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<p>Real-world biological and physical systems are often affected by uncertainty, memory effects, and time delays. Yet most existing models treat these aspects separately, limiting their predictive power. In this work, we develop a&#xa0;unified numerical framework for solving fuzzy mixed delay Volterra-Fredholm integral equations that arise, for example, in bio-thermal processes with imprecise parameters and physiological response delays. We consider both linear and nonlinear formulations, where the nonlinearity accounts for temperature-dependent blood perfusion—a&#xa0;key feature in realistic tissue heating. Using the <i>α</i>-cut representation, we transform the fuzzy problem into a&#xa0;system of deterministic integral equations and prove existence, uniqueness, and stability via Banach’s contraction mapping principle. The history function associated with the delay term is explicitly incorporated into the analysis. For the numerical solution, we adapt the Variational Iteration Method (VIM) to this setting and establish its convergence under mild Lipschitz conditions. An optimal Lagrange multiplier is introduced to accelerate the convergence of the iterative scheme. Numerical experiments on a&#xa0;bio-thermal model derived from Pennes’ equation demonstrate that VIM provides accurate, stable, and efficient approximations, while preserving the structure of fuzzy solutions. The proposed approach thus offers a&#xa0;reliable tool for uncertainty quantification in problems involving delay and nonlinear dynamics.</p>

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Variational iteration method for linear and nonlinear fuzzy mixed delay Volterra-Fredholm integral equations with bio-Thermal applications

  • 期刊论文

  • 发表日期：

    2026年5月21日

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  • Ch Subba Reddy,
  • T. L. Yookesh,
  • Sudam Sekhar Panda

摘要

Real-world biological and physical systems are often affected by uncertainty, memory effects, and time delays. Yet most existing models treat these aspects separately, limiting their predictive power. In this work, we develop a unified numerical framework for solving fuzzy mixed delay Volterra-Fredholm integral equations that arise, for example, in bio-thermal processes with imprecise parameters and physiological response delays. We consider both linear and nonlinear formulations, where the nonlinearity accounts for temperature-dependent blood perfusion—a key feature in realistic tissue heating. Using the α-cut representation, we transform the fuzzy problem into a system of deterministic integral equations and prove existence, uniqueness, and stability via Banach’s contraction mapping principle. The history function associated with the delay term is explicitly incorporated into the analysis. For the numerical solution, we adapt the Variational Iteration Method (VIM) to this setting and establish its convergence under mild Lipschitz conditions. An optimal Lagrange multiplier is introduced to accelerate the convergence of the iterative scheme. Numerical experiments on a bio-thermal model derived from Pennes’ equation demonstrate that VIM provides accurate, stable, and efficient approximations, while preserving the structure of fuzzy solutions. The proposed approach thus offers a reliable tool for uncertainty quantification in problems involving delay and nonlinear dynamics.