research-focus Research Focus International Scientific Journal Research Focus 2181-3833 Ilm-fan va innovatsiyalar akademiyasi 10.66073/researchfocus.v4i6.1664 REL-A-012990 01.00.00 RECOVERY OF A TIME-DEPENDENT SOURCE IN A FRACTIONAL LANGEVIN EQUATION ВОССТАНОВЛЕНИЕ ЗАВИСЯЩЕГО ОТ ВРЕМЕНИ ИСТОЧНИКА В ДРОБНОМ УРАВНЕНИИ ЛАНЖЕВЕНА KASR TARTIBLI LANJEVEN TENGLAMASIDA VAQTGA BOG‘LIQ MANBANI TIKLASH Bakhodirjon Toshtemirov Kokand University, Uzbekistan. 06 07 2025 2025 4 6 8 14 https://creativecommons.org/licenses/by/4.0 This work is licensed under a Creative Commons Attribution 4.0 International License.

In the current paper, we are interested in studying the time-dependent inverse source problem for the space-degenerate fractional Langevin-type PDE involving a bi-ordinal Hilfer fractional derivative. Sufficient conditions for the given data were established for the existence and uniqueness of the solution. The technique for showing the existence result is based on the uniform convergence of the series.

В текущей статье мы заинтересованы в изучении зависящей от времени обратной задачи источника для вырожденного по пространству дробного уравнения Ланжевена в частных производных с бипорядковой дробной производной Хильфера. Были установлены достаточные условия для заданных данных для существования и единственности решения. Методика демонстрации результата существования основана на равномерной сходимости ряда.

Ushbu maqolada bi-tartibli Hilfer kasr tartibli hosilasini o‘z ichiga olgan, fazo bo‘yicha degeneratsiyalangan kasr tartibli Lanjeven tipidagi xususiy hosilali differensial tenglama uchun vaqtga bog‘liq teskari manba masalasini o‘rganish qiziqish uyg‘otadi. Yechimning mavjudligi va yagonaligi uchun berilgan ma‘lumotlarga nisbatan yetarli shartlar o‘rnatildi. Mavjudlik natijasini ko‘rsatish usuli qatorning tekis yaqinlashishiga asoslangan.

 bi-ordinal Hilfer fractional derivative Fourier-Legendre series time-dependent inverse source problem бипорядковая дробная производная Хильфера ряд Фурье- Лежандра зависящая от времени обратная задача источника bi-ordinal Hilfer kasr tartibli hosilasi Fourier-Legendre qatori vaqtga bog‘liq teskari manba masalasi Stefanov, A. Vasy, M. Zworski, Inverse Problems and Applications, Amer Mathematical Society, -2014. N. Blaunstein, V. Yakubov, Electromagnetic and Acoustic Wave Tomography: Direct and Inverse Problems in Practical Applications, Chapman and Hall/CRC, -2018. A.S. Hendy, K. Van Bockstal, On a Reconstruction of a Solely Time-Dependent Source in a Time-Fractional Diffusion Equation with Non-smooth Solutions, J. Sci. Comput. -2022. -90, 41. K. Sakamoto, M. Yamamoto, Initial value/boundary value problems for fractional diffusion- wave equations and applications to some inverse problems, Journal of Mathematical Analysis and Applications, -2011, -Vol. 382(1), pp. 426-447, M. Slodička, D. Lesnic, T.T.M. Onyango, Determination of a time-dependent heat transfer coefficient in a nonlinear inverse heat conduction problem, Inverse Problems in Science and Engineering, -2009. -Vol. 18(1). -P. 65-81. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and applications of fractional differential equation, Elsevier, Amsterdam. -2006. E. Karimov, B. Toshtemirov, On a time-nonlocal boundary value problem for time-fractional partial differential equation, International Journal of Applied Mathematics, -2022, Vol. 35(3), pp. 423-438. doi:10.12732/ijam.v35i3.5 Kaplan W. Advanced Calculus. 5th Edition. Pearson, -2002.