Abstract

This research develops a generalized analytical framework to solve non-linear integro-differential equations of order n using a modified inverse operator. The study investigates two cases to demonstrate the versatility of the proposed operator. In the first case, a first-order model with a variable coefficient q(x) = x is analyzed, where the operator effectively integrates the linear structure by incorporating an exponential integrating factor into the framework. In the second case, the study addresses a second-order model with a constant coefficient q(x) = 1. By integrating this modified operator with the Adomian Decomposition Method (ADM), a robust recursive scheme is constructed that simplifies the computational process and preserves initial conditions. To overcome symbolic execution challenges and memory exhaustion, a discrete summation algorithm is implemented for the nonlinear terms. The methodology is supported by rigorous mathematical proofs of existence, uniqueness, and convergence based on the Banach fixed-point theorem. Numerical results via Mathematica software confirm high-order precision and rapid convergence, proving the scheme’s reliability for complex physical applications.