Inverse optimization problem for a fractional analog of the Barenblatt--Zheltov--Kochina equation

Document Type : Research Paper

Authors

1 1. Tashkent State Transport University, Temiryulchilar Street 1, Tashkent, 100167 Uzbekistan. 2. Alfraganus University, Yukori Karakamysh street 2A, Tashkent, 100190 Uzbekistan.

2 Universität Duisburg-Essen, Thea-Leymann-Straße 9, D-45127 Essen, Germany.

Abstract

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.

Keywords

Main Subjects

References

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Computational Methods for Differential Equations
Volume 14, Issue 2
April 2026
Pages 877-907

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History

  • Receive Date: 03 July 2024
  • Revise Date: 13 January 2025
  • Accept Date: 05 February 2025

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