Advancing solutions of integer and fractional order system of PDE’s using a modified transform method

Systems of partial differential equations (PDEs) serve as fundamental tools for modeling and solving complex, multidimensional problems involving interdependent processes. This paper introduces an efficient technique for solving linear and nonlinear systems of PDEs of both integer and fractional orders. The proposed method, referred to as the Modified Yang Transform Method (MYT), combines the Adomian Decomposition Method with the Yang Transform. Owing to its simplicity and versatility, this approach holds significant potential for application across diverse scientific and engineering fields. A generalized solution procedure is outlined in a step-by-step manner for both integer- and fractional-order systems. Theoretical analysis is performed to ensure the method’s convergence and stability. To demonstrate the method’s effectiveness, several illustrative examples are solved. Visual validation is provided through 2D and 3D plots that depict the behavior of the solutions, while numerical error analysis is presented in tabular form. These results reveal that the approximate solutions exhibit strong agreement with the exact ones, with improved accuracy as the number of iterations increases. Detailed discussions of the findings are included to further support the reliability and applicability of the proposed method.