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**Harmonic numbers operational matrix for solving fifth-order two point boundary value problems.**
*(English)*
Zbl 1434.65212

Summary: The principal purpose of this paper is to present and implement two numerical algorithms for solving linear and nonlinear fifth-order two point boundary value problems. These algorithms are developed via establishing a new Galerkin operational matrix of derivatives. The nonzero elements of the derived operational matrix are expressed explicitly in terms of the well-known harmonic numbers. The key idea for the two proposed numerical algorithms is based on converting the linear or nonlinear fifth-order two BVPs into systems of linear or nonlinear algebraic equations by employing Petrov-Galerkin or collocation spectral methods. Numerical tests are presented aiming to ascertain the high efficiency and accuracy of the two proposed algorithms.

### MSC:

65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |

65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |

35C10 | Series solutions to PDEs |

42C10 | Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) |

### Keywords:

harmonic numbers; Legendre polynomials; fifth-order BVPs; Galerkin and collocation methods; operational matrix
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\textit{Y. H. Youssri} and \textit{W. M. Abd-Elhameed}, Tbil. Math. J. 11, No. 2, 17--33 (2018; Zbl 1434.65212)

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