Asian Journal of Pure and Applied Mathematics

Fractional integral transforms have attracted substantial attention because of their adaptability and applications in physics, optics, electronics, communication engineering, and related areas. This study introduces a two-dimensional Fourier transform of fractional order formulated through fractional derivatives and the Mittag-Leffler function. The proposed transform extends the two-dimensional Fourier transform within a fractional-calculus framework and is accompanied by selected analytical properties. The formulation is based on the modified Riemann-Liouville fractional derivative and a fractional integral representation. The transform is then applied to obtain analytical solutions for selected two-dimensional linear fractional partial differential equations. Specifically, applications are developed for fractional dispersion, time-space heat-like, fractional telegraph, and time-space wave-like equations under the conditions stated in the manuscript. General transformed solutions are derived, and special cases are considered when the fractional parameters reduce to unity. In these cases, the resulting expressions recover forms associated with the corresponding classical two-dimensional equations. The study therefore presents a unified transform-based procedure for treating the selected linear fractional partial differential equations while retaining the structure of the proposed two-dimensional fractional Fourier framework. The analytical development and applications presented here are intended to support further investigation of fractional integral transforms and their use in solving fractional mathematical models within this framework.