An advanced meshless approach for the high-dimensional multi-term time-space-fractional PDEs on convex domains

2020/03/03 by Xiaogang Zhu, Zhu, Xiaogang, Yufeng Nie +5
Computer Science · Engineering · Mathematics · #35R11 #65D25 #65M9 #Applied mathematics #Computer science #Differential Equations and Numerical Methods #Discretization #FOS: Mathematics #Fractional Differential Equations Solutions #Fractional calculus #Geometry #Mathematical analysis #Mathematical optimization #Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering #Partial differential equation #Quadrature (astronomy) #Radial basis function #Regular polygon #Stability (learning theory) #Term (time) #cs.NA #math.NA #msc:35R11 #msc:65D25 #msc:65M9

paper · pdf · doi:10.48550/arxiv.2003.01336

published in arXiv (Cornell University) (Cornell University) · 22 pages, 26 figures

openalex publication_date 2020/03/03 · arxiv created 2021/01/27 · arxiv updated 2021/01/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

In this article, an advanced differential quadrature (DQ) approach is proposed for the high-dimensional multi-term time-space-fractional partial differential equations (TSFPDEs) on convex domains. Firstly, a family of high-order difference schemes is introduced to discretize the time-fractional derivative and a semi-discrete scheme for the considered problems is presented. We strictly prove its unconditional stability and error estimate. Further, we derive a class of DQ formulas to evaluate the fractional derivatives, which employs radial basis functions (RBFs) as test functions. Using these DQ formulas in spatial discretization, a fully discrete DQ scheme is then proposed. Our approach provides a flexible and high accurate alternative to solve the high-dimensional multi-term TSFPDEs on convex domains and its actual performance is illustrated by contrast to the other methods available in the open literature. The numerical results confirm the theoretical analysis and the capability of our proposed method finally.

Citations

• From continuous time random walks to the fractional Fokker-Planck equation
• Generalizing the finite element method: Diffuse approximation and diffuse elements
• Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—II solutions to parabolic, hyperbolic and elliptic partial differential equations
• Finite Element Method for the Space and Time Fractional Fokker–Planck Equation
• Element‐free Galerkin methods
• A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics

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