A hybrid radial basis functions collocation technique to numerically solve fractional advection–diffusion models

Abstract

In this work, we propose a hybrid radial basis functions (RBFs) collocation technique for the numerical solution of fractional advection-diffusion models. In the formulation of hybrid RBFs (HRBFs), there exist shape parameter (c*) and weight parameter () that control numerical accuracy and stability. For these parameters, an adaptive algorithm is developed and validated. The proposed HRBFs method is tested for numerical solutions of some fractional Black-Sholes and diffusion models. Numerical simulations performed for several benchmark problems verified the proposed method accuracy and efficiency. The quantitative analysis is made in terms of L ∞ , L 2 , L rms , and L rel error norms as well as number of nodes N over space domain and time-step t. Numerical convergence in space and time is also studied for the proposed method. The unconditional stability of the proposed HRBFs scheme is obtained using the von Neumann methodology. It is observed that the HRBFs method circumvented the ill-conditioning problem greatly, a major issue in the Kansa method.

Application of a collocation method based on linear barycentric interpolation for solving 2D and 3D Klein-Gordon-Schrödinger (KGS) equations numerically

“…Meshless methods can be broadly separated into two groups, namely, meshless methods based on strong form such as Kansa's-radial basis function (RBF) and collocation methods (Dehghan and Shokri, 2007;Hu et al, 2005;Jankowska et al, 2018;Kansa, 1990;Hussain and Haq, 2020;Lin et al, 2017;Singh et al, 2019;Xiong et al, 2018), smoothed particle hydrodynamics method (Liu et al, 2004) and finite point collocation method (Onate et al, 1996) and meshless methods based on weak form such as element-free Galerkin method (Belytschko et al, 1994), direct meshless local Petrov-Galerkin method (Ilati and Dehghan, 2017) and local radial point interpolation (MLRPI) method (Shivanian and Jafarabadi, 2018;Shivanian, 2015;Shivanian, 2016). In the methods of first group, by using collocation approach, governing equations and boundary conditions are discretized at the set of scattered nodes to get an algebraic system of equations.…”

A meshless finite point method for the improved Boussinesq equation using stabilized moving least squares approximation and Richardson extrapolation

“…2. Form A (0) and b (0) using Equations ( 32) and (35), respectively. Solve the linear system (31) for k = 0 to obtain u (1) .…”

“…The finite point method (FPM) [32] is a well-known meshless method especially in the field of fluid dynamics. Based on the MLS approximation [23,24,33] and the point collocation technique [34][35][36], the FPM possesses some outstanding merits such as truly meshless, no numerical quadrature, easy to implement, generating sparse and banded discrete systems, and the extensible to high dimensional problems. Theoretical error estimations for the FPM have been established in References [37][38][39].…”