Pioneering Numerical Techniques for Solving Differential Equations - A Comprehensive overview

The field of numerical analysis studies the application of mathematics to solve problems of practical importance. When solving differential equations derived from real-world scenarios, numerical techniques play a crucial role, particularly when a closed-form solution is unavailable or obtaining an exact/accurate solution is challenging. This paper’s main goal is to look into specific numerical techniques for solving ODEs that have initial conditions. With a primary focus on the Adomian Decomposition, Differential Transform, and Multistep approaches, this study investigates a variety of numerical strategies for solving differential equations. Several mathematicians discovered after a thorough examination of their work that these methods have greatly advanced the analysis of differential equations and are widely used in the fundamental sciences, engineering and economics. The study also emphasizes how essential it is to carry out advanced research in this field so as to create numerical approaches for solving differential equations that are more precise and effective. Research has also carried out on the creation of general-purpose numerical techniques and algorithms for solving the problems, with main focus on stability and convergence in multistep approaches. The two-dimensional nonlinear wave equation is solved using the Adomian Decomposition method, and a unique multistep approach is suggested for handling nonlinear differential equations. The results produced by various techniques are contrasted.