The Jacobi spectral collocation method for fractional integro-differential equations with non-smooth solutions

Yin Yang; Sujuan Kang; Vasiliy I. Vasil'ev; Yin Yang; Sujuan Kang; Vasiliy I. Vasil'ev

doi:10.3934/era.2020064

Electronic Research Archive

2020, Volume 28, Issue 3: 1161-1189. doi: 10.3934/era.2020064

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The Jacobi spectral collocation method for fractional integro-differential equations with non-smooth solutions

1.
Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Key Laboratory of Intelligent Computing & Information Processing of Ministry of Education, School of Mathematics and Computational Science, Xiangtan University, Xiangtan, MO 411105, China
2.
Xiangtan University, Xiangtan, MO 411105, China
3.
Ammosov North-Eastern Federal University, Yakutsk, MO 677000, Russia

Received: 01 March 2020 Revised: 01 May 2020
34A12, 74S25, 65L60, 65L70, 33C45

In recent years, many numerical methods have been extended to fractional integro-differential equations. But most of them ignore an important problem. Even if the input function is smooth, the solutions of these equations would exhibit some weak singularity, which leads to non-smooth solutions, and a deteriorate order of convergence. To overcome this problem, we first study in detail the singularity of the fractional integro-differential equation, and then eliminate the singularity by introducing some smoothing transformation. We can maximize the convergence rate by adjusting the parameters in the auxiliary transformation. We use the Jacobi spectral-collocation method with global and high precision characteristics to solve the transformed equation. A comprehensive and rigorous error estimation under the $ L^{\infty} $- and $ L^{2}_{\omega^{\alpha, \beta}} $-norms is derived. Finally, we give specific numerical examples to show the accuracy of the theoretical estimation and the feasibility and effectiveness of the proposed method.
- The fractional integro-differential equation,
- non-smooth solution,
- smoothing transformation,
- spectral collocation method,
- convergence analysis
Citation: Yin Yang, Sujuan Kang, Vasiliy I. Vasil'ev. The Jacobi spectral collocation method for fractional integro-differential equations with non-smooth solutions[J]. Electronic Research Archive, 2020, 28(3): 1161-1189. doi: 10.3934/era.2020064

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Abstract

In recent years, many numerical methods have been extended to fractional integro-differential equations. But most of them ignore an important problem. Even if the input function is smooth, the solutions of these equations would exhibit some weak singularity, which leads to non-smooth solutions, and a deteriorate order of convergence. To overcome this problem, we first study in detail the singularity of the fractional integro-differential equation, and then eliminate the singularity by introducing some smoothing transformation. We can maximize the convergence rate by adjusting the parameters in the auxiliary transformation. We use the Jacobi spectral-collocation method with global and high precision characteristics to solve the transformed equation. A comprehensive and rigorous error estimation under the $ L^{\infty} $- and $ L^{2}_{\omega^{\alpha, \beta}} $-norms is derived. Finally, we give specific numerical examples to show the accuracy of the theoretical estimation and the feasibility and effectiveness of the proposed method.

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