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.
The authors would like to thank the referees for carefully reading the paper and also for their constructive and valuable comments which have greatly improved the paper.
"Harmonic numbers operational matrix for solving fifth-order two point boundary value problems." Tbilisi Math. J. 11 (2) 17 - 33, June 2018. https://doi.org/10.32513/tbilisi/1529460019