Multipoint boundary value problems by differential quadrature method.

*(English)*Zbl 0999.65074Summary: This paper extends the application of the differential quadrature method (DQM) to high order (\(\geq 3^{\text{rd}}\)) ordinary differential equations with the boundary conditions specified at multiple points (\(\geq\) three different points). Explicit weighting coefficients for higher order derivatives have been derived using interpolating trigonometric polynomials. A three-point, linear third-order differential equation governing the shear deformation of sandwich beams is examined.

Two examples of four-point nonlinear fourth-order systems are also presented. Accurate results are obtained for the example problems. Since boundary conditions are usually specified only at two extreme ends and not at intermediate boundary points, the present work opens new areas of application of the DQM.

Two examples of four-point nonlinear fourth-order systems are also presented. Accurate results are obtained for the example problems. Since boundary conditions are usually specified only at two extreme ends and not at intermediate boundary points, the present work opens new areas of application of the DQM.

##### MSC:

65L10 | Numerical solution of boundary value problems involving ordinary differential equations |

74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |

34B05 | Linear boundary value problems for ordinary differential equations |

65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |

34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |

74S25 | Spectral and related methods applied to problems in solid mechanics |

##### Keywords:

numerical examples; differential quadraturemethod; generalized collocation method; multipoint boundary value problem; shear deformation of sandwich beams; four-point nonlinear fourth-order systems
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\textit{G. R. Liu} and \textit{T. Y. Wu}, Math. Comput. Modelling 35, No. 1--2, 215--227 (2002; Zbl 0999.65074)

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