In this paper, we propose a numerical scheme to solve generalized space fractional partial differential equations (GFPDEs). The proposed scheme uses Shifted Chebyshev fifth-kind polynomials with the spectral collocation approach. Besides, the proposed GFPDEs represent a great generalization of significant types of fractional partial differential equations (FPDEs) and their applications, which contain many previous reports as a special case. The fractional differential derivatives are expressed in terms of the Caputo sense. Moreover, the Chebyshev collocation method together with the finite difference method is used to reduce these types of differential equations to a system of differential equations which can be solved numerically. In addition, the classical fourth-order Runge-Kutta method is also used to treat the differential system with the collocation method which obtains a great accuracy. Numerical approximations performed by the proposed method are presented and compared with the results obtained by other numerical methods. The introduced numerical experiments are fractional-order mathematical physics models, as advection-dispersion equation (FADE) and diffusion equation (FDE). The results reveal that our method is a simple, easy to implement and effective numerical method.
Citation: Khalid K. Ali, Mohamed A. Abd El Salam, Mohamed S. Mohamed. Chebyshev fifth-kind series approximation for generalized space fractional partial differential equations[J]. AIMS Mathematics, 2022, 7(5): 7759-7780. doi: 10.3934/math.2022436
In this paper, we propose a numerical scheme to solve generalized space fractional partial differential equations (GFPDEs). The proposed scheme uses Shifted Chebyshev fifth-kind polynomials with the spectral collocation approach. Besides, the proposed GFPDEs represent a great generalization of significant types of fractional partial differential equations (FPDEs) and their applications, which contain many previous reports as a special case. The fractional differential derivatives are expressed in terms of the Caputo sense. Moreover, the Chebyshev collocation method together with the finite difference method is used to reduce these types of differential equations to a system of differential equations which can be solved numerically. In addition, the classical fourth-order Runge-Kutta method is also used to treat the differential system with the collocation method which obtains a great accuracy. Numerical approximations performed by the proposed method are presented and compared with the results obtained by other numerical methods. The introduced numerical experiments are fractional-order mathematical physics models, as advection-dispersion equation (FADE) and diffusion equation (FDE). The results reveal that our method is a simple, easy to implement and effective numerical method.
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Khalid K. Ali, Mohamed A. Abd El Salam, Mohamed S. Mohamed. Chebyshev fifth-kind series approximation for generalized space fractional partial differential equations[J]. AIMS Mathematics, 2022, 7(5): 7759-7780. doi: 10.3934/math.2022436
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