An elegant operational matrix based on harmonic numbers: Effective solutions for linear and nonlinear fourth-order two point boundary value problems

Abstract

This paper analyzes the solution of fourth-order linear and nonlinear two point boundary value problems. The suggested method is quite innovative and it is completely different from all previous methods used for solving such kind of boundary value problems. The method is based on employing an elegant operational matrix of derivatives expressed in terms of the well-known harmonic numbers. Two algorithms are presented and implemented for obtaining new approximate solutions of linear and nonlinear fourth-order boundary value problems. The two algorithms rely on employing the new introduced operational matrix for reducing the differential equations with their boundary conditions to systems of linear or nonlinear algebraic equations which can be efficiently solved by suitable solvers. For this purpose, the two spectral methods namely, Petrov-Galerkin and collocation methods are applied. Some illustrative examples are considered aiming to ascertain the wide applicability, validity, and efficiency of the two proposed algorithms. The obtained numerical results are satisfactory and the approximate solutions are very close to the analytical solutions and they are more accurate than those obtained by some other existing techniques in literature.

New formulas of thehigh‐orderderivatives offifth‐kindChebyshev polynomials: Spectral solution of theconvection–diffusionequation

“…The formulas are novel and they link the derivatives with their original polynomials. The expressions are given in terms of terminating hypergeometric functions of the type 4 F 3 (1). The derivatives are employed to introduce a new spectral solution for convection-diffusion equation based on the application of the spectral tau method.…”

“…The principal aim of this section is to derive new formulas of the high-order derivatives of Chebyshev polynomials of the fifth-kind in terms of their original polynomials. We will show that the derived formulas involve hypergeometric functions each of type 4 F 3 (1). These hypergeometric functions can be reduced in case of deriving the first derivative formula.…”

“…Proof. The elements of the matrix A can be obtained by making use of Lemma 4 along with the orthogonality relation (1). In virtue of Lemma 5 along with the orthogonality relation, the elements of the matrix E can be computed.…”

“…Tau method is more flexible than the Galerkin method due to the freedom of choosing the used basis functions, see for example [5,9,17,19]. The collocation method is common in the application as it can be applied to all kinds of differential equations, see for example [1,12,13,18,23,27,38].…”

A New First Finite Class of Classical Orthogonal Polynomials Operational Matrices: An Application for Solving Fractional Differential Equations

“…Overall, these recent papers highlight the continued usefulness of these methods in solving FDEs. Many studies have shown how useful it is to use operational matrices (OMs) of the derivatives of OPs to solve ordinary and FDEs numerically [22][23][24][25][26][27][28][29]. The main goal of this paper is to come up with new numerical spectral solutions for some types of FDE models.…”