Fixed-Point Theorems Based Evaluation of Analytical Solution in Fractional Diffusion Equations

The definition of fractional calculus is a careful study conducted by experts in a variety of technical and scientific fields. Fractional calculus is a topic that has gained the attention of researchers in engineering and science for its various applications. As a result, for the purpose of computing the solution to a fractional-order nonlinear differential equation, the Caputo-derivative based Iterative Adomian Decomposition (CIADM) technique has been devised in this study. In this research, we address two primary limitations, such as not modifying the nonlinear fractional differential equation in order to reduce the number of iterations required for the linear algebraic fractional equation. Some fixed-point theorems are taken into account while investigating this methodology. In addition, the uniqueness of the solution and its existence are evaluated using a variety of case studies. The effectiveness of this proposed strategy is analyzed taking into account both the modified Adomian decomposition method (ADM) and the traditional Adomian decomposition method.