Papers by Abbas Saadatmandi
In recent years, an increasing interest has focused on the use of Sinc methods as an essential to... more In recent years, an increasing interest has focused on the use of Sinc methods as an essential tools for solving some singular problems arising in different areas of applied sciences. Due to the presence of singularity, these problems raise difficulties in obtaining their analytic or numerical solutions, and various schemes have been proposed to overcome these difficulties. However, among existing approaches, the Sinc methods are well-suited for handling singularity and have high performance on boundary value problems (BVPs) such as problems on unbounded domains or problems with endpoint singularities. In this work, in several geometries and kinetics, an implementation of the Sinc-Galerkin scheme is used to approximate effectiveness factor and concentration profile of key component when a single independent reaction takes place in a porous catalyst structure where enzymes are immobilized. A comparison between the proposed approximated solution and numerical solution reveals that the Sinc-Galerkin method (SGM), as demonstrated with examples, is reliable, accurate and its convergence rate is high.
Computers & Mathematics with Applications, 2010
In this work we consider nonlinear reaction-diffusion equations arising in mathematical biology. ... more In this work we consider nonlinear reaction-diffusion equations arising in mathematical biology. We use the exp-function method in order to obtain conventional solitons and periodic solutions. The proposed scheme can be applied to a wide class of nonlinear equations.
Iranian journal of mathematical chemistry, Feb 1, 2012
In this paper, a Chebyshev finite difference method has been proposed in order to solve nonlinear... more In this paper, a Chebyshev finite difference method has been proposed in order to solve nonlinear two-point boundary value problems for second order nonlinear differential equations. A problem arising from chemical reactor theory is then considered. The approach consists of reducing the problem to a set of algebraic equations. This method can be regarded as a non-uniform finite difference scheme. The method is computationally attractive and applications are demonstrated through an illustrative example. Also a comparison is made with existing results.
A sinc-Gauss-Jacobi collocation method for solving Volterra's population growth model with fractional order
Tbilisi Mathematical Journal, Jun 1, 2018
An approximate solution of the MHD flows of UCM fluids over porous stretching sheets by rational Legendre collocation method
International Journal of Numerical Methods for Heat & Fluid Flow, Sep 5, 2016
Purpose The purpose of this paper is to develop an efficient method for solving the magneto-hydro... more Purpose The purpose of this paper is to develop an efficient method for solving the magneto-hydrodynamic (MHD) boundary layer flow of an upper-convected Maxwell (UCM) fluid over a porous isothermal stretching sheet. Design/methodology/approach The paper applied a collocation approach based on rational Legendre functions for solving the third-order non-linear boundary value problem, describing the MHD boundary layer flow of an UCM fluid over a porous isothermal stretching sheet. This method solves the problem on the semi-infinite domain without transforming domain of the problem to a finite domain. Findings This approach reduces the solution of a problem to the solution of a system of algebraic equations. The numerical values of the skin friction coefficient are presented and analyzed for various parameters of interest in the problem. The authors also compare the results of this work with some recent results and show that the new method is efficient and applicable. Originality/value The method solves this problem without use of discrete variables and linearization or small perturbation. Also it was confirmed by the theorem and figure of absolute coefficients that this approach has exponentially convergence rate.
Applied Mathematics and Computation, Nov 1, 2019
In this paper, for the first time, the shifted Legendre operational matrix of distributed order f... more In this paper, for the first time, the shifted Legendre operational matrix of distributed order fractional derivative has been derived. Also, this new operational matrix is used together with tau method for approximation of solutions of linear distributed order fractional differential equations and diffusion equations with distributed order in time. Moreover, eight numerical examples are implemented in order to show the validity and reliability of the suggested methods.
Applied Mathematics and Computation, Apr 1, 2018
In this paper, a direct collocation method based on rational Legendre functions is proposed for s... more In this paper, a direct collocation method based on rational Legendre functions is proposed for solving the magneto-hydrodynamic (MHD) boundary layer flow over a nonlinear stretching sheet. Here, we use rational Legendre-Gauss-Radau nodes and transformed Hermite-Gauss nodes as interpolation points. We present the comparison of this work with some other numerical results. Moreover, residual norm shows that the present solutions are accurate and applicable.
Mathematics Interdisciplinary Research, Dec 1, 2019
This paper provides the fractional derivatives of the Caputo type for the sinc functions. It allo... more This paper provides the fractional derivatives of the Caputo type for the sinc functions. It allows to use efficient numerical method for solving fractional differential equations. At first, some properties of the sinc functions and Legendre polynomials required for our subsequent development are given. Then we use the Legendre polynomials to approximate the fractional derivatives of sinc functions. Some numerical examples are introduced to demonstrate the reliability and effectiveness of the introduced method.
Computational Methods for Differential Equations, 2020
This work presents a new approximation approach to solve the linear/nonlinear distributed order f... more This work presents a new approximation approach to solve the linear/nonlinear distributed order fractional differential equations using the Chebyshev polynomials. Here, we use the Chebyshev polynomials combined with the idea of the collocation method for converting the distributed order fractional differential equation into a system of linear/nonlinear algebraic equations. Also, fractional differential equations with initial conditions can be solved by the present method. We also give the error bound of the modified equation for the present method. Moreover, four numerical tests are included to show the effectiveness and applicability of the suggested method.
