<![CDATA[The current paper presents a detailed study of the application of the Coupled Mahgoub-Adomian Decomposition Method (CMADM) to solve both linear and nonlinear Volterra and Fredholm IDEs. The current method includes the effectiveness of the application of the Mahgoub integral transform along with the efficiency of the use of the ADM to handle the nonlinearity in the problem. In addition, existence and uniqueness theorems for the solutions of IDEs via the Banach fixed point theorem and the Picard iteration process are proved for the first time. Convergence of the generated series via the current method is investigated via the Cauchy-estimate bound on the Adomian polynomials, which replaces the unjustified linear estimate used in a previous version of this paper, and norm estimates in a suitable Banach space. Conditions under which the accuracy of the result after truncation of the series can be achieved are derived. We specify what is new in comparison with the existing Laplace, Sumudu, and Elzaki Adomian methods; we fix two inaccuracies found in the referee review process: an erroneous “exact solution” of one nonlinear problem, and an arithmetic mistake in the Fredholm example; and we add one more nonlinear problem with an oscillatory kernel along with convergence rate tables and error graphs. As an illustration, four examples, including one linear and three nonlinear integro-differential equations, are considered.]]>
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