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Inverse Coefficient Problems for a Time-Fractional Wave Equation with the Generalized Riemann–Liouville Time Derivative

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Abstract

This paper considers the inverse problem of determining the time-dependent coefficient in the fractional wave equation with Hilfer derivative. In this case, the direct problem is initial-boundary value problem for this equation with Cauchy type initial and nonlocal boundary conditions. As overdetermination condition nonlocal integral condition with respect to direct problem solution is given. By the Fourier method, this problem is reduced to equivalent integral equations. Then, using the Mittag–Leffler function and the generalized singular Gronwall inequality, we get apriori estimate for solution via unknown coefficient which we will need to study of the inverse problem. The inverse problem is reduced to the equivalent integral of equation of Volterra type. The principle of contracted mapping is used to solve this equation. Local existence and global uniqueness results are proved.

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REFERENCES

  1. R. Hilfer, Applications of Fractional Calculus in Physics (World Scientific, Singapore, 2000). https://doi.org/10.1142/9789812817747

    Book  Google Scholar 

  2. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of their Applications, Ed. by I. Podlubny, Mathematics in Science and Engineering, Vol. 198 (Elsevier, New York, 1999). https://doi.org/10.1016/S0076-5392(13)60008-9

  3. R. Hilfer, Y. Luchko, and Z. Tomovski, “Operational method for the solution of fractional differential equations with generalized Riemann–Liouville fractional derivatives,” Fract. Calc. Appl. Anal. 12, 299–318 (2009).

    MathSciNet  Google Scholar 

  4. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier, Amsterdam, 2006). https://doi.org/10.1016/s0304-0208(06)80001-0

    Book  Google Scholar 

  5. B. M. Vinagre, I. Podlubny, A. Hernandez, and V. Feliu, “Some approximations of fractional order operators used in control theory and applications,” Fract. Calc. Appl. Anal. 3, 231–248 (2000).

    MathSciNet  Google Scholar 

  6. R. Ashurov, A. Cabada, and B. Turmetov, “Operator method for construction of solutions of linear fractional differential equations with constant coefficients,” Fractional Calculus Appl. Anal. 19, 229–252 (2016). https://doi.org/10.1515/fca-2016-0013

    Article  MathSciNet  Google Scholar 

  7. R. Ashurov and S. Umarov, “Determination of the order of fractional derivative for subdiffusion equations,” Fractional Calculus Appl. Anal. 23, 1647–1662 (2020). https://doi.org/10.1515/fca-2020-0081

    Article  MathSciNet  Google Scholar 

  8. Sh. Alimov and R. Ashurov, “Inverse problem of determining an order of the Caputo time-fractional derivative for a subdiffusion equation,” J. Inverse Ill-Posed Probl. 28, 651–658 (2020). https://doi.org/10.1515/jiip-2020-0072

    Article  MathSciNet  Google Scholar 

  9. P. Agarwal, A. Berdyshev, and E. Karimov, “Solvability of a non-local problem with integral transmitting condition for mixed type equation with caputo fractional derivative,” Results Math. 71, 1235–1257 (2017). https://doi.org/10.1007/s00025-016-0620-1

    Article  MathSciNet  Google Scholar 

  10. M. S. Salakhitdinov and E. T. Karimov, “Uniqueness of an inverse source non-local problem for fractional order mixed type equations,” Eurasian Math. J. 7 (1), 74–83 (2016).

    MathSciNet  Google Scholar 

  11. A. S. Berdyshev, E. T. Karimov, and N. S. Akhtaeva, “On a boundary-value problem for the parabolic-hyperbolic equation with the fractional derivative and the sewing condition of the integral form,” AIP Conf. Proc. 1611, 133–137 (2014). https://doi.org/10.1063/1.4893817

    Article  Google Scholar 

  12. E. Karimov, M. Mamchuev, and M. Ruzhansky, “Non-local initial problem for second order time-fractional and space-singular equation,” Hokkaido Math. J. 49, 349–361 (2020). https://doi.org/10.14492/hokmj/1602036030

    Article  MathSciNet  Google Scholar 

  13. D. K. Durdiev and Z. D. Totieva, “The problem of determining the one-dimensional matrix kernel of the system of viscoelasticity equations,” Math. Methods Appl. Sci. 41, 8019–8032 (2018). https://doi.org/10.1002/mma.5267

