Solvability and Numerical Analysis of Coupled Nonlinear Differential Equation Systems

Abstract

Coupled nonlinear and differential equations systems appear in a large variety of scientific and engineering problems, such as dynamical systems, biological interactions and control processes. Nevertheless, they are inherently nonlinear and interconnected, so the analytical solutions are not easily obtained, and the current research tends to consider solvability and numerical approximation independently, which means that there are no coherent frameworks where both theoretical and numerical rigour are used. To fill this gap, this paper outlines a vigorous strategy that unites solvability analysis with powerful numerical calculation. To state it, the existence and uniqueness of solutions are strictly proved by the fixed-point theory and appropriate continuity and Lipschitz conditions. It is on this theoretical basis that an efficient numerical scheme is proposed, which implements an iterative approach to the solution of this coupled system by means of an adaptive procedure aiming at achieving a solution at a higher level of stability and accuracy. The effectiveness of the proposed method is measured with the help of comprehensive analysis of errors and convergence, based on a standard set of metrics, including absolute error, root mean square error and residual error. Numerical experiments show that the method reduces the error greatly and has high convergence rates as compared to traditional methods, especially when dealing with highly nonlinear coupled systems. The findings substantiate that the suggested framework offers a sound and scalable tool to analyze and solve coupled nonlinear differential equations, with both theoretical and practical efficiency of implementing it in the real-world scenario.