Application of the modified double ara-sumudu decomposition method to partial differential equations

In this thesis, the modified double ARA–Sumudu transform method (MDARA-ST), developed to obtain analytical solutions of partial differential equations involving nonlinear and fractional terms, is examined. Partial differential equations (PDEs) serve as fundamental tools for modeling complex systems in numerous scientific fields such as physics, engineering, biology, and chemistry. However, solving these equations using classical analytical methods is often quite challenging. Therefore, the development of transform-based methods and semi-analytical approaches has become a significant research area. The Sumudu transform, due to its similarity to the Laplace transform and compatibility with initial conditions, stands out as an effective tool in analytical solution techniques. Furthermore, the integration of the Sumudu transform with the Adomian Decomposition Method (ADM) and the ARA has led to the development of the DARA-ST and subsequently the MDARA-ST methods, which offer notable advantages in terms of solution accuracy and convergence speed. Within the scope of this thesis, the mathematical foundations and theoretical background of the Sumudu, ARA, and DARA-ST transforms are presented in detail. Then, the MDARA-ST method is introduced along with explanations on how Adomian polynomials are utilized. The advantages provided by this hybrid method are evaluated through the solutions of various differential equations, and the obtained results are compared with those from classical methods. In particular, comparisons with solutions obtained via the Variational Iteration Method and the classical Adomian Decomposition Method demonstrate that the MDARA-ST method yields more precise solutions with fewer iterations. The results indicate that the proposed method is highly effective for both nonlinear and fractional PDEs.