A modification of the homotopy analysis method based on Chebyshev operational matrices

Shaban, M.; Kazem, Saeed; Rad, Jamal Amani

doi:10.1016/j.mcm.2012.09.024

Cited by 33 publications

(22 citation statements)

References 34 publications

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“…Martin [26] proposed an analytical algorithm to solve an integral-differential equation of the transport theory for stationary case using HAM. The solution of nonlinear boundary value problems (NBVPs) based on the Chebyshev operational matrices was investigated by Shaban et al [27]. They applied the Tau method to convert a set of algebraic equations so that the solution can be obtained iteratively.…”

Section: Homotopy Analysis Methods (Ham)mentioning

confidence: 99%

A Mathematical Approach Based on the Homotopy Analysis Method: Application to Solve the Nonlinear Harry-Dym (HD) Equation

Ghiasi¹,

Saleh²

2017

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In this paper, the homotopy analysis method (HAM) has been employed to obtain the approximate analytical solution of the nonlinear Harry-Dym (HD) equation, which is one of the most important soliton equations. Utilizing the HAM, thereby employing the initial approximation, variations of the 7th-order approximation of the Harry-Dym equation is obtained. It is found that effect of the nonzero auxiliary parameter on convergence rate of the series solution is undeniable. It is also shown that, to some extent, order of the fractional derivative plays a fundamental role in the prediction of convergence. The final results reported by the HAM have been compared with the exact solution as well as those obtained through the other methods.

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Section: Homotopy Analysis Methods (Ham)mentioning

confidence: 99%

A Mathematical Approach Based on the Homotopy Analysis Method: Application to Solve the Nonlinear Harry-Dym (HD) Equation

Ghiasi¹,

Saleh²

2017

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“…Also, the mentioned method has been employed to find the solution of the coupled Burgers equation. The authors of [31] presented a modified form of the homotopy analysis method based on Chebyshev operational matrices. This method is based on the operational matrix of Chebyshev polynomials and constructs the derivative and the product of the unknown function as a matrix.…”

Section: Research Literaturementioning

confidence: 99%

Operation matrix method based on Bernstein polynomials for the Riccati differential equation and Volterra population model

Parand

Hossayni

Rad

2016

Applied Mathematical Modelling

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Please cite this article as: K. Parand, Sayyed A. Hossayni, J.A. Rad, An operation matrix method based on Bernstein polynomials for Riccati differential equation and Volterra population model, Applied Mathematical Modelling (2015), Highlight• We present an exact formulation for the operational matrix.• This method transforms the original problem into a algebraic equations system.• Riccati equation and Volterra population model are solved by new method. AbstractIn this paper, we present a modified configuration, including exact formulation for the operational matrix form of the integration, differentiation and product operators to be applied in the Galerkin method. Up to now, so many studies have been conducted on the methods of obtaining operational matrices (derivative, integral and product) for Fourier, Chebyshev, Legendre and Jacobi polynomials and, sometimes, the non-orthogonal bases, which almost all of them operate approximately; but, in this research, we aim to obtain the exact operational matrices (EOMs), which can be used for many classes of orthogonal and non-orthogonal polynomials. Similar to the previous works, this method transforms the original problem into a system of nonlinear, algebraic equations. To keep the simplicity of the procedure, samples have been considered in one dimensional contexts; however, the proposed technique can be employed for two and three-dimensional problems, as well. For confirming the accuracy of the new approach and for showing the performance of EOMs over ordinary operational matrices (OOMs), two examples are presented. The corresponding results demonstrated the increased accuracy of the new method. Also the convergence of the EOM method is studied numerically and analytically, to prove the method efficiency.

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“…Spectral methods, in the context of numerical schemes for differential equations, generically belong to the family of weighted residual methods (WRMs) [6]. WRMs represent a particular group of approximation techniques, in which the residuals (or errors) are minimized in a certain way and thereby leading to specific methods including Galerkin, Petrov-Galerkin, collocation and Tau formulations [7][8][9][10][11][12][13].…”

Section: Introductionmentioning

confidence: 99%

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

Baharifard

Kazem

Parand

2015

Int. J. Appl. Comput. Math

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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.

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A modification of the homotopy analysis method based on Chebyshev operational matrices

Cited by 33 publications

References 34 publications

A Mathematical Approach Based on the Homotopy Analysis Method: Application to Solve the Nonlinear Harry-Dym (HD) Equation

A Mathematical Approach Based on the Homotopy Analysis Method: Application to Solve the Nonlinear Harry-Dym (HD) Equation

Operation matrix method based on Bernstein polynomials for the Riccati differential equation and Volterra population model

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

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