Numerical solutions of Troesch's problem based on a faster iterative scheme with an application

Junaid Ahmad; Muhammad Arshad; Zhenhua Ma; Junaid Ahmad; Muhammad Arshad; Zhenhua Ma

doi:10.3934/math.2024446

AIMS Mathematics

2024, Volume 9, Issue 4: 9164-9183. doi: 10.3934/math.2024446

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Research article

Numerical solutions of Troesch's problem based on a faster iterative scheme with an application

1.
Department of Mathematics and Statistics, International Islamic University, Islamabad 44000, Pakistan
2.
Department of Mathematics and Physics, Hebei University of Architecture, Zhangjiakou 075024, China

Received: 23 November 2023 Revised: 27 January 2024 Accepted: 31 January 2024 Published: 06 March 2024
MSC : 34B27, 47H09, 47H10

The purpose of this manuscript was to introduce a new iterative approach based on Green's function for approximating numerical solutions of the Troesch's problem. A Banach space of continuous functions was considered for establishing the main outcome. First, we set an integral operator using a Green's function and embedded this new operator into a three-step iterative scheme. We proved the main convergence result with the help of some mild assumptions on the parameters involved in our scheme and in the problem. Moreover, we proved that the new iterative approach was weak $ w^{2} $-stable. The high accuracy and stability of the scheme was confirmed by several numerical simulations. As an application of the main result, we solved a class of fractional boundary value problems (BVPs). The main results improved and unified several known results of the literature.
- Troesch's problem,
- fractional BVPs,
- numerical solution,
- iterative scheme,
- Green's function
Citation: Junaid Ahmad, Muhammad Arshad, Zhenhua Ma. Numerical solutions of Troesch's problem based on a faster iterative scheme with an application[J]. AIMS Mathematics, 2024, 9(4): 9164-9183. doi: 10.3934/math.2024446

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Abstract

The purpose of this manuscript was to introduce a new iterative approach based on Green's function for approximating numerical solutions of the Troesch's problem. A Banach space of continuous functions was considered for establishing the main outcome. First, we set an integral operator using a Green's function and embedded this new operator into a three-step iterative scheme. We proved the main convergence result with the help of some mild assumptions on the parameters involved in our scheme and in the problem. Moreover, we proved that the new iterative approach was weak $ w^{2} $-stable. The high accuracy and stability of the scheme was confirmed by several numerical simulations. As an application of the main result, we solved a class of fractional boundary value problems (BVPs). The main results improved and unified several known results of the literature.

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