Differential quadrature solutions of eighth-order boundary-value differential equations.

*(English)*Zbl 1001.65085Summary: 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.

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

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

65L12 | Finite difference and finite volume methods for ordinary differential equations |

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

34B15 | Nonlinear boundary value problems for ordinary differential equations |

65L20 | Stability and convergence of numerical methods for ordinary differential equations |

##### Keywords:

differential quadrature method; finite difference method; eighth-order boundary-value problem; hydrodynamic stability; pseudospectral method; collocation method; numerical results; polynomial splines; convergence
PDF
BibTeX
XML
Cite

\textit{G. R. Liu} and \textit{T. Y. Wu}, J. Comput. Appl. Math. 145, No. 1, 223--235 (2002; Zbl 1001.65085)

Full Text:
DOI

##### References:

[1] | Bellomo, N., Nonlinear models and problems in applied sciences from differential quadrature to generalized collocation methods, Math. comput. modelling, 26, 13-34, (1997) · Zbl 0898.65074 |

[2] | Bert, C.W.; Malik, M., Differential quadrature method in computational mechanics: a review, Appl. mech. rev., 49, 1-27, (1996) |

[3] | Boutayeb, A.; Twizell, E.H., Finite-difference methods for the solution of special eighth-order boundary-value problems, Internat. J. comput. math., 48, 63-75, (1993) · Zbl 0820.65046 |

[4] | G.R. Liu, T.Y. Wu, Application of Generalized Differential Quadrature Rule in Blasius and Onsager equations, Internat. J. Numer. Methods Eng. 52 (2001) 1013-1027. · Zbl 0996.76072 |

[5] | Shu, C.; Richards, B.E., Application of generalized differential quadrature to solve two-dimensional incompressible navier – stokes equation, Internat. J. numer. meth. fluids, 15, 791-798, (1992) · Zbl 0762.76085 |

[6] | Siddiqi, S.S.; Twizell, E.H., Spline solutions of linear eighth-order boundary-value problems, Comput. methods appl. mech. eng., 131, 309-325, (1996) · Zbl 0881.65076 |

[7] | Siddiqi, S.S.; Twizell, E.H., Spline solutions of linear twelfth-order boundary-value problems, Comput. J. comput. appl. math., 78, 371-390, (1997) · Zbl 0865.65059 |

[8] | Wu, T.Y.; Liu, G.R., The differential quadrature as a numerical method to solve the differential equation, Comput. mech., 24, 197-205, (1999) · Zbl 0976.74557 |

[9] | Wu, T.Y.; Liu, G.R., A generalized differential quadrature rule for initial-value differential equations, J. sound vib., 233, 195-213, (2000) · Zbl 1237.65018 |

[10] | T.Y. Wu, G.R. Liu, The generalized differential quadrature rule for fourth-order differential equations, Internat. J. Numer. Methods Eng. 50 (2001) 1907-1929. · Zbl 0999.74120 |

[11] | T.Y. Wu, G.R. Liu, Application of the generalized differential quadrature rule to sixth-order differential equations, Comm. Numer. Methods Eng. 16 (2000) 777-784. · Zbl 0969.65070 |

[12] | Michelsen, M.L.; Villadsen, J., A convenient computational procedure for collocation constants, J. chem. eng., 4, 64-68, (1972) |

[13] | Wu, T.Y.; Liu, G.R., Multipoint boundary value problems by the differential quadrature method, Math. comput. modelling, 35, 215-227, (2002) · Zbl 0999.65074 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.