Existence, Uniqueness, and Numerical Experiments for a Caputo Fractional PDE With Variable‐Exponent Diffusion in Image Denoising

ABSTRACT We propose a novel nonlinear evolution model that integrates a Caputo time‐fractional derivative with a variable exponent diffusion operator for image decomposition and noise removal. The Caputo derivative captures memory effects through a fractional order, while the spatially adaptive exponent allows the diffusion process to adjust locally to image features. From a theoretical perspective, we establish well–posedness results by proving existence and uniqueness of solutions via the Faedo–Galerkin method in variable exponent Sobolev spaces. We further show that our model preserves positivity, a key property for image processing applications. On the numerical side, extensive experiments on grayscale and medical images demonstrate the robustness of the method under high noise levels. The results highlight the influence of both the fractional order and the variable exponent, confirming that our approach achieves stronger noise reduction and better feature preservation than state‐of‐the‐art techniques.