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Jacobi polynomial truncations and approximate solutions to classes of nonlinear differential equations Dodd, Ronald Edward

Abstract

Solutions to classes of second-order, nonlinear differential equations of the form [formula omitted] + f(x) + 0, x(0) = 1, x(∙)(0) = 0 are approximated in this work. The techniques which are developed involve the replacement of the characteristic, f(x), in the nonlinear model by piecewise-linear or piecewise-cubic approximations. From these, closed-form time solutions in terms of the circular trigonometric functions or the Jacobian elliptic functions may be obtained. Particular examples in which f(x) is grossly nonlinear and asymmetric are considered. The orthogonal Jacobi and shifted Jacobi polynomials are introduced for the approximation in order to satisfy criteria which are imposed on the error and on the use of symmetry. Error bounds are then developed which demonstrate that the maximum error in the normalized time solution is bounded, no matter how large the coefficients of the non-linear terms in the model become. Because of these error-bound results, an heuristic measure of the departure from linearity is defined for classes of symmetric oscillations, and the weighting of convergence of the Jacobi and shifted Jacobi polynomial expansions is set according to this measure. For asymmetric conservative models, shifted Chebychev polynomials are used to obtain near-uniform approximations to the characteristic in the nonlinear differential equation. Based on the equivalence of the classical approximation techniques which is given for the symmetric, conservative models, extension of the polynomial approximation to classes of non-conservative models is considered. Throughout the work, by comparison with classical approximation methods, the polynomial approximation techniques are shown to provide an improved, direct and more general attack on the approximation problem with a decrease in tedious labor.

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