University of Tabriz Computational Methods for Differential Equations 2345-3982 14 2 2026 04 01 Inverse optimization problem for a fractional analog of the Barenblatt--Zheltov--Kochina equation 877 907 19281 10.22034/cmde.2025.62338.2740 EN Tursun K. Yuldashev 1. Tashkent State Transport University, Temiryulchilar Street 1, Tashkent, 100167 Uzbekistan. 2. Alfraganus University, Yukori Karakamysh street 2A, Tashkent, 100190 Uzbekistan. Aysel Ramazanova Universität Duisburg-Essen, Thea-Leymann-Straße 9, D-45127 Essen, Germany. Journal Article 2024 07 03 The generalized solvability of a nonlinear optimal control for thermal and diffusion processes in a mixed inverse problem for a Barenblatt-Zheltov-Kochina differential equation with Hilfer fractional operator is studied. The inverse problem is considered with spectral and intermediate conditions. Eigenvalues, eigenfunctions, and associated functions of the spectral problem are found and the corresponding adjoint problem is solved. Countable systems of fractional order differential equations with final value conditions are obtained. The necessary optimality conditions for nonlinear control are formulated. The determination of the optimal control function is reduced to solve a complicated nonlinear functional integral equation, and the process of solving consists of solving separately taken two nonlinear functional-integral equations. Nonlinear functional integral equations are solved by the method of successive approximations and the unique solvability of these equations is proved by the method of contracting mapping. Approximate calculations for the optimal control function, the redefinition function, and the state function of the controlled process are obtained. The absolute and uniform convergence of the obtained Fourier series are proved. https://cmde.tabrizu.ac.ir/article_19281_e3891eba32cbfda4ba49fe459e2535bd.pdf