Meshless collocation method by Delta-shaped basis functions for default barrier model

In this paper we first approximate a ‘nearly singular’ function, which tends to be the Dirac-delta function, to high degree of accuracy by using a recently developed Delta-shaped basis function. The Hermite-based meshless collocation method based on radial basis functions is then applied to solve a default barrier model, which is a time-dependent boundary value problem with a singularity at the initial condition. For numerical verification on the accuracy and efficiency of the newly proposed method, we compare the results with an analytical solution of the default barrier model under an assumption on the affine boundary. Numerical results indicate that the proposed method has potential advantage to solve problems with Dirac-type singularities.