An iterative spectral strategy for fractional-order weakly singular integro-partial differential equations with time and space delays

M. Usman; T. Zubair; J. Imtiaz; C. Wan; W. Wu; M. Usman; T. Zubair; J. Imtiaz; C. Wan; W. Wu

doi:10.3934/era.2022090

Electronic Research Archive

2022, Volume 30, Issue 5: 1775-1798. doi: 10.3934/era.2022090

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An iterative spectral strategy for fractional-order weakly singular integro-partial differential equations with time and space delays

M. Usman ¹,
T. Zubair ²,
J. Imtiaz ³,
C. Wan ⁴,
W. Wu ^{5
,
,}

1.
Department of Mathematics, National University of Modern Languages, Islamabad 44790, Pakistan
2.
School of Mathematical Sciences, Peking University, Beijing 100871, China
3.
Department of Electrical Engineering, Bahria University, Islamabad, Pakistan
4.
Department of Mathematics, Southern University of Science and Technology, China
5.
Department of Mathematics, Hong Kong Baptist University, Hong Kong SAR, China

Academic Editor: Jingzhi Li

Received: 22 December 2021 Revised: 14 March 2022 Accepted: 15 March 2022 Published: 30 March 2022

This study aims at extending and implementing an iterative spectral scheme for fractional-order unsteady nonlinear integro-partial differential equations with weakly singular kernel. In this scheme, the unknown function u(x, t) is estimated by using shifted Gegenbauer polynomials vector Λ(x, t), and Picard iterative scheme is used to handle underlying nonlinearities. Some novel operational matrices are developed for the first time in order to approximate the singular integral like, $ \int_0^x {\int_0^y {u(p{a_1} + {b_1}, q{a_2} + {b_2}, t)/{{({x^{{\rho _1}}} - {p^{{\rho _1}}})}^{{\alpha _1}}}{{({y^{{\rho _2}}} - {q^{{\rho _2}}})}^{{\alpha _2}}}{\text{d}}q{\text{d}}p} } $ and $ \int_0^t {{u^\gamma }({\bf{x}}, \xi)/{{({t^{{\rho _3}}} - {\xi ^{{\rho _3}}})}^{{\alpha _3}}}{\text{d}}\xi } $, where ρ's > 1, 0 < α's < 1 by means of shifted Gegenbauer polynomials vector. The advantage of this extended method is its ability to convert nonlinear problems into systems of linear algebraic equations. A computer program in Maple for the proposed scheme is developed for a sample problem, and we validate it to compare the results with existing results. Six new problems are also solved to illustrate the effectiveness of this extended computational method. A number of simulations are performed for different ranges of the nonlinearity n, α, fractional-order, ρ, and convergence control M, parameters. Our results demonstrate that the extended scheme is stable, accurate, and appropriate to find solutions of complex problems with inherent nonlinearities.
- spectral methods,
- shifted gegenbauer polynomials,
- fractional calculus,
- weakly singular integral equations
Citation: M. Usman, T. Zubair, J. Imtiaz, C. Wan, W. Wu. An iterative spectral strategy for fractional-order weakly singular integro-partial differential equations with time and space delays[J]. Electronic Research Archive, 2022, 30(5): 1775-1798. doi: 10.3934/era.2022090

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Abstract

This study aims at extending and implementing an iterative spectral scheme for fractional-order unsteady nonlinear integro-partial differential equations with weakly singular kernel. In this scheme, the unknown function u(x, t) is estimated by using shifted Gegenbauer polynomials vector Λ(x, t), and Picard iterative scheme is used to handle underlying nonlinearities. Some novel operational matrices are developed for the first time in order to approximate the singular integral like, $ \int_0^x {\int_0^y {u(p{a_1} + {b_1}, q{a_2} + {b_2}, t)/{{({x^{{\rho _1}}} - {p^{{\rho _1}}})}^{{\alpha _1}}}{{({y^{{\rho _2}}} - {q^{{\rho _2}}})}^{{\alpha _2}}}{\text{d}}q{\text{d}}p} } $ and $ \int_0^t {{u^\gamma }({\bf{x}}, \xi)/{{({t^{{\rho _3}}} - {\xi ^{{\rho _3}}})}^{{\alpha _3}}}{\text{d}}\xi } $, where ρ's > 1, 0 < α's < 1 by means of shifted Gegenbauer polynomials vector. The advantage of this extended method is its ability to convert nonlinear problems into systems of linear algebraic equations. A computer program in Maple for the proposed scheme is developed for a sample problem, and we validate it to compare the results with existing results. Six new problems are also solved to illustrate the effectiveness of this extended computational method. A number of simulations are performed for different ranges of the nonlinearity n, α, fractional-order, ρ, and convergence control M, parameters. Our results demonstrate that the extended scheme is stable, accurate, and appropriate to find solutions of complex problems with inherent nonlinearities.

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