Theoretical and Numerical Investigations of Integro Differential Equations with Applications to Image Processing

Abstract

Integral and integro-differential equations serve as powerful mathematical tools for modeling a wide range of physical phenomena across science and engineering. Recently, integer and fractional-order integro-differential equations have attracted attention for modeling complex systems with memory effects. Since exact solutions are often unattainable, research focuses on existence-uniqueness results and the development of reliable numerical methods. Fixed point theorems, such as those proposed by Banach, Krasnoselskii, Schaefer and Arzela-Ascoli, are widely used to establish theoretical solvability conditions. Once solvability is ensured, numerical methods play a crucial role in approximating the solutions, with particular emphasis on analyzing their convergence and stability. This thesis presents a comprehensive investigation into a class of nonlinear VolterraFredholm integro-differential equations (VFIDEs), with a special focus on fractionalorder forms, delayed arguments and higher-dimensional systems. The research integrates both qualitative analysis and numerical experimentation and further extends into practical applications in image processing.