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**Sextic spline solutions of fifth order boundary value problems.**
*(English)*
Zbl 1125.65071

Summary: The sextic spline is used for numerical solutions of the fifth order linear special case boundary value problems. End conditions for the definition of the spline are derived, consistent with the fifth order boundary value problem. The algorithm developed approximates the solutions, and their higher order derivatives. The method is compared with that developed by H. N. Caglar, S. H. Caglar, and E. H. Twizell [Appl. Math. Lett. 12, 25–30 (1999; Zbl 0941.65073)], which is first order convergent, while the method developed in this work is observed to be second order convergent. Two examples are considered for the numerical illustration of the method developed.

### MSC:

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

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

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

### Keywords:

sextic spline; boundary value problems; consistency relations; interpolatory spline; end conditions; numerical examples; convergence; algorithm### Citations:

Zbl 0941.65073
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\textit{S. S. Siddiqi} and \textit{G. Akram}, Appl. Math. Lett. 20, No. 5, 591--597 (2007; Zbl 1125.65071)

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### References:

[1] | Davies, A.R.; Karageorghis, A.; Phillips, T.N., Spectral Galerkin methods for the primary two-point boundary-value problem in modelling viscoelastic flows, Internat. J. numer. methods engrg., 26, 647-662, (1988) · Zbl 0635.73091 |

[2] | Karageorghis, A.; Phillips, T.N.; Davies, A.R., Spectral collocation methods for the primary two-point boundary-value problem in modelling viscoelastic flows, Internat. J. numer. methods engrg., 26, 805-813, (1988) · Zbl 0637.76008 |

[3] | Caglar, H.N.; Caglar, S.H.; Twizell, E.H., The numerical solution of fifth-order boundary-value problems with sixth-degree \(B\)-spline functions, Appl. math. lett., 12, 25-30, (1999) · Zbl 0941.65073 |

[4] | Papamichael, N.; Worsey, A.J., A cubic spline method for the solution of a linear fourth-order two-point boundary value problem, J. comput. appl. math., 7, 187-189, (1981) · Zbl 0461.65063 |

[5] | Usmani, R.A.; Sakai, M., Quartic spline solutions for two-point boundary value problems involving third order differential equations, J. math. phys. sci., 18, 4, (1984), (India) · Zbl 0583.65052 |

[6] | Agarwal, R.P., Boundary value problems for high order differential equations, (1986), World Scientific Singapore · Zbl 0598.65062 |

[7] | S.S. Siddiqi, G. Akram, End conditions for interpolatory sextic splines, Intern. J. Comput. Math. (in press) · Zbl 1127.41003 |

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