Finite difference spectral collocation schemes for the solutions of boundary value problems

A O Adewumi; A A Aderogba; S O Akindeinde; O O Fabelurin; R S Lebelo

doi:10.1016/j.heliyon.2023.e23453

Finite difference spectral collocation schemes for the solutions of boundary value problems

Heliyon. 2023 Dec 7;10(1):e23453. doi: 10.1016/j.heliyon.2023.e23453. eCollection 2024 Jan 15.

Authors

A O Adewumi¹, A A Aderogba¹, S O Akindeinde², O O Fabelurin¹, R S Lebelo³

Affiliations

¹ Department of Mathematics, Obafemi Awolowo University, 220005, Ile-Ife, Nigeria.
² Department of Mathematics and Statistical Sciences, Botswana International University of Science and Technology (BIUST), Private Bag 016, Palapye, Botswana.
³ Faculty of Human Sciences, Vaal University of Technology, Vanderbijlpark, South Africa.

Abstract

This article introduces novel numerical approaches utilizing both standard and nonstandard finite difference methods to solve one-dimensional Bratu's problems. Using the quasilinearization technique, the original problem is converted into a sequence of linear problems. Chebyshev polynomials are employed to approximate the second derivative of the function $y (x)$ , after which Sumudu transform is applied to obtain a new form of trial function. The obtained trial function is then substituted into a linearized and discretized Bratu's equations. We discuss the convergence of the schemes and compare the numerical outcomes to those derived using other relevant methods. We further modify one of the new schemes and apply it to solve boundary value problem with associated Robin conditions. The results show that the proposed schemes yield accurate approximations to the solutions of the problems considered.

Keywords: 65L12; 65M06; 65N06; 97N40; Boundary value problem; Bratu's problems; Chebyshev polynomials; Quasilinearization technique; Robin conditions; Standard and nonstandard finite difference schemes; Sumudu transform.