A new operational matrix based on Müntz–Legendre polynomials for solving distributed order fractional differential equations
Mathematics and Computers in Simulation, 2021
The construction of a new operational matrix of the distributed-order fractional derivative using Chebyshev polynomials and its applications
International Journal of Computer Mathematics, 2021
In this paper, the properties of Chebyshev polynomials and the Gauss–Legendre quadrature rule are... more In this paper, the properties of Chebyshev polynomials and the Gauss–Legendre quadrature rule are employed to construct a new operational matrix of distributed-order fractional derivative. This operational matrix is applied for solving some problems such as distributed-order fractional differential equations, distributed-order time-fractional diffusion equations and distributed-order time-fractional wave equations. Our approach easily reduces the solution of all these problems to the solution of some set of algebraic equations. We also discuss the error analysis of approximation distributed-order fractional derivative by using this operational matrix. Finally, to illustrate the efficiency and validity of the presented technique five examples are given. Abbreviations: DFDEs: distributed-order fractional differential equations; DTFDEs: distributed-order time-fractional diffusion equations; DTFWEs: distributed-order time-fractional wave equations; OMFD: operational matrix of fractional derivative; SCP: shifted Chebyshev polynomial
Paper: CHEBYSHEV FINITE DIFFERENCE METHOD FOR A TWO-POINT BOUNDARY VALUE PROBLEMS WITH APPLICATIONS TO CHEMICAL REACTOR THEORY
Transformed Hermite functions on a finite interval and their applications to a class of singular boundary value problems
Computational and Applied Mathematics, 2015
In this paper, a weighted orthogonal system on finite interval based on the transformed Hermite f... more In this paper, a weighted orthogonal system on finite interval based on the transformed Hermite functions is introduced. Some results on approximations using the Hermite functions on finite interval are obtained from corresponding approximations on infinite interval via a conformal map. To illustrate the potential of the new basis, we apply it to the collocation method for solving a class of singular two-point boundary value problems. The numerical results show that our new scheme is very effective and convenient for solving singular boundary value problems.
Applied Mathematics and Computation, 2007
The Chebyshev finite difference method is presented for solving a nonlinear system of second-orde... more The Chebyshev finite difference method is presented for solving a nonlinear system of second-order boundary value problems. Our approach consists of reducing the problem to a set of algebraic equations. This method can be regarded as a non-uniform finite difference scheme. Some numerical results are also given to demonstrate the validity and applicability of the presented technique and a comparison is made with the existing results. The method is easy to implement and yields very accurate results.
Applied Mathematics and Computation, 2018
In this paper, a direct collocation method based on rational Legendre functions is proposed for s... more In this paper, a direct collocation method based on rational Legendre functions is proposed for solving the magneto-hydrodynamic (MHD) boundary layer flow over a nonlinear stretching sheet. Here, we use rational Legendre-Gauss-Radau nodes and transformed Hermite-Gauss nodes as interpolation points. We present the comparison of this work with some other numerical results. Moreover, residual norm shows that the present solutions are accurate and applicable.
A sinc-Gauss-Jacobi collocation method for solving Volterra's population growth model with fractional order
Tbilisi Mathematical Journal, 2018
Applied Mathematics and Computation, 2017
An important class of fluids commonly used in industries is non-Newtonian fluids. In this paper, ... more An important class of fluids commonly used in industries is non-Newtonian fluids. In this paper, two numerical techniques based on rational Legendre functions and Chebyshev polynomials are presented for solving the flow of a third-grade fluid in a porous half space. This problem can be reduced to a nonlinear two-point boundary value problem on semi-infinite interval. Our methods are utilized to reduce the computation of this problem to some algebraic equations. The comparison of the results with the other methods and residual norm show very good accuracy and rate of convergence of our approach.
International Journal of Numerical Methods for Heat & Fluid Flow, 2012
PurposeRosenau‐Hyman equation was discovered as a simplified model to study the role of nonlinear... more PurposeRosenau‐Hyman equation was discovered as a simplified model to study the role of nonlinear dispersion on pattern formation in liquid drops. Also, this equation has important roles in the modelling of various problems in physics and engineering. The purpose of this paper is to present the solution of Rosenau‐Hyman equation.Design/methodology/approachThis paper aims to present the solution of the Rosenau‐Hyman equation by means of semi‐analytical approaches which are based on the homotopy perturbation method (HPM), variational iteration method (VIM) and Adomian decomposition method (ADM).FindingsThese techniques reduce the volume of calculations by not requiring discretization of the variables, linearization or small perturbations. Numerical solutions obtained by these methods are compared with the exact solutions, revealing that the obtained solutions are of high accuracy. These results reveal that the proposed methods are very effective and simple to perform.Originality/value...
Zeitschrift für Naturforschung A, 2010
In this paper, the homotopy analysis method is applied to solve linear fractional problems. Based... more In this paper, the homotopy analysis method is applied to solve linear fractional problems. Based on this method, a scheme is developed to obtain approximation solution of fractional wave, Burgers, Korteweg-de Vries (KdV), KdV-Burgers, and Klein-Gordon equations with initial conditions, which are introduced by replacing some integer-order time derivatives by fractional derivatives. The fractional derivatives are described in the Caputo sense. So the homotopy analysis method for partial differential equations of integer order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique.
The Fractional Linear Systems of Equations Within an Operational Approach
Journal of Computational and Nonlinear Dynamics, 2012
Fractional calculus is a rapidly going area from both experimental and theoretical points of view... more Fractional calculus is a rapidly going area from both experimental and theoretical points of view. As a result new methods and techniques should be developed in order to deal with new types of fractional differential equations. In this paper the operational matrix of fractional derivative together with the τ method are used to solve the linear systems of fractional differential equations. The results of this method are shown by solving three illustrative examples. By comparing the obtained results with the analytic solutions and with the ones provided by three standard methods for solving the fractional differential equations we conclude that our method gave comparable results.
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Papers by Abbas Saadatmandi