Well-Posedness of Problems for the Heat Equation with a Fractional-Loaded Term and Memory
Umida BaltaevaDepartment of Applied Mathematics and Mathematical Physics, Urgench State University, Urgench 220100, UzbekistanBobur KhasanovDepartment of Exact Sciences, Khorezm Mamun Academy, Khiva 220900, UzbekistanOmongul EgamberganovaHigher and Applied Mathematics, Tashkent State University of Economics, Tashkent 100066, UzbekistanHamrobek HayitbayevDepartment of Information Technology, Mamun University, Khiva 220900, Uzbekistan
ABI
Abstract
We investigate the Cauchy problem for a heat equation incorporating variable diffusion coefficients and fractional memory effects modeled by a separable convolution kernel. By employing the fundamental solution of the associated parabolic equation, the problem is reformulated as a Volterra-type integral equation. Under appropriate regularity assumptions, we establish existence and uniqueness of classical solutions. Furthermore, we address an inverse problem aimed at simultaneously recovering the memory kernel and the solution. Using a differentiability-based approach, we derive a stable and well-posed formulation that enables the identification of memory effects in fractional heat models.
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