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<p>This study investigates the existence and uniqueness of solutions for a class of two-dimensional fractional-order Volterra-Fredholm integro-differential equations (2D-FVFIDEs). The Krasnoselskii fixed-point theorem is employed to prove the existence of solutions, while Banach’s fixed-point theorem is used to establish their uniqueness. To compute numerical solutions, we adopt the discrete Laplace-Adomian Decomposition Method (LADM), which effectively handles the complexities introduced by fractional and integral operators. A series of illustrative numerical examples validate the theoretical results and demonstrate the robustness of the proposed approach. We have also compared the numerical results with those obtained using existing methods. Moreover, the developed integro-differential model is applied to the field of image processing for enhancement tasks. The model’s practical utility is showcased through experiments on various medical and real-time images, confirming its potential in real-world applications.</p>

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Theoretical and Numerical Analysis of 2D Fractional Volterra-Fredholm Integro-Differential Equations with Applications to Image Enhancement

  • 期刊论文

  • 发表日期：

    2026年7月7日

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  • Bapan Ali Miah,
  • Mausumi Sen,
  • R. Murugan

摘要

This study investigates the existence and uniqueness of solutions for a class of two-dimensional fractional-order Volterra-Fredholm integro-differential equations (2D-FVFIDEs). The Krasnoselskii fixed-point theorem is employed to prove the existence of solutions, while Banach’s fixed-point theorem is used to establish their uniqueness. To compute numerical solutions, we adopt the discrete Laplace-Adomian Decomposition Method (LADM), which effectively handles the complexities introduced by fractional and integral operators. A series of illustrative numerical examples validate the theoretical results and demonstrate the robustness of the proposed approach. We have also compared the numerical results with those obtained using existing methods. Moreover, the developed integro-differential model is applied to the field of image processing for enhancement tasks. The model’s practical utility is showcased through experiments on various medical and real-time images, confirming its potential in real-world applications.