An efficient mesh-free method for nonlinear reaction-diffusion equations

The purpose of this paper is to develop a highly efficient mesh-free method for solving nonlinear diffusion-reaction equations in R-d, d = 2,3. Using various time difference schemes, a given time-dependent problem can be reduced to solving a series of inhomogeneous Helmholtz-type equations. The solution of these problems can then be further reduced to evaluating particular solutions and the solution of related homogeneous equations. Recently, radial basis functions have been successfully implemented to evaluate particular solutions for Possion-type equations. A more general approach has been developed in extending this capability to obtain particular solutions for Helmholtz-type equations by using polyharmonic spline interpolants. The solution of the homogeneous equation may then be solved by a variety of boundary methods, such as the method of fundamental solutions. Preliminary work has shown that an increase in efficiency can be achieved compared to more traditional finite element, finite difference and boundary element methods without the need of either domain or surface meshing.