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

  • 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
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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Published
25.07.2023
How to Cite
AshurovR., and ShakarovaM. “Time-Dependent Source Identification Problem for a Fractional Schrödinger Equation With the Riemann–Liouville Derivative”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 75, no. 7, July 2023, pp. 871 -87, doi:10.37863/umzh.v75i7.7155.
Section
Research articles