Distributed Order Fractional Calculus in Two Dimensions With Sonin Kernels: A Transmutation Approach

This article introduces a unified framework for transmuted general fractional derivatives of distributed order in two dimensions based on parameter‐dependent Sonin kernels. General Riemann–Liouville and Caputo‐type operators are first formulated and extended to the distributed‐order setting via suitable weight functions, enabling the modeling of multi‐scale nonlocal behavior. The proposed operators are shown to be well‐defined, and their analytical properties are established using two‐dimensional convolution and Laplace transform techniques. Fundamental results, including relationships between operator types, inverse properties, and solution representations for associated problems, are derived. Furthermore, by incorporating transmutation operators, a broader class of fractional operators in two dimensions is obtained, preserving key structural properties such as commutativity and semigroup behavior. The framework unifies several existing approaches and provides a flexible tool for analyzing multi‐dimensional fractional systems, with potential extensions to higher‐dimensional settings.