References

  1. Alsaedi A., Kirane M., Lassoued R. Global existence and asymptotic behavior for a time fractional reaction–diffusion system. Comput. Math. Appl. 2017, 73 (6), 951–958. doi:10.1016/j.camwa.2016.05.006
  2. Bagley R.L., Torvik P.J. A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheol. 1983, 27, 201–210. doi:10.1122/1.549724
  3. Djrbashian M.M., Nersesian A.B. Fractional derivatives and Cauchy problem for differential equations of fractional order. Fract. Calc. Appl. Anal. 2020, 23, 1810–1836. doi:10.1515/fca-2020-0090
  4. Duan J.-S. Time- and space-fractional partial differential equations. J. Math. Phys. 2005, 46, 013504. doi:10.1063/1.1819524
  5. Eidelman S.D., Ivasyshen S.D., Kochubei A.N. Analytic methods in the theory of differential and pseudo-differential equations of parabolic type. Basel-Boston-Berlin, Birkhauser Verlag, 2004.
  6. Fox C. The \(G\) and \(H\) functions as symmetrical Fourier kernels. Trans. Amer. Math. Soc. 1961, 98 (3), 395–429. doi:10.1090/S0002-9947-1961-0131578-3
  7. Fudjita Y. Integrodifferential equation which interpolates the heat equation and the wave equation. Osaka J. Math. 1990, 27 (2), 309–321. doi:10.18910/4060
  8. Gelfand I.M., Shilov G.E. Generalized Functions, Vol. 2. Spaces of Fundamental and Generalized Functions. AMS Chelsea Publ., 2016.
  9. Güner Ö., Bekir A. Exact solutions of some fractional differential equations arising in mathematical biology. Int. J. Biomath. 2015, 8 (1), 1550003. doi:10.1142/S1793524515500035
  10. Hilfer R. Fractional time evolution. In: Hilfer R. (Ed.) Applications of Fractional Calculus in Physics. World Sci., Singapore, 2020, 87–130. doi:10.1142/9789812817747_0002
  11. Ismailov M.I., Çiçek M. Inverse source problem for a time-fractional diffusion equation with nonlocal boundary conditions. Appl. Math. Model. 2016, 40 (7–8), 4891–4899. doi:10.1016/j.apm.2015.12.020
  12. Janno J., Kasemets K. Uniqueness for an inverse problem for a semilinear time-fractional diffusion equation. Inverse Probl. Imaging 2017, 11 (1), 125–149. doi:10.3934/ipi.2017007
  13. Kilbas A.A., Srivastava H.M., Trujillo J.J. Theory and Applications of fractional differential equations. Elsevier, Amsterdam, 2006.
  14. Kirane M., Lopushansky A., Lopushanska H. Determination of two unknown functions of different variables in a time-fractional differential equation. Math. Meth. Appl. Sci. 2024, 48 (4), 4185–4194. doi:10.1002/mma.10539
  15. Kirane M., Lopushansky A., Lopushanska H. Inverse problem for a time-fractional differential equation with a time- and space-integral conditions. Math. Meth. Appl. Sci. 2023, 46 (15), 16381–16393. doi:10.1002/mma.9453
  16. Kirane M., Samet B., Torebek B.T. Determination of an unknown source term temperature distribution for the sub-diffusion equation at the initial and final data. Electron. J. Diff. Equ. 2017, 2017 (257), 1–13.
  17. Kochubei A.N. Fractional-hyperbolic systems. Fract. Calc. Appl. Anal. 2013, 16 (4), 860–873. doi:10.2478/s13540-013-0053-4
  18. Lopushanska H., Lopushansky A. Inverse problems for a time fractional diffusion equation in the Schwartz-type distributions. Math. Methods Appl. Sci. 2021, 44 (3), 2381–2392. doi:10.1002/mma.5894
  19. Lopushanska H., Lopushansky A. Inverse problem with a time-integral condition for a fractional diffusion equation. Math. Meth. Appl. Sci. 2019, 42 (9), 3327–3340. doi:10.1002/mma.5587
  20. Lopushanska H., Lopushansky A., Myaus O. Inverse problem in a space of periodic spatial distributions for a time fractional diffusion equation Electron. J. Differ. Equ. 2016, 2016 (14), 1–9.
  21. Lopushans’ka H.P., M’yaus O.M. Restoration of initial data in the problem for a diffusion equation with fractional derivative with respect to time. J. Math. Sci. 2018, 229 (2), 187–199. doi:10.1007/s10958-018-3670-y
  22. Lopushansky A.O., Lopushanska H.P. Inverse problem for \(2b\)-order differential equation with a time-fractional derivative. Carpathian Math. Publ. 2019, 11 (1), 107–118. doi:10.15330/cmp.11.1.107-118
  23. Luchko Yu., Mainardi F. Cauchy and signaling problems for the time-fractional diffusion-wave equation. J. Vib. Acoust. 2014, 136 (5), 050904. doi:10.1115/1.4026892
  24. Luchko Yu., Yamamoto M. Comperison principles for the linear and semilinear time-fractional diffusion equations with the Robin boundary condition. arXiv:2208.04606 [math.AP] doi:10.48550/arXiv.2208.04606
  25. Mainardi F. The fundamental solutions for the fractional diffusion-wave equation. Appl. Math. Lett. 1996, 9 (6), 23–28.
  26. Podlubny I. Fractional differential equations. San Diego, Acad. Press, 1999.
  27. Povstenko Y. Linear fractional diffusion-wave equation for scientists and engeneers. New-York, Birkhauser, 2015.
  28. Sakamoto K., Yamamoto M. Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl. 2011, 382 (1), 426–447. doi:10.1016/j.jmaa.2011.04.058
  29. Sokolov I.M. and Klafter J. From diffusion to anomalous diffusion: A century after Einstein’s Brownian motion. Chaos. 2005, 15 (2), 026103. doi:10.1063/1.1860472
  30. Trong D.D., Hai D.N.D. Backward problem for time-space fractional diffusion equations in Hilbert scales. Comput. Math. Appl. 2021, 93 (1), 253–264. doi:10.1016/j.camwa.2021.04.018
  31. Voroshylov A.A., Kilbas A.A. Conditions of the existence of classical solution of the Cauchy problem for diffusion-wave equation with Caputo partial derivative. Dokl. Ak. Nauk. 2007, 414 (4), 1–4.
  32. Wang J.-G., Ran Y.-H. An iterative method for an inverse source problem of time-fractional diffusion equation. Inverse Probl. Sci. Eng. 2018, 26 (10), 1509–1521. doi:10.1080/17415977.2017.1417406
  33. Wei T., Jan X.B. Recovering a space-dependent source term in a time-fractional diffusion-wave equation. J. Appl. Anal. Comput. 2019, 9 (5), 1801–1821. doi:10.11948/20180318
  34. Wen J., Cheng J.-F. The method of fundamental solution for the inverse source problem for the space-fractional diffusion equation. Inverse Probl. Sci. Eng. 2018, 26 (7), 925–941. doi:10.1080/17415977.2017.1369537
  35. Xian J., Wei T. Determination of the initial data in a time-fractional diffusion-wave problem by a final time data. Comput. Math. Appl. 2019, 78 (8), 2525–2540. doi:10.1016/j.camwa.2019.03.056
  36. Yang F., Zhang Y., Li X.-X. Landweber iterative method for identifying the initial value problem of the time-space fractional diffusion-wave equation. Numer. Algorithms 2020, 83, 1509–1530. doi:10.1007/s11075-019-00734-6