Modified two-dimensional differential transform method for solving proportional delay partial differential equations

In this study, we develop a modified version of the two-dimensional differential transform (TDDT) method for solving proportional delay partial differential equations (PDPDEs) that frequently arise in engineering and scientific models. This modification is achieved by integrating the TDDT method with the Laplace transform and the Padé approximant, thereby leveraging the strengths of each technique to improve overall performance. Theorems are provided in a general manner to cover various types of PDEs, with constant or variable coefficients. To validate the approach, we apply it to three test problems, demonstrating its effectiveness in extending the convergence domain of the traditional TDDT approach, reducing computational complexity, and yielding analytic solutions with fewer computational steps. Results indicate that the method is a viable alternative for addressing PDPDEs, especially in scenarios where traditional analytic solutions are challenging to obtain. This combination opens new avenues for efficiently solving complex delayed systems in engineering and science, potentially outperforming existing numerical and analytical techniques in both speed and reliability.