Innovative Approaches in Differential Equation Analysis Using the Enhanced Differential Transform and Homotopy Perturbation Method

Ordinary differential equations (ODEs) are very basic when it comes to modeling dynamic systems in various fields of science and engineering. However, solving high-dimensional, nonlinear, and stiff ODEs is still a major challenge given the limitations of existing numerical methods, which tend to have difficulties in terms of accuracy and efficiency. This paper presents a new numerical approach called the enhanced differential transform homotopy perturbation method (E-DTHPM), dedicated to overcoming the above difficulties. E-DTHPM presents a combination of the benefits of the differential transform method (DTM) and homotopy perturbation method (HPM) with an adaptive step sizing mechanism that is dynamic in step sizing based on local truncation errors (LTEs). The combination of these methods addresses an important research gap by offering an efficient and accurate solution to the IVP of ODEs with nonsmooth behaviors (singularities and discontinuity behaviors). The proposed method is an improvement over conventional techniques and will provide increased computational efficiency and precision for complicated high-dimensional systems. Numerical experiments show that E-DTHPM is better than existing methods and is useful for applications that need to be dynamic enough and accurate enough for fields such as physics, biology, and engineering.