A Self-Adaptive Integral Transform Framework for Robust Solution of Nonlinear Differential Equations

Abstract

Integral transforms have long served as powerful analytical tools in engineering and applied mathematics, offering elegant and computationally efficient solutions to a broad class of differential equations. However, classical transform formulations are inherently static, and their performance often deteriorates when applied to strongly nonlinear, time-dependent, or dynamically evolving engineering systems. In modern applications characterized by sharp gradients, Multiphysics coupling, and rapidly changing boundary conditions, fixed transform kernels can lead to numerical instability, reduced spectral accuracy, and distortion of key physical features.

This study introduces a self-evolving integral transform framework in which artificial intelligence is embedded directly within the transform kernel. Rather than functioning as a passive mathematical operator, the transform becomes adaptive: kernel parameters are continuously updated during computation based on residual feedback and system response indicators. Through this learning-driven mechanism, the transform dynamically adjusts its internal structure in real time, improving convergence behavior, stabilizing inverse operations, and better preserving physically meaningful solution characteristics.

The proposed method is evaluated using the relative error norm and residual reduction metrics, and benchmarked against classical fixed-kernel integral transforms, spectral methods, and the finite element method (FEM). Numerical experiments demonstrate average error reductions of 37.6%, 34.2%, and 29.8%, respectively. Although the adaptive framework introduces a modest computational overhead—less than 12% compared with traditional transform techniques—it provides significantly improved spectral stability and overall solution accuracy.