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Differential quadrature solutions of eighth-order boundary-value differential equations. (English) Zbl 1001.65085

Summary: Special cases of linear eighth-order boundary-value problems have been solved using polynomial splines. However, divergent results were obtained at points adjacent to boundary points. This paper presents an accurate and general approach to solve this class of problems, utilizing the generalized differential quadrature rule (GDQR) proposed recently by the authors. Explicit weighting coefficients are formulated to implement the GDQR for eighth-order differential equations. A mathematically important by-product of this paper is that a new kind of Hermite interpolation functions is derived explicitly for the first time.

Linear and non-linear illustrations are given to show the practical usefulness of the approach developed. Using Fréchet derivatives, non-linear eighth-order problems are also solved for the first time. Numerical results obtained using even only seven sampling points are of excellent accuracy and convergence in an entire domain. The present GDQR has shown clear advantages over the existing methods and demonstrated itself as a general, stable, and accurate numerical method to solve high-order differential equations.

MSC:
65L10Boundary value problems for ODE (numerical methods)
34B05Linear boundary value problems for ODE
65L12Finite difference methods for ODE (numerical methods)
65L60Finite elements, Rayleigh-Ritz, Galerkin and collocation methods for ODE
34B15Nonlinear boundary value problems for ODE
65L20Stability and convergence of numerical methods for ODE