Wavelets collocation method for singularly perturbed differential–difference equations arising in control system

Abstract

In this paper, we present a wavelet collocation method for efficiently solving singularly perturbed differential–difference equations (SPDDEs) and one-parameter singularly perturbed differential equations (SPDEs) taking into account the singular perturbations inherent in control systems. These equations represent a class of mathematical models that exhibit a combination of differential and difference equations, making their analysis and solution challenging. The terms that include negative and positive shifts were approximated using Taylor series expansion. The main aim of this technique is to convert the problems by using operational matrices of integration of Haar wavelets into a system of algebraic equations that can be solved using Newton’s method. The adaptability and multi-resolution properties of wavelet functions offer the ability to capture system behavior across various scales, effectively handling singular perturbations present in the equations. Numerical experiments were conducted to showcase the effectiveness and accuracy of the wavelet collocation method, demonstrating its potential as a reliable tool for analyzing and solving SPDDEs in control system.

Keywords

• Collocation point
• Control system
• Haar wavelet
• Multiresolution analysis
• Singularly perturbed differential–difference equations