Automatic particular solutions of arbitrary high-order splines associated with polyharmonic and poly-Helmholtz equations

The explicit analytical particular solutions of the splines and monomials for the polyharmonic and poly-Helmholtz operators and their products were derived in the author's previous study. These solutions enable an implementation for automatically approximating particular solutions of arbitrary high-order splines. The automatic particular solutions obtained by the proposed implementation are extremely accurate. In the case of a polyharmonic equation, these solutions are more accurate than the numerical solutions obtained using the multiquadrics within the limits of the IEEE double precision. In the case of poly-Helmholtz and product equations, this implementation can also result in very accurate solutions despite the fact that an analytical particular solution for the multiquadrics does not exist. After particular solutions are obtained, boundary-type numerical methods, such as the boundary element method, the method of fundamental solutions and the Trefftz method, can be applied to solve the homogeneous differential equations.