Chansangiam, Pattrawut

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Chansangiam, Pattrawut
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Chansangiam, P.
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pattrawut.ch@kmitl.ac.th

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    Numerical solutions of the space-time fractional diffusion equation via a gradient-descent iterative procedure
    (2023-01-01)
    Tansri, Kanjanaporn
    ;
    Kittisopaporn, Adisorn
    ;
    Chansangiam, Pattrawut
    A one-dimensional space-time fractional diffusion equation describes anomalous diffusion on fractals in one dimension. In this paper, this equation is discretized by finite difference schemes based on the Grünwald-Letnikov approximation for Riemann-Liouville and Caputo’s fractional derivatives. It turns out that the discretized equations can be put into a compact form, i.e., a linear system with a block lower-triangular coefficient matrix. To solve the linear system, we formulate a matrix iterative algorithm based on gradient-descent technique. In particular, we work out for the space fractional diffusion equation. Theoretically, the proposed solver is always applicable with satisfactory convergence rate and error estimates. Simulations are presented numerically and graphically to illustrate the accuracy, the efficiency, and the performance of the algorithm, compared to other iterative procedures for linear systems.
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    Approximate solutions of the 2D space-time fractional diffusion equation via a gradient-descent iterative algorithm with Grünwald-Letnikov approximation
    (2022-01-01)
    Kittisopaporn, Adisorn
    ;
    Chansangiam, Pattrawut
    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.