×
Ahmad, Anwar; Ali, Muhammad; Malik, Salman A.

Inverse problems for diffusion equation with fractional Dzherbashian-Nersesian operator. (English) Zbl 1498.26006

Fract. Calc. Appl. Anal. 24, No. 6, 1899-1918 (2021).
Summary: Fractional Dzherbashian-Nersesian operator is considered and three famous fractional order derivatives named after Riemann-Liouville, Caputo and Hilfer are shown to be special cases of the earlier one. The expression for Laplace transform of fractional Dzherbashian-Nersesian operator is constructed. Inverse problems of recovering space dependent and time dependent source terms of a time fractional diffusion equation with involution and involving fractional Dzherbashian-Nersesian operator are considered. The results on existence and uniqueness for the solutions of inverse problems are established. The results obtained here generalize several known results.

Cited in 9 Documents

MSC:

26A33 Fractional derivatives and integrals
35R11 Fractional partial differential equations
35R30 Inverse problems for PDEs

Keywords:

fractional Dzherbashian-Nersesian operator; inverse problems; fractional differential equations; Mittag-Leffler function
Cite Review PDF
Full Text: DOI arXiv

References:

[1] A.R. Aftabizadeh, Y.K. Huang, J. Wiener, Bounded solutions for differential equations with reflection of the argument. J. Math. Anal. Appl. 135, No 1 (1988), 31-37. · Zbl 0655.34030
[2] M. Ali, S. Aziz, S.A. Malik, Inverse problem for a multi-term fractional differential equation. Fract. Calc. Appl. Anal. 23, No 3 (2020), 799-821; ; https://www.degruyter.com/journal/key/fca/23/3/html. · Zbl 1488.65379 · doi:10.1515/fca-2020-0040
[3] M. Ali, S. Aziz, S.A. Malik, Inverse source problem for a space-time fractional diffusion equation. Fract. Calc. Appl. Anal. 21, No 3 (2018), 844-863; ; https://www.degruyter.com/journal/key/fca/21/3/html. · Zbl 1412.80006 · doi:10.1515/fca-2018-0045
[4] M. Ali, S. Aziz, S.A. Malik, Inverse problem for a space-time fractional diffusion equation: Application of fractional Sturm-Liouville operator. Math. Meth. Appl. Sci. 41, No 7 (2018), 2733-2747; . · Zbl 1388.35205 · doi:10.1002/mma.4776
[5] M. Ali, S.A. Malik, An inverse problem for a family of two parameters time fractional diffusion equations with nonlocal boundary conditions. Math. Meth. Appl. Sci. 40, No 18 (2017), 7737-7748; . · Zbl 1387.80006 · doi:10.1002/mma.4558
[6] M. Ali, S.A. Malik, An inverse problem for a family of time fractional diffusion equations. Inverse Probl. Sci. Eng. 25, No 9 (2017), 1299-1322; . · Zbl 1398.65232 · doi:10.1080/17415977.2016.1255738
[7] N. Al-Salti, S. Kerbal, M. Kirane, Initial-boundary value problems for a time-fractional differential equation with involution perturbation. Math. Model. Nat. Phenom. 14, No 3 (2019), # 312; . · Zbl 1419.35235 · doi:10.1051/mmnp/2019014
[8] A.A. Andreev, Analogs of classical boundary value problems for a second-order differential equation with deviating argument. Differ. Equ. 40, No 8 (2004), 1192-1194. · Zbl 1084.35108
[9] S. Aziz, S.A. Malik, Identification of an unknown source term for a time fractional fourth order parabolic equation. Electron. J. Differ. Equ. 2016 (2016), # 293, 1-20. · Zbl 1358.80004
[10] M. Caputo, Linear models of dissipation whose Q is almost frequency independent II. Geophys J Intl. 13, No 5 (1967), 529-539.
[11] M.M. Dzherbashian, A.B. Nersesian, Fractional derivatives and Cauchy problem for differential equations of fractional order. Fract. Calc. Appl. Anal. 23, No 6 (2020), 1810-1836; ; https://www.degruyter.com/journal/key/fca/23/6/html. · Zbl 1474.35018 · doi:10.1515/fca-2020-0090
[12] M.M. Dzhrbashyan, A.B. Nersesyan, Fractional derivatives and the Cauchy problem for fractional differential equations. Izv. Akad. Nauk Armyan. SSR. 3, No 1 (1968), 3-29. · Zbl 0165.40801
[13] R. Gorenflo, A.A. Kilbas, F. Mainardi, S.V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications. Springer, Berlin-Heidelberg (2014), 2nd Ed. (2020). · Zbl 1451.33001
[14] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam (2006). · Zbl 1092.45003
[15] M. Kirane, N. Al-Salti, Inverse problems for a nonlocal wave equation with an involution perturbation. J. Nonlinear Sci. Appl. 9, (2016), 1243-1251. · Zbl 1327.35440
[16] Y. Luchko, Fractional derivatives and the fundamental theorem of fractional calculus. Fract. Calc. Appl. Anal. 23, No 4 (2020), 939-966; ; https://www.degruyter.com/journal/key/fca/23/4/html. · Zbl 1474.26024 · doi:10.1515/fca-2020-0049
[17] R. Metzler, S. Schick, H.G. Kilian, T.F. Nonnenmacher, Relaxation in filled polymers: A fractional calculus approach. J. Chem. Phys. 103, No 16 (1995), 7180-7186.
[18] M.D. Ortigueira and J.A. Tenreiro Machado, What is a fractional derivative. J. Comput. Phys. 293 (2015), 4-13. · Zbl 1349.26016
[19] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego (1999). · Zbl 0918.34010
[20] J. Sabatier, O.P. Agrawal, J.A.T. Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht (2007), 323-332. · Zbl 1116.00014
[21] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, Amsterdam (1993). · Zbl 0818.26003
[22] B.T. Torebek, R. Tapdigoglu, Some inverse problems for the nonlocal heat equation with Caputo fractional derivative. Math. Meth. Appl. Sci. 40, No 18 (2017), 6468-6479; . · Zbl 1387.35625 · doi:10.1002/mma.4468
[23] V.E. Tarasov, Leibniz rule and fractional derivatives of power functions, J. Comput. Nonlinear Dyn. 11, No 3 (2016), # 031014.
[24] V.E. Tarasov, No nonlocality. No fractional derivative. Commun. Nonlinear Sci. 62 (2018), 157-163; . · Zbl 1470.26014 · doi:10.1016/j.cnsns.2018.02.019
[25] V.E. Tarasov, Local fractional derivatives of differentiable functions are integer-order derivatives or zero. Intl. J. Appl. Comput. Math. 2, No 2 (2016), 195-201; . · Zbl 1420.26006 · doi:10.1007/s40819-015-0054-6
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.