## 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.)
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### References:

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