Meshless Galerkin method based on RBFs and reproducing Kernel for quasi-linear parabolic equations with dirichlet boundary conditions

• Published in: Mathematical modelling and analysis Vol. 26; no. 2; pp. 318 - 336
• Main Authors: Mesrizadeh, Mehdi, Shanazari, Kamal
• Format: Journal Article
• Language: English
• Published: Vilnius Vilnius Gediminas Technical University 26.05.2021
• Subjects: Analysis; Boundary conditions; Differential equations; Dirichlet problem; Error analysis; Exact solutions; Finite element method; Galerkin meshless method; Galerkin method; Hilbert space; Kernels; Mathematical analysis; Meshless methods; Methods; Parameterization; Partial differential equations; quasi-linear parabolic equations; Radial basis function; radial basis functions; reproducing kernel Hilbert space method; Solution space; radial basis functions; Galerkin meshless method; quasi-linear parabolic equations; reproducing kernel Hilbert space method
• ISSN: 1392-6292; 1648-3510
• DOI: 10.3846/mma.2021.11436

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.