Application of a hybrid pseudospectral method to a new two-dimensional multi-term mixed sub-diffusion and wave-diffusion equation of fractional order

Farman Ali Shah; Kamran; Dania Santina; Nabil Mlaiki; Salma Aljawi; Farman Ali Shah; Kamran; Dania Santina; Nabil Mlaiki; Salma Aljawi

doi:10.3934/nhm.2024003

Networks and Heterogeneous Media

2024, Volume 19, Issue 1: 44-85. doi: 10.3934/nhm.2024003

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Application of a hybrid pseudospectral method to a new two-dimensional multi-term mixed sub-diffusion and wave-diffusion equation of fractional order

1.
Department of Mathematics, Islamia College Peshawar, Jamrod Road, Peshawar 25120, Khyber PakhtunKhwa, Pakistan
2.
Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia
3.
Department of Mathematical Sciences, Princess Nourah Bint Abdulrahman University, Riyadh, PO Box 84428, Saudi Arabia

Received: 12 November 2023 Revised: 27 December 2023 Accepted: 08 January 2024 Published: 12 January 2024

In the current study, a novel multi-term mixed sub-diffusion and wave-diffusion model was considered. The new model has a unique time-space coupled derivative in addition to having the diffusion-wave and sub-diffusion terms concurrently. Typically, an elliptic equation in the space variable is obtained by applying a finite difference time-stepping procedure. The severe stability restrictions are the main disadvantage of the finite difference method in time. It has been demonstrated that the Laplace transform is an excellent choice for solving diffusion problems and offers a substitute to the finite difference approach. In this paper, a method based on Laplace transform coupled with the pseudospectral method was developed for the novel model. The proposed method has three main steps: First, the model was reduced to a time-independent model via Laplace transform; second, the pseudospectral method was employed for spatial discretization; and finally, the inverse Laplace transform was applied to transform the obtained solution in Laplace transform domain back into a real domain. We also presented the numerical scheme's stability and convergence analysis. To demonstrate our method's efficacy, four problems were examined.
- mixed sub-diffusion and diffusion-wave equation,
- Caputo's derivative,
- Laplace transform,
- pseudospectral method,
- Talbot's method
Citation: Farman Ali Shah, Kamran, Dania Santina, Nabil Mlaiki, Salma Aljawi. Application of a hybrid pseudospectral method to a new two-dimensional multi-term mixed sub-diffusion and wave-diffusion equation of fractional order[J]. Networks and Heterogeneous Media, 2024, 19(1): 44-85. doi: 10.3934/nhm.2024003

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Abstract

In the current study, a novel multi-term mixed sub-diffusion and wave-diffusion model was considered. The new model has a unique time-space coupled derivative in addition to having the diffusion-wave and sub-diffusion terms concurrently. Typically, an elliptic equation in the space variable is obtained by applying a finite difference time-stepping procedure. The severe stability restrictions are the main disadvantage of the finite difference method in time. It has been demonstrated that the Laplace transform is an excellent choice for solving diffusion problems and offers a substitute to the finite difference approach. In this paper, a method based on Laplace transform coupled with the pseudospectral method was developed for the novel model. The proposed method has three main steps: First, the model was reduced to a time-independent model via Laplace transform; second, the pseudospectral method was employed for spatial discretization; and finally, the inverse Laplace transform was applied to transform the obtained solution in Laplace transform domain back into a real domain. We also presented the numerical scheme's stability and convergence analysis. To demonstrate our method's efficacy, four problems were examined.

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