Existence and Uniqueness of the Solution of Two-dimensional Fuzzy Volterra Integral Equation with Piecewise Kernel
Samad Noeiaghdam
TL;DR
This work addresses the existence and uniqueness of solutions to two-dimensional fuzzy Volterra integral equations with piecewise-discontinuous kernels, extending analysis to both linear and nonlinear cases. A collocation framework is developed using 2-D fuzzy Lagrange interpolation and 2-D fuzzy Gauss-Legendre integration after mapping to the domain $[-1,1]^2$, yielding discrete nonlinear and linear systems at collocation points. The main theoretical contribution is a rigorous existence/uniqueness result under a Lipschitz condition on the nonlinear term $\phi$ and positivity/continuity of the kernels $k_p$, supported by a 2-D Gronwall-based contraction argument with the Hausdorff distance $D$. The numerical construction is complemented by an iterative scheme that converges to the unique fuzzy solution, with implications for energy-storage modeling and other applications where FVIEs with discontinuous kernels arise.
Abstract
This study investigates the existence and uniqueness of solutions to Volterra integral equations with discontinuous kernels in both linear and nonlinear cases. The problem is two-dimensional, and the collocation method is employed to analyze the equations. The research aims to provide a comprehensive understanding of the solution properties of these integral equations, which are crucial in various mathematical and physical applications. By examining the existence and uniqueness of solutions, this study contributes to the development of numerical methods for solving Volterra integral equations with discontinuous kernels. The findings of this research have the potential to impact various fields, including physics, engineering, and economics, where integral equations play a significant role in modeling complex phenomena.