Meshless Generalized Finite Difference Method For Cauchy Inverse Problem Of Variable Coefficient Partial Differential Equations

As a new meshless numerical method,the generalized finite difference method(GFDM)has become popular in recent years,and compared with the traditional mesh method,this method has obvious advantages in dealing with the initial-boundary value problems,which mainly shows that it can avoid the time-consuming and laborintensive mesh generation and complicated numerical quadrature.In modern engineering application,there is an increasing demand for research on thermo-elasticity,heat conduction of functionally graded materials(FGMs)and inverse problems.The research on the abovementioned problems is not only the desire of the development of modern high-tech fields,but also the require of the advancement of society.This paper makes the first effort to use the GFDM to solve partial differential equations with variable coefficients,and documents the first attempt to apply this method for the inverse problem of thermal elasticity and steady-state heat conduction of FGMs.The numerical method is based on the coupling of the Taylor series expansion and the moving least square method,then the unknown partial derivative term at each point can be approximated by a linear combination of the function values at neighboring nodes,so that the partial differential equation can be discretized into a algebraic matrix system.In view of the aforementioned contents,preliminary research results have been obtained.Using GFDM to solve above problems is actually calculating a set of partial differential equations with constant or variable coefficients,and this paper analyzes several calculated examples.It can be observed that putting different levels of noise into the exact data,adjusting the number of supporting nodes or changing the ratio parameter of the available boundary of inverse problem will not make the numerical results change dramatically,and the final numerical solutions are in good agreement with the corresponding exact results,and the relative errors can keep within the acceptable scope of the inverse problem.Thus,it can be concluded that the GFDM can solve the Cauchy inverse problem of partial differential equations with variable coefficients in a stable,accurate and efficient manner.