Exploring a Novel Multi-Stage Differential Transform Method Coupled with Adomian Polynomials for Solving Implicit Nonlinear ODEs with Analytical Solutions

In engineering, physics, and other fields, implicit ordinary differential equations are essential to simulate complex systems. However, because of their intrinsic nonlinearity and difficulty separating higher-order derivatives, implicit ordinary differential equations pose substantial challenges. When applied to these types of equations, traditional numerical methods frequently have problems with convergence or require a significant amount of computing power. In this work, we present the multi-stage differential transform method, a novel semi-analytical approach for effectively solving first- and second-order implicit ordinary differential systems, in conjunction with Adomian polynomials. The main contribution of this method is that it simplifies the solution procedure and lowers processing costs by enabling the differential transform method to be applied directly to implicit systems without transforming them into explicit or quasi-linear forms. We obtain straightforward and effective algorithms that build solutions incrementally utilizing the characteristics of Adomian polynomials, providing benefits in theory and practice. By solving several implicit ODE systems that are difficult for traditional software programs such as Maple 2024, Mathematica 14, or Matlab 24.1, we validate our approach. The multi-stage differential transform method’s contribution includes expanded convergence intervals for numerical results, more accurate approximate solutions for wider domains, and the efficient calculation of exact solutions as a convergent power series. Because of its ease of implementation in educational computational tools and substantial advantages in terms of simplicity and efficiency, our method is suitable for researchers and practitioners working with complex implicit differential equations.