Statistical analysis for solution of non-linear integro-differential equation by using odinary and accerlated technique of Kamal-Adomian Decomposition

The integro-differential equations have a significant role in presenting the daily life phenomenon and researchers used different approaches for the solution of such problems. But it has been noticed that nonlinear integro-differential equations lack closed-form solutions. Consequently, approximate approaches are essential for determining approximate solutions to these models. The foremost goal of this research is to offer an innovative combined method by employing the Kamal transform and the Adomian decomposition method (ADM) for extracting analytical estimated, closed form, and numerical results of the non-linear differential-integral equations. The proposed method is named the Kamal Adomian Decomposition Method (KADM). To evaluate its efficiency and consistency the outcomes attained by the offered method are compared with the Laplace decomposition method which shows that our method is more efficient, that reveals the reliability of the presented method. The KADM will be used in the future to analyze new systems that emerge in many branches of science. The convergence analysis of the series of solutions is also presented. Furthermore, we provide some interesting non-trivial examples to show the validity of our main results.