We obtain explicit analytical particular solutions for Helmholtz-type operators, using higher order splines. These results generalize those in Golberg, Chen and Rashed (1998) and Chen and Rashed (1998) for thin plate splines. This enables one to substantially improve the accuracy of algorithms for solving boundary value problems for Helmholtz-type equations.
In this article, we consider a variant of the Dual Reciprocity Method (DRM) for solving boundary value problems based on approximating source terms by polynomials other than the traditional basis functions. The use of pseudo-spectral approximations and symbolic methods enables us to obtain highly accurate results without solving the often ill-conditioned equations that occur when radial basis function approximations are used. When the given partial differential equation is either Poisson's equation or an inhomogeneous Helmholtz-type equation, we are able to obtain either closed form particular solutions or efficient recursive algorithms. Using the particular solutions, we convert the inhomogeneous equations to homogeneous. The resulting homogeneous equations are then amenable to solution by boundary-type methods such as the Boundary Element Method (BEM) or the Method of Fundamental Solutions (MFS). Using the MFS, we provide numerical solutions to a variety of boundary value problems in R 2 and R 3 . Using this approach, we can achieve high accuracy with a modest number of interpolation and collocation points.
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