A method for solving partial differential equations via radial basis functions: Application to the heat equation

In this work a technique is proposed for solving partial differential equations using radial basis functions. The approach is different from the traditional schemes. The radial basis functions are very suitable instruments for solving partial differential equations of various types. However, the matrices which result from the discretization of the equations are usually ill-conditioned especially in higher-dimensional problems. In the current paper, a stable method will be proposed for solving the partial differential equations and will be generalized to solve higher-dimensional problems. To the contrast of most existing methods, the new technique provides a closed form approximation for the solution. Another advantage of the developed method is that it can be applied to problems with nonregular geometrical domains.