On spectral numerical method for variable-order partial differential equations

Kamal Shah; Hafsa Naz; Muhammad Sarwar; Thabet Abdeljawad; Kamal Shah; Hafsa Naz; Muhammad Sarwar; Thabet Abdeljawad

doi:10.3934/math.2022581

AIMS Mathematics

2022, Volume 7, Issue 6: 10422-10438. doi: 10.3934/math.2022581

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Research article

On spectral numerical method for variable-order partial differential equations

1.
Department of Mathematics and Sciences, Prince Sultan University, 11586, Riyadh, Saudi Arabia
2.
Department of Mathematics, University of Malakand, Chakdara Dir(L), 18000, Khyber Pakhtunkhwa, Pakistan
3.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan

Received: 29 January 2022 Revised: 28 February 2022 Accepted: 07 March 2022 Published: 28 March 2022
MSC : 26A33, 35A02, 35A25

In this research article, we develop a powerful algorithm for numerical solutions to variable-order partial differential equations (PDEs). For the said method, we utilize properties of shifted Legendre polynomials to establish some operational matrices of variable-order differentiation and integration. With the help of the aforementioned operational matrices, we reduce the considered problem to a matrix type equation (equations). The resultant matrix equation is then solved by using computational software like Matlab to get the required numerical solution. Here it should be kept in mind that the proposed algorithm omits discretization and collocation which save much of time and memory. Further the numerical scheme based on operational matrices is one of the important procedure of spectral methods. The mentioned scheme is increasingly used for numerical analysis of various problems of differential as well as integral equations in previous many years. Pertinent examples are given to demonstrate the validity and efficiency of the method. Also some error analysis and comparison with traditional Haar wavelet collocations (HWCs) method is also provided to check the accuracy of the proposed scheme.
- variable-order derivative,
- matrix equation,
- multi variable Legendre polynomials,
- FPDEs.
Citation: Kamal Shah, Hafsa Naz, Muhammad Sarwar, Thabet Abdeljawad. On spectral numerical method for variable-order partial differential equations[J]. AIMS Mathematics, 2022, 7(6): 10422-10438. doi: 10.3934/math.2022581

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Abstract

In this research article, we develop a powerful algorithm for numerical solutions to variable-order partial differential equations (PDEs). For the said method, we utilize properties of shifted Legendre polynomials to establish some operational matrices of variable-order differentiation and integration. With the help of the aforementioned operational matrices, we reduce the considered problem to a matrix type equation (equations). The resultant matrix equation is then solved by using computational software like Matlab to get the required numerical solution. Here it should be kept in mind that the proposed algorithm omits discretization and collocation which save much of time and memory. Further the numerical scheme based on operational matrices is one of the important procedure of spectral methods. The mentioned scheme is increasingly used for numerical analysis of various problems of differential as well as integral equations in previous many years. Pertinent examples are given to demonstrate the validity and efficiency of the method. Also some error analysis and comparison with traditional Haar wavelet collocations (HWCs) method is also provided to check the accuracy of the proposed scheme.

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