4) 72–83Abstract:Lane-Emden equation is a nonlinear singular equation that plays an important role in the astrophysics. In this paper, we have applied the collocation method based on rational Chebyshev functions to solve Lane-Emden type equations. The method reduces solving the nonlinear ordinary differential equation to solving a system of nonlinear algebraic equations. The comparison of the results with the other numerical methods shows that our solutions are highly accurate and the method is convergent.

#### Quasilinearization approach for solving Volterra's population model

K. Parand, M. Gasemi, S. Rezazadeh, A. Peiravi Applied and Computational Mathematics 2010(9) 197-201Abstract:In this paper, we have applied the homotopy perturbation method (HPM) for obtaining the analytical solution of unsteady flow of gas through a porous medium and we have also compared the findings of this research with some other analytical results. Results showed a very good agreement between results of HPM and the numerical solutions of the problem rather than other analytical solutions which have previously been applied. The results of homotopy perturbation method are of high accuracy and the method is very effective and succinct.

##### 2009

#### Generalized Laguerre Polynomials and Rational Chebyshev Collocation Method for Solving Unsteady Gas Equation

K. Parand, M. Shahini and A. Taghavi Int. J. Contemp. Math. Sciences 2009(4) 95-103Abstract:The study used the Quasilinearization method to solve Volterra's model for popu-lation growth of a species within a closed system is proposed. This model is a nonlinear integro-differential where the integral term represents the effect of toxin. First we convert this model to a nonlinear ordinary differential equation, then approximate the solution of this equation by treating the nonlinear terms as a perturbation about the linear ones. Finally we compare this method with the other methods and come to the conclusion that the Quasilinearization method gives excellent results.

#### Rational Chebyshev pseudospectral approach for solving Thomas–Fermi equation

K. Parand , M. Shahini Physics Letters A 2009(373) 1005 - 1011Abstract:In this paper we propose, a collocation method to solve unsteady gas equation which is a nonlinear ordinary differential equation on semiinfnite interval. This approach is based on generalized Laguerre polynomials and rational Chebyshev functions. This method reduces the solution of this problem to the solution of a system of algebraic equations. We also present the comparison of this work with some other numerical results. It shows that the present solution is highly accurate.

#### COMPARISON BETWEEN RATIONAL CHEBYSHEV AND MODIFIED GENERALIZED LAGUERRE FUNCTIONS PSEUDOSPECTRAL METHODS FOR SOLVING LANE–EMDEN AND UNSTEADY GAS EQUATIONS

K. Parand, A. Taghavi, M. Shahini ACTA PHYSICA POLONICA B 2009(40 ) 210–213Abstract:In this Letter we propose a pseudospectral method for solving Thomas–Fermi equation which is a nonlinear ordinary differential equation on semi-infinite interval. This approach is based on rational Chebyshev pseudospectral method. This method reduces the solution of this problem to the solution of a system of algebraic equations. Comparison with some numerical solutions shows that the present solution is highly accurate.

#### Sinc-collocation method for solving the Blasius equation

K Parand, Mehdi Dehghan, A Pirkhedri Physics Letters A 2009(373) 4060-4065Abstract:In this paper we provide a pseudospectral method for Lane–Emden equation which models many phenomena in mathematical physics and astrophysics. We also use this method for solving unsteady gas equation which model unsteady flow of a gas through a semi-infinite porous medium. This approach is based on some orthogonal functions which will be defined. Pseudospectral method reduces the solution of these problems to the solution of systems of algebraic equations. We also compare this work with some other numerical results.

#### Lagrangian method for solving unsteady gas equation

Amir Taghavi, Hosein Fani International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering 2009(3) 991-995Abstract:Abstract Sinc-collocation method is applied for solving Blasius equation which comes from boundary layer equations. It is well known that sinc procedure converges to the solution at an exponential rate. Comparison with Howarth and Asaithambi's numerical solutions reveals that the proposed method is of high accuracy and reduces the solution of Blasius' equation to the solution of a system of algebraic equations.

