Homotopy method of fundamental solutions for solving nonlinear heat conduction problems

Abstract

In this study, we propose a meshless and boundary-type numerical method, namely the homotopy method of fundamental solutions (HMFS), to solve the steady-state nonlinear heat conduction problems in two dimensions. The HMFS is composed by the homotopy analysis method (HAM) and the method of fundamental solutions (MFS). In the solution procedure, the Kirchhoff transformation is employed to transform the nonlinear governing partial differential equation into the Laplace equation with nonlinear boundary conditions. Sequentially, the HAM is applied to convert the Laplace equation with nonlinear boundary conditions into a sequence of the Laplace equation with linear boundary conditions, which can be solved by the MFS. In order to solve strongly nonlinear problems, a convergence control parameter is introduced to ensure the solution convergence of the prescribed sequence of problems. Several numerical experiments were carried out to validate the proposed method. In addition, a multiple-precision computing is performed to demonstrate the exponential convergence of the HMFS in both the spatial and homotopy coordinates for solving nonlinear heat conduction problems. Finally, bi-material and irregular-domain problems are also considered.