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\


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.


A. V. Pskhu, Fractional partial differential equations} (in Russian), Nauka, Moscow (2005).

S. Umarov, Introduction to fractional and pseudo-differential equations with singular symbols, Springer (2015).

R. Ashurov, O. Muhiddinova, Initial-boundary value problem for a time-fractional subdiffusion equation with an arbitrary elliptic differential operator, Lobachevskii J. Math., 42, № 3, 517–525 (2021).

A. Ashyralyev, M. Urun, Time-dependent source identification problem for the Schrödinger equation with nonlocal boundary conditions, AIP Conf. Proc., 2183, Article 070016, Amer. Inst. Phys. (2019).

A. Ashyralyev, M. Urun, On the Crank–Nicolson difference scheme for the time-dependent source identification problem, Bull. Karaganda Univ., Math., 102, № 2, 35–44 (2021).

A. Ashyralyev, M. Urun, Time-dependent source identification Schr{"o}dinger type problem, Int. J. Appl. Math., 34, № 2, 297–310 (2021).

Y. Liu, Z. Li, M. Yamamoto, Inverse problems of determining sources of the fractional partial differential equations, Handbook of Fractional Calculus with Applications, vol. 2, De Gruyter (2019), p. 411–430.

S. I. Kabanikhin, Inverse and ill-posed problems. Theory and applications, De Gruyter (2011).

A. I. Prilepko, D. G. Orlovsky, I. A. Vasin, Methods for solving inverse problems in mathematical physics, Marcel Dekkers, New York (2000).

K. Sakamoto, M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. and Appl., 382, 426–447 (2011).

P. Niu, T. Helin, Z. Zhang, An inverse random source problem in a stochastic fractional diffusion equation, Inverse Problems, 36, № 4, Article 045002 (2020).

M. Slodichka, Uniqueness for an inverse source problem of determining a space-dependent source in a non-autonomous time-fractional diffusion equation, Fract. Calc. and Appl. Anal., 23, № 6, 1703–1711 (2020); DOI: 10.1515/fca-2020-0084.

M. Slodichka, K. Sishskova, V. Bockstal, Uniqueness for an inverse source problem of determining a space dependent source in a time-fractional diffusion equation, Appl. Math. Lett., 91, 15–21 (2019).

Y. Zhang, X. Xu, Inverse scource problem for a fractional differential equations, Inverse Problems, 27, № 3, 31–42 (2011).

M. Ismailov, I. M. Cicek, Inverse source problem for a time-fractional diffusion equation with nonlocal boundary conditions, Appl. Math. Model., 40, 4891–4899 (2016).

M. Kirane, A. S. Malik, Determination of an unknown source term and the temperature distribution for the linear heat equation involving fractional derivative in time, Appl. Math. and Comput., 218, 163–170 (2011).

M. Kirane, B. Samet, B. T. Torebek, Determination of an unknown source term and the temperature distribution for the subdiffusion equation at the initial and final data, Electron. J. Different. Equat., 217, 1–13 (2017).

H. T. Nguyen, D. L. Le, V. T. Nguyen, Regularized solution of an inverse source problem for a time fractional diffusion equation, Appl. Math. Model., 40, 8244–8264 (2016).

B. T. Torebek, R. Tapdigoglu, Some inverse problems for the nonlocal heat equation with Caputo fractional derivative, Math. Methods Appl. Sci., 40, 6468–6479 (2017).

R. Ashurov, Yu. Fayziev, Determination of fractional order and source term in a fractional subdiffusion equation}; https: //www.researchgate.net/publication/354997348.

Z. Li, Y. Liu, M. Yamamoto, Initial-boundary value problem for multi-term time-fractional diffusion equation with positive constant coefficients, Appl. Math. and Comput., 257, 381–397 (2015).

W. Rundell, Z. Zhang, Recovering an unknown source in a fractional diffusion problem, J. Comput. Phys., 368, 299–314 (2018).

N. A. Asl, D. Rostamy, Identifying an unknown time-dependent boundary source in time-fractional diffusion equation with a non-local boundary condition, J. Comput. and Appl. Math., 335, 36–50 (2019).

L. Sun, Y. Zhang, T. Wei, Recovering the time-dependent potential function in a multi-term time-fractional diffusion equation, Appl. Numer. Math., 135, 228–245 (2019).

S. A. Malik, S. Aziz, An inverse source problem for a two parameter anomalous diffusion equation with nonlocal boundary conditions, Comput. and Math. Appl., 3, 7–19 (2017).

M. Ruzhansky, N. Tokmagambetov, B. T. Torebek, Inverse source problems for positive operators. I, Hypoelliptic diffusion and subdiffusion equations, J. Inverse Ill-Possed Probl., 27, 891–911 (2019).

R. Ashurov, O. Muhiddinova, Inverse problem of determining the heat source density for the sub-diffusion equation, Different. Equat., 56, № 12, 1550–1563 (2020).

K. M. Furati, O. S. Iyiola, M. Kirane, An inverse problem for a generalized fractional diffusion, Appl. Math. and Comput., 249, 24–31 (2014).

M. Kirane, A. M. Salman, A. Mohammed Al-Gwaiz, An inverse source problem for a two dimensional time fractional diffusion equation with nonlocal boundary conditions, Math. Methods Appl. Sci. (2012); DOI: 10.1002/mma.2661.

A. Muhammad, A. M. Salman, An inverse problem for a family of time fractional diffusion equations, Inverse Probl. Sci. and Eng., 25, № 9, 1299–1322 (2016); DOI: 10.1080/17415977.2016.1255738.

Zh. Shuang, R. Saima, R. Asia, K. Khadija, M. A. Abdullah, Initial boundary value problems for a multi-term time fractional diffusion equation with generalized fractional derivatives in time, AIMS Math., 6, № 11, 12114–12132 (2021); DOI, 10.3934/math.2021703.

R. Ashurov, Y. Fayziev, On the nonlocal problems in time for time-fractional subdiffusion equations, Fractal and Fractional, 6, 41 (2022); https: //doi.org/10.3390/fractalfract6010041.

R. Ashurov, Yu. Fayziev, Uniqueness and existence for inverse problem of determining an order of time-fractional derivative of subdiffusion equation, Lobachevskii J. Math., 42, № 3, 508–516 (2021).

R. Ashurov, Yu. Fayziev, Inverse problem for determining the order of the fractional derivative in the wave equation, Math. Notes, 110, № 6, 842–852 (2021).

M. M. Dzherbashian, Integral transforms and representation of functions in the complex domain} (in Russian), Nauka, Moscow (1966).

R. Gorenflo, A. A. Kilbas, F. Mainardi, S. V. Rogozin, Mittag-Leffler functions, related topics and applications, Springer (2014).

R. Ashurov, A. Cabada, B. Turmetov, Operator method for construction of solutions of linear fractional differential equations with constant coefficients, Fract. Calc. Appl. Anal., 1, 229–252 (2016).

R. R. Ashurov, Yu. E. Fayziev, On construction of solutions of linear fractional differentional equations with constant coefficients and the fractional derivatives, Uzb. Math. J., 3, 3–21 (2017).

A. Zygmund, Trigonometric series, vol. 1, Cambridge (1959).

A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Math. Stud., vol. 204 (2006).

How to Cite
Ashurov, R., and M. Shakarova. “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.
Research articles