A Numerical Approach for Solving Load-Leveling Energy System Models Governed by First-Kind Volterra Integral Equations

This paper presents an accurate and efficient spectral collocation method for solving first-kind Volterra integral equations with piecewise continuous kernels. These equations frequently arise in applied mathematics and engineering, particularly in modeling systems with memory effects and cumulative processes, but they are known to be ill-posed and computationally challenging. To address these difficulties, the proposed method constructs an approximate solution based on shifted Legendre polynomial expansions, which provide high accuracy and spectral convergence properties.

The numerical procedure employs Legendre–Gauss quadrature rules combined with a collocation strategy to discretize the integral equation. This transformation reduces the original continuous problem into a finite-dimensional system of algebraic equations. To further enhance the robustness and accuracy of the computational framework, the resulting system is solved using a Physics-Informed Neural Network (PINN) approach, which is used to determine the unknown expansion coefficients efficiently while maintaining consistency with the governing integral equation.

To demonstrate the effectiveness and applicability of the proposed method, an engineering application related to load leveling in energy storage systems is investigated. In this context, the first-kind Volterra integral equation models the charge–discharge dynamics of an energy-storage device required to satisfy time-varying electrical load demands. Numerical experiments show that the proposed hybrid spectral–PINN framework achieves high accuracy and stability, even in the presence of discontinuities in the kernel. The results confirm the efficiency of the method and its potential for solving a wide range of ill-posed integral equations arising in engineering and applied sciences.