Qualitative Studies of a Novel Three-Dimensional FractionalIntegro-Differential Equation with Application in Image Representation

Three-dimensional fractional integro-differential equations (3D-FIDEs) offer a robust mathematical framework with wide-ranging implications in image processing and physics. This study provides valuable insights into the existence, uniqueness and Hyers–Ulam stability of solutions to 3D-FIDEs using the Banach fixed point theorem, Arzela–Ascoli theorem and Krasnoselskii fixed point theorem. Moreover, this study successfully derives positive solutions as well as maximal and minimal solutions for the problem. Furthermore, the paper includes a set of numerical examples to illustrate the theoretical findings and verify their validity. We also use this IDE as an image-representation model for image processing applications and apply it to different types of images. The results contribute to advancing knowledge in this area and lay the foundation for further research and applications in fractional calculus.