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

Keywords: Schr\

Abstract

UDC 517.9

We consider a Schrödinger equation $i \partial_t^\rho u(x,t)-u_{xx}(x,t) = p(t)q(x) + f(x,t),$ $0<t\leq T,$ $0<\rho<1,$ with the Riemann–Liouville derivative. An inverse problem is investigated in which, parallel with $u(x,t),$ a time-dependent factor $p(t)$ of the source function is also unknown. To solve this inverse problem, we use an additional condition $ B [u (\cdot,t)] = \psi (t) $ with an arbitrary bounded linear functional $ B $. The existence and uniqueness theorem for the solution to the problem under consideration is proved. The stability inequalities are obtained. The applied method make it possible to study a similar problem by taking, instead of $d^2/dx^2,$ an arbitrary elliptic differential operator $A(x, D)$ with compact inverse.

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