A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations

Abstract:In this paper, two-dimensional Genocchi polynomials and the Ritz-Galerkin method were developed to investigate the Fractional Diffusion Wave Equation (FDWE) and the Fractional Klein-Gordon Equation (FKGE). A satisfier function that satisfies all the initial and boundary conditions was used. A linear system of algebraic equations was obtained for the considered equation with the help of two-dimensional Genocchi polynomials along with the Ritz-Galerkin method. The FDWE and FKGE, including the nonlinear case, were reduced to solve the linear system of the algebraic equation. Hence, the proposed method was able to greatly reduce the complexity of the problems and provide an accurate solution. The effectiveness of the proposed technique is demonstrated through several examples.

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“…Ref. [3] had proposed the spectral tau method based on the Jacobi operational matrix to solve the FDWE. Ref.…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Fractional Diffusion Wave Equation and Fractional Klein–Gordon Equation via Two-Dimensional Genocchi Polynomials with a Ritz–Galerkin Method

2018

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“…In [10,11], Sinc-finite difference method and Sinc-Chebyshev method were employed for solving the fractional diffusion-wave equations respectively. Recently methods based on operational matrix of Jacobi and Chebyshev polynomials were proposed to deal with the fractional diffusion-wave equations ( [12][13][14]). In [15], the authors applied fractional order Legendre functions method depending on the choices of two parameters to solve the fractional diffusion-wave equations.…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Time-Fractional Diffusion-Wave Equations via Chebyshev Wavelets Collocation Method

Zhou

2017

Advances in Mathematical Physics

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The second-kind Chebyshev wavelets collocation method is applied for solving a class of time-fractional diffusion-wave equation. Fractional integral formula of a single Chebyshev wavelet in the Riemann-Liouville sense is derived by means of shifted Chebyshev polynomials of the second kind. Moreover, convergence and accuracy estimation of the second-kind Chebyshev wavelets expansion of two dimensions are given. During the process of establishing the expression of the solution, all the initial and boundary conditions are taken into account automatically, which is very convenient for solving the problem under consideration. Based on the collocation technique, the second-kind Chebyshev wavelets are used to reduce the problem to the solution of a system of linear algebraic equations. Several examples are provided to confirm the reliability and effectiveness of the proposed method.

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“…In the case q ¼ 2, this equation is named telegraph equation. Recently, considerable amount of papers have been proposed methods for solving the FDWE [2][3][4][5][6][7][8][9][10][11][12][13]. Chen et al [1] obtained the analytical solution by the method of separation of variables and proposed the numerical solution with finite difference method.…”

Section: Introductionmentioning

confidence: 99%

“…Chen et al [1] obtained the analytical solution by the method of separation of variables and proposed the numerical solution with finite difference method. In [2], Bhrawy et al applied a spectral tau method based on the Jacobi operational matrix to solve the problem. Liu et al [3] proposed the fractional predictor-corrector method to solve this problem.…”

Section: Introductionmentioning

confidence: 99%

Two-dimensional Bernoulli wavelets with satisfier function in the Ritz–Galerkin method for the time fractional diffusion-wave equation with damping

Barikbin

2017

Math Sci

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In this paper, the two-dimensional Bernoulli wavelets (BWs) with Ritz-Galerkin method are applied for the numerical solution of the time fractional diffusionwave equation. In this way, a satisfier function which satisfies all the initial and boundary conditions is derived. The two-dimensional BWs and Ritz-Galerkin method with satisfier function are used to transform the problem under consideration into a linear system of algebraic equations. The proposed scheme is applied for numerical solution of some examples. It has high accuracy in computation that leads to obtaining the exact solutions in some cases.

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A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations

Cited by 185 publications

References 29 publications

Numerical Solution of Fractional Diffusion Wave Equation and Fractional Klein–Gordon Equation via Two-Dimensional Genocchi Polynomials with a Ritz–Galerkin Method

Numerical Solution of Fractional Diffusion Wave Equation and Fractional Klein–Gordon Equation via Two-Dimensional Genocchi Polynomials with a Ritz–Galerkin Method

Numerical Solution of Time-Fractional Diffusion-Wave Equations via Chebyshev Wavelets Collocation Method

Two-dimensional Bernoulli wavelets with satisfier function in the Ritz–Galerkin method for the time fractional diffusion-wave equation with damping

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