Time-dependent source identification problem for a fractional Schrödinger equation with the Riemann–Liouville derivative

Ravshan Ashurov; Marjona Shakarova

doi:10.37863/umzh.v75i7.7155

Ravshan Ashurov Institute of Mathematics, Uzbekistan Academy of Science, Tashkent and School of Engineering, Central Asian University, Uzbekistan
Marjona Shakarova National University of Uzbekistan named after Mirzo Ulugbek, Tashkent

DOI: https://doi.org/10.37863/umzh.v75i7.7155

Анотація

УДК 517.9

Залежна від часу задача ідентифікації джерела для дробового рівняння Шредінгера з похідною Рімана–Ліувілля

Розглянуто рівняння Шредінгера $i \partial_t^\rho u(x,t)-u_{xx}(x,t) = p(t)q(x) + f(x,t),$ $0<t\leq T,$ $0<\rho<1,$ з похідною Рімана–Ліувілля. Досліджено обернену задачу, в якій крім $u(x,t)$ також невідомий залежний від часу множник $p(t)$ функції джерела. Для розв'язання оберненої задачі введено додаткову умову $B [u (\cdot,t)] = \psi (t) $ для довільного обмеженого лінійного функціонала $B $. Доведено теорему існування та єдиності розв'язку задачі, що розглядається. Отримано нерівності щодо стійкості. Застосований метод дає змогу дослідити аналогічну задачу, в якій замість $d^2/dx^2$ фігурує довільний еліптичний диференціальний оператор $A(x, D),$ що має компактний обернений оператор.

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