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Highly Accurate Methods For Solving Elliptic And Parabolic Partial Differential Equations


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E. J. Kansa


This paper is a study of two approaches to obtain very high accuracy in timedependent parabolic partial differential equations (PDEs) with the use of the C∞ multiquadric (MQ) radial basis functions (RBFs). For the spatial part of the solution, the MQ-RBF is generalized having the form, φj(x) = {(x-xj)2 +c2j }β and β > −1/2 can be either a half integer, or any number, excluding a whole integer. The other shape parameter, c2j , is allowed to be different on the boundary and the interior, and is permitted to vary with odd and even values of the index, j. The temporal and spatial variations of the solution, U(x,t) are treated by the separation of variables in which the temporal portion is accounted by the expansion coefficients and the spatial portion is accounted by theMQ-RBFs. It was observed that the PDE on the interior is really a system of time dependent ordinary differential equations (ODEs) with either stationary or non-stationary constraints on the boundary. The solution of the time advanced expansion coefficients both on the interior and on the boundary can be accomplished by analytical methods, rather than by low order time advanced schemes. 1 Introduction The interest in mesh-free methods to solve PDEs has grown considerably in the past 15 years. The two principal reasons are: (1) Mesh generation over complicated two and three dimension domains may require weeks or months to produce a well behaved mesh, and (2) The convergence rate of traditional methods are typically second order in space and time. The mesh-free radial basis functions (RBFs) have been shown to be particularly attractive by Fedoseyev et al. [1], and Cheng et al. [2] because of the exponential convergence of certain C∞ RBFs that has been