Operator Theoretic Analysis of Coupled Decomposition Methods

Abstract

Coupled Kharrat-Toma-Mohand Transform and Adomian Decomposition Techniques for Nonlinear Differential and Integro-Differential Equations have been studied using a combination of an integral transform with Adomian Decomposition Method. The convergence property has been proved for specific problems to date. In this paper, we give a general proof for this issue. Assuming boundedness of the operators in the Banach space of continuous functions, we will prove that the proposed combination technique is nothing but Picard Iteration of an operator T. By virtue of the contraction property of the operator T, existence, uniqueness, error estimate, and stability of the method are proved. Some logistic and diffusion problem examples illustrate the divergence of the constant at blow-up point.