MESHLESS GALERKIN METHOD BASED ON RBFS AND REPRODUCING KERNEL FOR QUASI-LINEAR PARABOLIC EQUATIONS WITH DIRICHLET BOUNDARY CONDITIONS

Mehdi Mesrizadeh,Kamal Shanazari

doi:10.3846/mma.2021.11436

Abstract

The main aim of this paper is to present a hybrid scheme of both meshless Galerkin and reproducing kernel Hilbert space methods. The Galerkin meshless method is a powerful tool for solving a large class of multi-dimension problems. Reproducing kernel Hilbert space method is an extremely efficient approach to obtain an analytical solution for ordinary or partial differential equations appeared in vast areas of science and engineering. The error analysis and convergence show that the proposed mixed method is very efficient. Since the solution space spanned by radial basis functions do not directly satisfy essential boundary conditions, an auxiliary parameterized technique is employed. Theoretical studies indicate that this new method is very stable, though a parameterized problem is employed instead of the main problem.

Highlights

During the past three decades, considerable effort has been devoted to developing the meshfree methods based on radial basis functions (RBFs) for solvingCopyright c 2021 The Author(s)
We present a new method combining both Galerkin RBFs (GRBFs) and reproducing kernel Hilbert space method (RKHS) methods
We present a new hybrid method of both meshless based on RBFs and RKHS

Summary

Introduction

During the past three decades, considerable effort has been devoted to developing the meshfree methods based on radial basis functions (RBFs) for solving. Authors in [9,10] applied RBFs to Galerkin method for solving elliptic problems with Dirichlet boundary conditions by making an artificial Neumann boundary condition via some auxiliary parameters. They proved the convergence and obtained an error estimate of their method in the weak form. Dehghan et al [14] solved the time fractional diffusion-wave equation by the method of lines using this technique They first discretized the main problem by employing the Crank Nicolson method and applied the meshless Galerkin method by the use of the auxiliary parameters technique presented in [9, 10].

Hybrid meshless Galerkin method and reproducing kernel Hilbert space

Implementation of new method

Numerical analysis

Numerical investigation

Conclusions