Fuzzy Volterra Integral Equation with Piecewise Continuous Kernel: Theory and Numerical Solution

TL;DR

The paper addresses fuzzy Volterra integral equations with piecewise continuous kernels, formulating the problem as $Z(v) = Y(v) \oplus (\mathcal{FR}) \sum_{t=1}^{m'} \int_{\theta_{t-1}(v)}^{\theta_t(v)} K_t(r,v) \odot G(Z(r))\,dr$ and proving existence and uniqueness via a Banach fixed-point argument in a fuzzy metric space. The main contribution is the contraction-based framework that yields an explicit error bound for the successive approximations $Z_m$, along with a discretized numerical scheme on a grid that converges to the unique solution as the mesh size $h$ tends to zero. The paper provides detailed error estimates involving moduli of continuity of the data and kernels, and validates the theory through linear and nonlinear numerical examples with graphical illustrations. This work extends FVIE theory to piecewise kernels and demonstrates practical numerical solvability for applications in balance problems of hereditary dynamics.

Abstract

This study aims to discuss the existence and uniqueness of solution of fuzzy Volterra integral equation with piecewise continuous kernel. Such problems appears in many balance problems for hereditary dynamic systems, e.g. in electric load leveling. The method of successive approximations is applied and the main theorems are proved based on the method. Some examples are discussed and the results are presented for different values of $μ$ by plotting several graphs.

Figures (5)

• Figure 1: Graphical plot of fuzzy approximate solution.
• Figure 2: Fuzzy approximate solution for various $\mu$.
• Figure 3: Error for (a) $\underline{F}(v), m=5, \mu=0.5$ (b) $\overline{F}(v), m=10, \mu=0.5$.
• Figure 4: Fuzzy approximate solution for various $\mu$.
• Figure 5: Error for (a) $\underline{F}(v), m=20, \mu=0.5$ (b) $\overline{F}(v), m=20, \mu=0.5$.

Theorems & Definitions (17)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1
• Theorem 2
• Definition 4
• Definition 5
• Theorem 3
• Definition 6
• Lemma 1