#### Rational Legendre pseudospectral approach for solving nonlinear differential equations of Lane–Emden type

Kourosh Parand, Mehdi Shahini, Mehdi Dehghan Journal of Computational Physics 2009(228) 8830-8840Abstract:Abstract

#### Rational scaled generalized Laguerre function collocation method for solving the Blasius equation

K. Parand1 , A. Taghavi Journal of Computational and Applied Mathematics 2009(233) 980–989Abstract:Abstract Lane–Emden equation is a nonlinear singular equation in the astrophysics that corresponds to the polytropic models. In this paper, a pseudospectral technique is proposed to solve the Lane–Emden type equations on a semi-infinite domain. The method is based on rational Legendre functions and Gauss–Radau integration. The method reduces solving the nonlinear ordinary differential equation to solve a system of nonlinear algebraic equations. The comparison of the results with the other numerical methods shows the efficiency and accuracy of this method.

##### 2008

#### Generalized Laguerre Polynomials Collocation Method for Solving Lane-Emden Equation

K. Parand and A. Taghavi Applied Mathematical Sciences 2008(2) 2955 - 2961Abstract:In this paper we propose, a collocation method for solving nonlinear singular Lane-Emden equation which is a nonlinear ordinary differential equation on semi-infnite interval. This approach is based on a generalized Laguerre polynomial collocation method. This method reduces the solution of this problem to the solution of a system of algebraic equations.We also present the comparison of this work with some well-known results and show that the present solution is highly accurate.

#### Solving Volterra's Population Model Using New Second Derivative Multistep Methods

K. Parand and G. Hojjati American Journal of Applied Sciences 2008(5) 1019-1022Abstract:In this study new second derivative multistep methods (denoted SDMM) are used to solve Volterra's model for population growth of a species within a closed system. This model is a nonlinear integro-differential where the integral term represents the effect of toxin. This model is first converted to a nonlinear ordinary differential equation and then the new SDMM, which has good stability and accuracy properties, are applied to solve this equation. We compare this method with the others and show that new SDMM gives excellent results.

##### 2004

#### Rational Chebyshev tau method for solving Volterra’s population model

K. Parand, M. Razzaghi Applied Mathematics and Computation 2004(149) 893-900Abstract:An approximate method for solving Volterra’s population model for population growth of a species in a closed system is proposed. Volterra’s model is a nonlinear integro-differential equation where the integral term represents the effect of toxin. The approach is based on a rational Chebyshev tau method. The Volterra’s population model is first converted to a nonlinear ordinary differential equation. The operational matrices of derivative and product of rational Chebyshev functions are presented. These matrices together with the tau method are then utilized to reduce the solution of the Volterra’s model to the solution of a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique and a comparison is made with existing results.

#### Rational Legendre approximation for solving some physical problems on semi-infinite intervals

K. Parand1, M. Razzaghi2 Physica Scripta 2004(69) , 353–357Abstract:A numerical technique for solving some physical problems on a semi-infinite interval is presented. Two nonlinear examples are proposed. In the first example the Volterra's population model growth is formulated as a nonlinear differential equation, and in the second example the Lane–Emden nonlinear differential equation is considered. The approach is based on a rational Legendre tau method. The operational matrices of derivative and product of rational Legendre functions are presented. These matrices together with the tau method are utilized to reduce the solution of these physical problems to the solution of systems of algebraic equations. The method is easy to implement and yields very accurate results

#### Rational Chebyshev tau method for solving higher-order ordinary differential equations

K. Parand, M. Razzaghi International Journal of Computer Mathematics 2004(80) 73-80Abstract:An approximate method for solving higher-order ordinary differential equations is proposed. The approach is based on a rational Chebyshev (RC) tau method. The operational matrices of the derivative and product of RC functions are presented. These matrices together with the tau method are utilized to reduce the solution of the higher-order ordinary differential equations to the solution of a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

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