*(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 FrĂ©chet 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:

65L10 | Boundary value problems for ODE (numerical methods) |

34B05 | Linear boundary value problems for ODE |

65L12 | Finite difference methods for ODE (numerical methods) |

65L60 | Finite elements, Rayleigh-Ritz, Galerkin and collocation methods for ODE |

34B15 | Nonlinear boundary value problems for ODE |

65L20 | Stability and convergence of numerical methods for ODE |