Startseite Inverse problems for diffusion equation with fractional Dzherbashian-Nersesian operator
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Inverse problems for diffusion equation with fractional Dzherbashian-Nersesian operator

  • Anwar Ahmad , Muhammad Ali und Salman A. Malik EMAIL logo
Veröffentlicht/Copyright: 22. November 2021
Veröffentlichen auch Sie bei De Gruyter Brill

Abstract

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.

MSC 2010: 80A23; 34A08; 34A12; 33E12

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), 31–37.10.1016/0022-247X(88)90139-4Suche in Google Scholar

[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; 10.1515/fca-2020-0040; https://www.degruyter.com/journal/key/fca/23/3/html.Suche in Google Scholar

[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; 10.1515/fca-2018-0045; https://www.degruyter.com/journal/key/fca/21/3/html.Suche in Google Scholar

[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; 10.1002/mma.4776.Suche in Google Scholar

[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; 10.1002/mma.4558.Suche in Google Scholar

[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; 10.1080/17415977.2016.1255738.Suche in Google Scholar

[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; 10.1051/mmnp/2019014.Suche in Google Scholar

[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), 1192–1194.10.1023/B:DIEQ.0000049836.04104.6fSuche in Google Scholar

[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.Suche in Google Scholar

[10] M. Caputo, Linear models of dissipation whose Q is almost frequency independent II. Geophys J Intl. 13, No 5 (1967), 529–539.10.1111/j.1365-246X.1967.tb02303.xSuche in Google Scholar

[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; 10.1515/fca-2020-0090; https://www.degruyter.com/journal/key/fca/23/6/html.Suche in Google Scholar

[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), 3–29.10.1515/fca-2020-0090Suche in Google Scholar

[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).10.1007/978-3-662-43930-2Suche in Google Scholar

[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).Suche in Google Scholar

[15] M. Kirane, N. Al-Salti, Inverse problems for a nonlocal wave equation with an involution perturbation. J. Nonlinear Sci. Appl. 9, (2016), 1243–1251.10.22436/jnsa.009.03.49Suche in Google Scholar

[16] Y. Luchko, Fractional derivatives and the fundamental theorem of fractional calculus. Fract. Calc. Appl. Anal. 23, No 4 (2020), 939–966; 10.1515/fca-2020-0049; https://www.degruyter.com/journal/key/fca/23/4/html.Suche in Google Scholar

[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), 7180–7186.10.1063/1.470346Suche in Google Scholar

[18] M.D. Ortigueira and J.A. Tenreiro Machado, What is a fractional derivative. J. Comput. Phys. 293 (2015), 4–13.10.1016/j.jcp.2014.07.019Suche in Google Scholar

[19] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego (1999).Suche in Google Scholar

[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), 323–332.10.1007/978-1-4020-6042-7Suche in Google Scholar

[21] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, Amsterdam (1993).Suche in Google Scholar

[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; 10.1002/mma.4468.Suche in Google Scholar

[23] V.E. Tarasov, Leibniz rule and fractional derivatives of power functions, J. Comput. Nonlinear Dyn. 11, No 3 (2016), # 031014.10.1115/1.4031364Suche in Google Scholar

[24] V.E. Tarasov, No nonlocality. No fractional derivative. Commun. Nonlinear Sci. 62 (2018), 157–163; 10.1016/j.cnsns.2018.02.019.Suche in Google Scholar

[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; 10.1007/s40819-015-0054-6.Suche in Google Scholar

Received: 2021-04-04
Revised: 2021-10-26
Published Online: 2021-11-22
Published in Print: 2021-12-20

© 2021 Diogenes Co., Sofia

Heruntergeladen am 16.11.2025 von https://www.degruyterbrill.com/document/doi/10.1515/fca-2021-0082/pdf?lang=de
Button zum nach oben scrollen