Harmonic numbers operational matrix for solving fifth-order two point boundary value problems

Y.H. Youssri$^{1}$ and W.M. Abd-Elhameed$^{2}$

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. 

Tbilisi Mathematical Journal, Vol. 11(2) (2018), pp. 17-33