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Time-Dependent Source Identification Problem for a Fractional Schrödinger Equationwith the Riemann–Liouville Derivative

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Ukrainian Mathematical Journal Aims and scope

We consider a Schrödinger equation \(i{\partial }_{t}^{\rho }u\left(x,t\right)-{u}_{xx}\left(x,t\right)=p\left(t\right)q\left(x\right)+f\left(x,t\right),0<t\le T,0<\rho <1,\) 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(∙, t)] =ψ(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 makes it possible to study a similar problem by taking, instead of d2/dx2, an arbitrary elliptic differential operator A(x,D) with compact inverse.

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Authors and Affiliations

  1. Institute of Mathematics, Uzbekistan Academy of Science, Tashkent, Uzbekistan

    Ravshan Ashurov

  2. School of Engineering, Central Asian University, Tashkent, Uzbekistan

    Ravshan Ashurov

  3. M. Ulugbek National University of Uzbekistan, Tashkent, Uzbekistan

    Marjona Shakarova

Authors
  1. Ravshan Ashurov
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  2. Marjona Shakarova
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Correspondence to Ravshan Ashurov.

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, No. 7, pp. 871–887, July, 2023. Ukrainian DOI: https://doi.org/10.37863/umzh.v75i7.7155.

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Ashurov, R., Shakarova, M. Time-Dependent Source Identification Problem for a Fractional Schrödinger Equationwith the Riemann–Liouville Derivative. Ukr Math J 75, 997–1015 (2023). https://doi.org/10.1007/s11253-023-02243-1

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  • DOI: https://doi.org/10.1007/s11253-023-02243-1

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