Volterra Integral Equations: A Numerical Solution Method Using Shifted Chebyshev Polynomial

Abstract: This study presents a numerical method for solving Volterra integral equations of the second kind using shifted Chebyshev polynomials. Volterra integral equations arise in various scientific and engineering applications, including population dynamics, physics, and control systems. Due to their complexity, obtaining analytical solutions is often challenging, making numerical techniques crucial. We employ shifted Chebyshev polynomials as basis functions to approximate the solution, transforming the integral equation into a system of algebraic equations. The shifted Chebyshev polynomials offer excellent approximation properties, improving convergence rates and accuracy. The proposed method is analyzed for stability and efficiency, and numerical experiments demonstrate its effectiveness in solving different classes of Volterra integral equations. The results highlight the advantages of the approach compared to traditional numerical methods.