A novel iterative scheme for solving delay differential equations and third order boundary value problems via Green's functions

Godwin Amechi Okeke; Akanimo Victor Udo; Rubayyi T. Alqahtani; Melike Kaplan; W. Eltayeb Ahmed; Godwin Amechi Okeke; Akanimo Victor Udo; Rubayyi T. Alqahtani; Melike Kaplan; W. Eltayeb Ahmed

doi:10.3934/math.2024315

AIMS Mathematics

2024, Volume 9, Issue 3: 6468-6498. doi: 10.3934/math.2024315

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Research article

A novel iterative scheme for solving delay differential equations and third order boundary value problems via Green's functions

1.
Functional Analysis and Optimization Research Group Laboratory (FANORG), Department of Mathematics, School of Physical Sciences, Federal University of Technology Owerri, P.M.B. 1526 Owerri, Imo State, Nigeria
2.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box 90950 Riyadh, Saudi Arabia
3.
Department of Computer Engineering, Faculty of Engineering and Architecture, Kastamonu University, Kastamonu, Türkiye

Received: 29 November 2023 Revised: 21 January 2024 Accepted: 30 January 2024 Published: 05 February 2024
MSC : 34A08, 47J25, 47J26

In this paper, we constructed a novel fixed point iterative scheme called the Modified-JK iterative scheme. This iteration process is a modification of the JK iterative scheme. Our scheme converged weakly to the fixed point of a nonexpansive mapping and strongly to the fixed point of a mapping satisfying condition (E). We provided some examples to show that the new scheme converges faster than some existing iterations. Stability and data dependence results were proved for this iteration process. To substantiate our results, we applied our results to solving delay differential equations. Furthermore, the newly introduced scheme was applied in approximating the solution of a class of third order boundary value problems (BVPs) by embedding Green's functions. Moreover, some numerical examples were presented to support the application of our results to BVPs via Green's function. Our results extended and generalized other existing results in literature.
- fixed point,
- rate of convergence,
- Garcia-Falset mapping,
- condition (E),
- boundary value problems,
- $ \mathcal{J} $-stability,
- Modified-JK iterative scheme,
- delay differential equations
Citation: Godwin Amechi Okeke, Akanimo Victor Udo, Rubayyi T. Alqahtani, Melike Kaplan, W. Eltayeb Ahmed. A novel iterative scheme for solving delay differential equations and third order boundary value problems via Green's functions[J]. AIMS Mathematics, 2024, 9(3): 6468-6498. doi: 10.3934/math.2024315

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Abstract

In this paper, we constructed a novel fixed point iterative scheme called the Modified-JK iterative scheme. This iteration process is a modification of the JK iterative scheme. Our scheme converged weakly to the fixed point of a nonexpansive mapping and strongly to the fixed point of a mapping satisfying condition (E). We provided some examples to show that the new scheme converges faster than some existing iterations. Stability and data dependence results were proved for this iteration process. To substantiate our results, we applied our results to solving delay differential equations. Furthermore, the newly introduced scheme was applied in approximating the solution of a class of third order boundary value problems (BVPs) by embedding Green's functions. Moreover, some numerical examples were presented to support the application of our results to BVPs via Green's function. Our results extended and generalized other existing results in literature.

References

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