Non-polynomial splines approach to the solution of sixth-order boundary-value problems.

*(English)*Zbl 1130.65077Summary: Non-polynomial splines, which are equivalent to seven-degree polynomial splines, are used to develop a class of numerical methods for computing approximations to the solution of sixth-order boundary-value problems with two-point boundary conditions. Second-, fourth- and sixth-order convergence is obtained by using standard procedures. It is shown that the present methods give approximations, which are better than those produced by other spline and domain decomposition methods. Numerical examples are given to illustrate the practical usefulness of the new approach.

##### 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:

sixth-order BVP; finite-difference methods; non-polynomial splines; convergence; numerical examples
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\textit{Siraj-ul-Islam} et al., Appl. Math. Comput. 195, No. 1, 270--284 (2008; Zbl 1130.65077)

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##### References:

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

[2] | Akram, G.; Siddiqi, S.S., Solution of linear sixth-order boundary-value problems using non-polynomial spline technique, Appl. math. comput., 1, 708-720, (2006) · Zbl 1155.65361 |

[3] | Baldwin, P., A localized instability in a bernard layer, Appl. anal., 24, 117-156, (1987) · Zbl 0588.76076 |

[4] | Baldwin, P., Asymptotic estimates of the eigenvalues of a sixth-order boundary-value problem obtained by using global-phase integral methods, Phil. trans. R. soc. lond. A, 322, 281-305, (1987) · Zbl 0625.76043 |

[5] | Boutayeb, A.; Twizell, E.H., Numerical methods for the solution of special sixth-order boundary-value problems, Int. J. comput. math., 45, 207-223, (1992) · Zbl 0773.65055 |

[6] | S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Clarendon Press, Oxford1961 (Reprinted: Dover Books, New York, 1981). |

[7] | Chawala, M.M.; P Katti, C., Finite difference methods for two-point boundary-value problems involving higher-order differential equations, Bit, 19, 27-33, (1979) · Zbl 0401.65053 |

[8] | Gamel, M.E.; Cannon, J.R.; I Zayed, J.LatourA., Sinc-Galerkin method for solving linear sixth-order boundary-value problems, Math. comput., 73, 247, 1325-1343, (2003) · Zbl 1054.65085 |

[9] | Glatzmaier, G.A., Numerical simulations of stellar convective dynamics III: at the base of the convection zone, Geophys. astrophys. fluid dyn., 31, 137-150, (1985) |

[10] | Hoskins, W.D.; Ponzo, P.J., Some properties of a class of band matrices, Math. comput., 26, 118, 390-400, (1972) · Zbl 0248.15008 |

[11] | Siddiqi, S.S.; Twizell, E.H., Spline solutions of linear sixth-order boundary-value problems, Int. J. comput. math., 45, 295-304, (1996) · Zbl 1001.65523 |

[12] | Toomre, J.; Zahn, J.R.; Latour, J.; Spiegel, E.A., Stellar convection theory II: single-mode study of the second convection zone in A-type stars, Astrophys. J., 207, 545-563, (1976) |

[13] | Twizell, E.H., A second order convergent method for sixth-order boundary-value problems, (), 495-506, (Singapore) |

[14] | Twizell, E.H.; Boutayeb, Numerical methods for the solution of special and general sixth-order boundary-value problems, with applications to benard layer eigenvalue problems, Proc. R. soc. lond. A., 431, 433-450, (1990) · Zbl 0722.65042 |

[15] | Wazwaz, A.M., The numerical solution of sixth-order boundary-value problems by modified decomposition method, Appl. math. comput., 118, 311-325, (2001) · Zbl 1023.65074 |

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