“YANGI AVLOD ILMIY TADQIQOTLARI” respublika ilmiy-amaliy konferensiyasi
THE INVERSE-SOURCE PROBLEM FOR MIXED-TYPE EQUATIONS
Fergana State University, Fergana, Uzbekistan
Mixed-type differential equations combining hyperbolic and parabolic parts arise naturally in many important processes of mathematical physics, including heat conduction, diffusion phenomena, fluid dynamics, and wave propagation in heterogeneous media. Such equations describe systems in which different physical regimes coexist in separate subdomains and are connected through transmitting conditions. A special place in the theory of mixed-type equations is occupied by problems with fractional-order derivatives, since fractional models allow a more accurate description of memory and hereditary properties of complex media. In recent years, considerable attention has been devoted to boundary value and inverse problems for fractional mixed-type equations. The method of solution is based on spectral expansion with respect to the eigenfunctions of the associated Sturm–Liouville problem, reduction to systems of fractional ordinary differential equations, and the application of Mittag-Leffler functions. Explicit formulas for the Fourier coefficients of the solution and unknown functions are derived, and sufficient conditions for existence and uniqueness are established. The obtained results extend previous studies on inverse problems for mixed-type equations and contribute to the further development of fractional differential equation theory.