A Comparative study of solution techniques for linear and nonlinear systems of differential equations

This study presents a comprehensive comparison of three advanced techniques for solving different systems of ordinary differential equations (ODEs), which include both linear and nonlinear systems. The proposed methods are the Multi-Step Differential Transform Method (MSDTM), the Adomian Decomposition Method (ADM), and the Residual Power Series Method (RPSM). These methods present the solutions without requiring discretization, linearization, or perturbation. Moreover, their solutions take the form of rapidly convergent series with easily computable components. The theoretical foundation, computational effectiveness, and suitability for various ODE systems are examined for each method. All algorithms are implemented entirely in MATLAB (version R2016b) using custom-built functions, without relying on built-in solvers or symbolic computation. The proposed methods were tested on a number of real-world problems, showing their superior performance over other existing methods. The paper discusses the strengths and limitations of each method, focusing on their performance in terms of convergence, stability, and ease of implementation. The findings provide useful insights for researchers and practitioners in selecting approximate solution techniques for certain ODE systems, hence contributing to the improvement of numerical and semi-analytical methods in applied mathematics.