    Article  MathSciNet  Google Scholar 

  14. D. K. Durdiev, “On the uniqueness of kernel determination in the integro-differential equation of parabolic type,” Vestn. Samar. Gos. Tekh. Univ., Ser. Fiz.-Mat. Nauki 19, 658–666 (2015). https://doi.org/10.14498/vsgtu1444

    Article  Google Scholar 

  15. V. V. Kharat, D. B. Dhaigude, and D. R. Hasabe, “On nonlinear mixed fractional integrodifferential inclusion with four-point nonlocal Riemann–Liouville integral boundary conditions,” Indian J. Pure Appl. Math. 50, 937–951 (2019). https://doi.org/10.1007/s13226-019-0365-0

    Article  MathSciNet  Google Scholar 

  16. H. Gou and T. Wang, “The method of lower and upper solution for Hilfer evolution equations with non-instantaneous impulses,” Indian J. Pure Appl. Math. 54, 499–523 (2023). https://doi.org/10.1007/s13226-022-00271-4

    Article  MathSciNet  Google Scholar 

  17. D. K. Durdiev and Kh. Kh. Turdiev, “The problem of finding the kernels in the system of integro-differential Maxwell’s equations,” J. Appl. Ind. Math. 15, 190–211 (2021). https://doi.org/10.1134/S1990478921020022

    Article  MathSciNet  Google Scholar 

  18. D. K. Durdiev and Kh. Kh. Turdiev, “Inverse problem for a first-order hyperbolic system with memory,” Differ. Equations 56, 1634–1643 (2020). https://doi.org/10.1134/S00122661200120125

    Article  MathSciNet  Google Scholar 

  19. D. K. Durdiev, A. A. Rahmonov, and Z. R. Bozorov, “A two-dimensional diffusion coefficient determination problem for the time-fractional equation,” Math. Methods Appl. Sci. 44, 10753–10761 (2021). https://doi.org/10.1002/mma.7442

    Article  MathSciNet  Google Scholar 

  20. U. Durdiev and Z. Totieva, “A problem of determining a special spatial part of 3D memory kernel in an integro-differential hyperbolic equation,” Math. Methods Appl. Sci. 42, 7440–7451 (2019). https://doi.org/10.1002/mma.5863

    Article  MathSciNet  Google Scholar 

  21. J. Damirchi, R. Pourgholi, T. R. Shamami, H. Zeidabadi, and A. Janmohammadi, “Identification of a time dependent source function in a parabolic inverse problem via finite element approach,” Indian J. Pure Appl. Math. 51, 1587–1602 (2020). https://doi.org/10.1007/s13226-020-0483-8

    Article  MathSciNet  Google Scholar 

  22. D. K. Durdiev, “Inverse coefficient problem for the time-fractional diffusion equation,” Eurasian J. Math. Comput. Appl. 9 (1), 44–54 (2021). https://doi.org/10.32523/2306-6172-2021-9-1-44-54

    Article  MathSciNet  Google Scholar 

  23. U. D. Durdiev, “Problem of determining the reaction coefficient in a fractional diffusion equation,” Differ. Equations 57, 1195–1204 (2021). https://doi.org/10.1134/s0012266121090081

    Article  MathSciNet  Google Scholar 

  24. D. Durdiev and A. Rahmonov, “A multidimensional diffusion coefficient determination problem for the time-fractional equation,” Turk. J. Math. 46, 2250–2263 (2022). https://doi.org/10.55730/1300-0098.3266

    Article  MathSciNet  Google Scholar 

  25. D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Vol. 840 (Springer, Berlin, 1981). https://doi.org/10.1007/BFb0089647

  26. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier, Amsterdam, 2006).

    Google Scholar 

  27. T. Sandev and Ž. Tomovski, Fractional Equations and Models, Developments in Mathematics, Vol. 61 (Springer, Cham, 2019). https://doi.org/10.1007/978-3-030-29614-8

  28. A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis (Dover, New York, 1976).

    Google Scholar 

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This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.

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  1. Bukhara Branch, Romanovskiy Institute of Mathematics, Academy of Sciences of the Republic of Uzbekistan, 200118, Bukhara, Uzbekistan

    H. H. Turdiev

  2. Bukhara State University, 200118, Bukhara, Uzbekistan

    H. H. Turdiev

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  1. H. H. Turdiev
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Turdiev, H.H. Inverse Coefficient Problems for a Time-Fractional Wave Equation with the Generalized Riemann–Liouville Time Derivative. Russ Math. 67, 14–29 (2023). https://doi.org/10.3103/S1066369X23100092

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