Spectral meshless radial point interpolation (SMRPI) method to two‐dimensional fractional telegraph equation

Abstract

H. Ammari In this article, an innovative technique so‐called spectral meshless radial point interpolation (SMRPI) method is proposed and, as a test problem, is applied to a classical type of two‐dimensional time‐fractional telegraph equation defined by Caputo sense for (1 < α≤2). This new methods is based on meshless methods and benefits from spectral collocation ideas, but it does not belong to traditional meshless collocation methods. The point interpolation method with the help of radial basis functions is used to construct shape functions, which play as basis functions in the frame of SMRPI method. These basis functions have Kronecker delta function property. Evaluation of high‐order derivatives is not difficult by constructing operational matrices. In SMRPI method, it does not require any kind of integration locally or globally over small quadrature domains, which is essential of the finite element method (FEM) and those meshless methods based on Galerkin weak form. Also, it is not needed to determine strict value for the shape parameter, which plays an important role in collocation method based on the radial basis functions (Kansa's method). Therefore, computational costs of SMRPI method are less expensive. Two numerical examples are presented to show that SMRPI method has reliable rates of convergence. Copyright © 2015 John Wiley & Sons, Ltd.

Error and stability analysis of numerical solution for the time fractional nonlinear Schrödinger equation on scattered data of general‐shaped domains

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Local radial basis function interpolation method to simulate 2D fractional‐time convection‐diffusion‐reaction equations with error analysis

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Numerical simulation based on meshless technique to study the biological population model

“…for more details about RPIM shape functions and the way they are constructed, the readers are referred to see [42].…”

“…In the current work, we testify spectral meshless radial point interpolation (SMRPI) method [42][43][44] on the problem (1) and then make simulations on two numerical experiments which leads to satisfactory results. A technique based on radial point interpolation is adopted to construct shape functions, also called basis functions, using radial basis functions.…”