A meshless radial basis function method for steady-state advection-diffusion-reaction equation in arbitrary 2D domains

In this paper, we present a new meshless method for the simulation of 2D linear and non-linear steady-state advection-diffusion-reaction equations (ADRE). The proposed method is simple and straight forward. The solution to the problem is separated into the approximation of the boundary conditions and the approximation of the ADRE inside the solution domain. The approximation of the boundary conditions is approximated by the chosen basis functions, and the approximation of the ADRE inside the solution domain is approximated by the basis functions which satisfy the homogeneous boundary conditions of the problem. Each basis function used in the algorithm is a sum of a radial basis function (RBF) and a special correcting function which is chosen to satisfy the corresponding homogeneous boundary conditions of the problem. The final approximated solution is given in the form which satisfies the boundary conditions of the initial problem with any choice of free parameters. Then these free parameters are obtained using the collocation techniques. The numerical examples demonstrate the high accuracy and efficiency of the proposed method in solving 2D ADRE in arbitrary domains.