An Efficient Localized Meshless Method Based on the Space–Time Gaussian RBF for High-Dimensional Space Fractional Wave and Damped Equations

Abstract

In this paper, an efficient localized meshless method based on the space–time Gaussian radial basis functions is discussed. We aim to deal with the left Riemann–Liouville space fractional derivative wave and damped wave equation in high-dimensional space. These significant problems as anomalous models could arise in several research fields of science, engineering, and technology. Since an explicit solution to such equations often does not exist, the numerical approach to solve this problem is fascinating. We propose a novel scheme using the space–time radial basis function with advantages in time discretization. Moreover this approach produces the (n + 1)-dimensional spatial-temporal computational domain for n-dimensional problems. Therefore the local feature, as a remarkable and efficient property, leads to a sparse coefficient matrix, which could reduce the computational costs in high-dimensional problems. Some benchmark problems for wave models, both wave and damped, have been considered, highlighting the proposed method performances in terms of accuracy, efficiency, and speed-up. The obtained experimental results show the computational capabilities and advantages of the presented algorithm.

Modified space‐time radial basis function collocation method for long‐time simulation of transient heat conduction in3Danisotropic composite materials

“…Using the space-time distance defined in space-time domain, a simple space-time RBF is given as follows, [40][41][42] Φ(x, t; 𝛏, 𝜏)…”

“…are nodes subjected to Dirichlet and Neumann boundary conditions (red points in Figure 1A). In the standard space-time RBF collocation method, 41,42 N st source points { 𝛏 (k) , 𝜏 (k) } N st k=1 with the same space-time coordinates as the collocation points are arranged inside the domain and on the boundary, and then the solution T(x, t) can be approximated as,…”

“…Using the space‐time distance defined in space‐time domain, a simple space‐time RBF is given as follows, 40–42 $$$$ \Phi \left(\mathbf{x},t;\boldsymbol{\upxi}, \tau \right)=\sqrt{\sum \limits_{i=1}^d{\left({x}_i-{\xi}_i\right)}^2+{\left(t-\tau \right)}^2+{c}^2}, $$$$ where d is the spatial dimension, c represents the shape parameter, $$$ \left(\mathbf{x},t\right) $$$ and $$$ \left(\boldsymbol{\upxi}, \tau \right) $$$ denote space‐time points.…”