Approximate solutions of the $ 2 $D space-time fractional diffusion equation via a gradient-descent iterative algorithm with Grünwald-Letnikov approximation

Adisorn Kittisopaporn; Pattrawut Chansangiam; Adisorn Kittisopaporn; Pattrawut Chansangiam

doi:10.3934/math.2022472

AIMS Mathematics

2022, Volume 7, Issue 5: 8471-8490. doi: 10.3934/math.2022472

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Approximate solutions of the $ 2 $D space-time fractional diffusion equation via a gradient-descent iterative algorithm with Grünwald-Letnikov approximation

Department of Mathematics, School of Science, King Mongkut's Institute of Technology Ladkrabang, Bangkok 10520, Thailand

Received: 31 December 2021 Revised: 12 February 2022 Accepted: 21 February 2022 Published: 28 February 2022
MSC : 65F10, 65N22, 34A08, 15A60

We consider the two-dimensional space-time fractional differential equation with the Caputo's time derivative and the Riemann-Liouville space derivatives on bounded domains. The equation is subjected to the zero Dirichlet boundary condition and the zero initial condition. We discretize the equation by finite difference schemes based on Grünwald-Letnikov approximation. Then we linearize the discretized equations into a sparse linear system. To solve such linear system, we propose a gradient-descent iterative algorithm with a sequence of optimal convergence factor aiming to minimize the error occurring at each iteration. The convergence analysis guarantees the capability of the algorithm as long as the coefficient matrix is invertible. In addition, the convergence rate and error estimates are provided. Numerical experiments demonstrate the efficiency, the accuracy and the performance of the proposed algorithm.
- fractional derivatives,
- fractional diffusion equation,
- Grünwald-Letnikov approximation,
- gradient descent,
- iterative method
Citation: Adisorn Kittisopaporn, Pattrawut Chansangiam. Approximate solutions of the $ 2 $D space-time fractional diffusion equation via a gradient-descent iterative algorithm with Grünwald-Letnikov approximation[J]. AIMS Mathematics, 2022, 7(5): 8471-8490. doi: 10.3934/math.2022472

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Abstract

We consider the two-dimensional space-time fractional differential equation with the Caputo's time derivative and the Riemann-Liouville space derivatives on bounded domains. The equation is subjected to the zero Dirichlet boundary condition and the zero initial condition. We discretize the equation by finite difference schemes based on Grünwald-Letnikov approximation. Then we linearize the discretized equations into a sparse linear system. To solve such linear system, we propose a gradient-descent iterative algorithm with a sequence of optimal convergence factor aiming to minimize the error occurring at each iteration. The convergence analysis guarantees the capability of the algorithm as long as the coefficient matrix is invertible. In addition, the convergence rate and error estimates are provided. Numerical experiments demonstrate the efficiency, the accuracy and the performance of the proposed algorithm.

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