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**Chebyshev finite difference approximation for the boundary value problems.**
*(English)*
Zbl 1027.65098

Summary: This paper presents a numerical technique for solving linear and non-linear boundary value problems for ordinary differential equations. This technique is based on using matrix operator expressions which applies to the differential terms. It can be regarded as a non-uniform finite difference scheme. The values of the dependent variable at the Gauss-Lobatto points are the unknown one solves for.

The application of the method to boundary value problems leads to algebraic systems. The method permits the application of iterative method in order to solve the algebraic systems. The effective application of the method is demonstrated by four examples.

The application of the method to boundary value problems leads to algebraic systems. The method permits the application of iterative method in order to solve the algebraic systems. The effective application of the method is demonstrated by four examples.

### MSC:

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

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

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

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

### Keywords:

Chebyshev approximation; boundary value problems; incomplete LU-decomposition; numerical examples; non-uniform finite difference scheme; iterative method
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\textit{E. M. E. Elbarbary} and \textit{M. El-Kady}, Appl. Math. Comput. 139, No. 2--3, 513--523 (2003; Zbl 1027.65098)

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### References:

[1] | Canuto, C.; Hussaini, M.Y.; Quarterini, A.; Zang, T.A., Spectral methods in fluid dynamics, (1988), Springer-Verlag Berlin |

[2] | Cabos, C.H., A preconditioning of the tau operator for ordinary differential equations, Zamm, 74, 521-532, (1994) · Zbl 0824.65068 |

[3] | Clenshaw, C.W.; Curtis, A.R., A method for numerical integration on an automatic computer, Numer. math., 2, 197-205, (1960) · Zbl 0093.14006 |

[4] | D. Gottlieb, S.A. Orszag, Numerical analysis of spectral methods: theory and applications, in: CBMS-NSF Regional Conference Series, Applied Mathematics 26, SIAM, Philadelphia, PA, 1977 · Zbl 0412.65058 |

[5] | E.F. Kaasschieter, The solution of non-symmetric linear systems by bi-conjugate gradients or conjugate gradients squared, Techn. Rep. Report 86-21, Delft University of Technology, Delft, 1986 |

[6] | Shen, J., Efficient spectral-Galerkin method II direct solver of second and fourth-order equations using Chebyshev polynomials, SIAM J. sci. comput., 16, 74-87, (1995) · Zbl 0840.65113 |

[7] | Fox, L.; Parker, I.B., Chebyshev polynomials in numerical analysis, (1968), Clarendon Press Oxford · Zbl 0153.17502 |

[8] | Hiegemann, M.; Strauß, K., On Chebyshev matrix operator method for ordinary linear differential equations with non-constant coefficients, Acta mech., 105, 227-232, (1994) · Zbl 0809.65089 |

[9] | Hiegemann, M., Chebyshev matrix operator method for the solution of integrated forms of linear ordinary differential equations, Acta mech., 122, 231-242, (1997) · Zbl 0873.65073 |

[10] | Voigt, R.G.; Gottlieb, D.; Hussaini, M.Y., Spectral methods for partial differential equations, (1984), SIAM Philadelphia, PA |

[11] | El-Gendi, S.E., Chebyshev solution of differential integral and integro-differential equations, Comput. J., 12, 282-287, (1969) · Zbl 0198.50201 |

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