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

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.

Keywords

• fixed point
• rate of convergence
• garcia-falset mapping
• condition (e)
• boundary value problems
• $ \mathcal{j} $-stability
• modified-jk iterative scheme
• delay differential equations