Solving differential equations using the laplace projected differential transformation method

In various fields of science and engineering, many problems are expressed using linear and nonlinear partial differential equations when a more realistic modeling approach is desired. Since finding analytical solutions to such equations is often challenging, there has been a growing need in recent years for methods that are easier to algorithmize, programmable, and yield much faster results for solving both linear and nonlinear problems. In this study, solution methods for differential equations were initially examined. Subsequently, numerical solution methods such as the Differential Transform Method (DTM), the Projected Differential Transform Method (PDTM), and a hybrid method, the Laplace Projected Differential Transform Method (LPDTM), were analyzed in terms of their properties, advantages, and disadvantages. Finally, the solutions of Klein-Gordon equations, the Kawahara equation, and its modified form were investigated using the LPDTM. The data obtained in this process has shown that, depending on the structure of the equation, solutions are reached in a shorter time with the LPDTM, while more precise results are obtained at most points compared to the PDTM. The method's rapid and efficient performance indicates that it can serve as a useful tool for solving a wide range of complex, linear, and nonlinear partial differential equations and systems of equations.