Solution of non‐linear differential equations by discrete least squares

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Abstract
A general‐purpose technique has been developed for solving non‐linear partial differential equations. A set of approximating functions with undetermined parameters is used to evaluate the differential equation and boundary conditions at discrete points, forming a set of residuals to be minimized. The parameters which minimize the sum of squared residuals are determined by a non‐linear least‐squares minimization technique. Initial value problems are solved by integrating the equations with respect to time at the fitting points by a predictor‐corrector algorithm. The resulting formulation is independent of the form of the problem and the approximating functions, so that a broad class of problems may be solved with a single computer program. The technique is applied to several boundary and initial value problems in one and two spatial dimensions. The tecnique is applied to several boundary and initial value problems in one and two spatial dimensions.