This is a novel, meshless, high accuracy global function method for the solutions of PDE,
proposed by Ed Kansa in 1990. It was used before for a scattered data interpolation.

We just started to work with this method with a goal to use it for 2D/3D viscous fluid
flow, thermal and thermo-vibrational convection and bifurcation-stability analysis.
More details about this method can be found at our RBF-PDE Web page.

Our first publication of the MQ method for the Continuation for Nonlinear PDE is to
appear in Int.J.Bifur.Chaos (Preprint (preliminary article version) is available at E-PRINT, LANL 1998 (gzipped postscript at http://xxx.lanl.gov/ps/math.NA/9812013)), here is the title and abstract:

A.I. Fedoseyev*, M.J. Friedman** and E.J. Kansa***
*Center for Microgravity and Materials Research and **Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, AL 35899
***Lawrence Livermore National Laboratory, Livermore, CA 94551.
"Continuation for Nonlinear Elliptic Partial Differential Equations Discretized by the Multiquadric Method"
Abstract. The Multiquadric Radial Basis Function (MQ) Method is a meshless collocation method with global basis functions. It is known to have exponentional convergence for interpolation problems. We descretize nonlinear elliptic PDEs by the MQ method. This results in modest size systems of nonlinear algebraic equations which can be efficiently continued by standard continuation software such as AUTO and CONTENT. Examples are given of detection of bifurcations in 1D and 2D PDEs. These examples show high accuracy with small number of unknowns, as compared with known results from the literature.
Keywords: Continuation, bifurcations, nonlinear elliptic PDEs, multiquadric radial basis function method.

EXAMPLE 1. Here is presented the interpolation of the function atan(10*(x-1/2)) by the MQ method. This is a difficult case, as the function has regions of slow and fast change. Nevertheless we are able to reach a good accuracy even with the equidistant knots (nodes) with the maximal error in function interpolation is 3E-5, and 2.27E-3 for the derivative). The derivative is obtained directly by the differentiation of the MQ interpolant for the function. You can not see the red line (the original function and derivative), the results are impossible to distinguish on a plot.

EXAMPLE 2. Smooth functions are interpolated by MQ with a very high accuracy, e.g.
EXP(x) has interpolation error 1E-10 with 5 MQ functions.

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