Abstract
Inverse problem for a family of multi-term time fractional differential equation with non-local boundary conditions is studied. The spectral operator of the considered problem is non-self-adjoint and a bi-orthogonal set of functions is used to construct the solution. The operational calculus approach has been used to obtain the solution of the multi-term time fractional differential equations. Integral type over-determination condition is considered for unique solvability of the inverse problem. Different estimates of multinomial Mittag-Leffler functions alongside Banach fixed point theorem are used to prove the unique local existence of the solution of the inverse problem. Stability of the solution of the inverse problem on the given datum is established.
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References
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, (2018), 7737–7748.
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, (2018), 2733–2744.
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; DOI:10.1515/fca-2018-0045 https://www.degruyter.com/view/journals/fca/21/3/fca.21.issue-3.xml.
R. Almeida, N.R.O. Bastos, M.T.T. Monteiro, Modeling some real phenomena by fractional differential equations. Math. Meth. Appl. Sci. 39, (2016), 4846–4855.
S. Aziz, S.A. Malik, Identification of source term in fourth order parabolic equation. Electr. J. of Diff. Equations 2016, (2016), #293, 20.
H. Brunner, L. Ling, M. Yamamoto, Numerical simulations of 2D fractional subdiffusion problems. J. of Comput. Physics 229, (2010), 6613–6622.
M. Ciesielski, M. Klimek, T. Blaszczyk, The fractional Sturm-Liouville problem-Numerical approximation and application in fractional diffusion. J. of Comput. and Appl. Math. 317, (2017), 573–588.
K. Diethelm, A fractional calculus based model for the simulation of an outbreak of dengue fever. Nonlinear Dynamics 71, (2013), 613–619.
X. Ding, Y. Jiang, Analytical solutions for multi-term time-space coupling fractional delay partial differential equations with mixed boundary conditions. Commun. Nonlinear Sci. Numer. Simul. 68, (2018), 231–247.
X. Ding, J. J. Nieto, Analytical solutions for multi-term time-space fractional partial differential equations with nonlocal damping. Fract. Calc. Appl. Anal. 21, No 2 (2018), 312–335; DOI:10.1515/fca-2018-0019 https://www.degruyter.com/view/journals/fca/21/2/fca.21.issue-2.xml.
W. Fan, F. Liu, X. Jiang, I. Turner, Some novel numerical techniques for an inverse problem of the multi-term time fractional partial differential equation. J. of Comput. and Appl. Math. 25, (2017), 1618–1638.
R. Garra, R. Gorenflo, F. Polito, Z. Tomovski, Hilfer-Prabhakar derivatives and some applications. Appl. Math. and Comput. 242, (2014), 576–589.
R. Gorenflo, Y. Luchko, Operational method for solving generalized Abel integral of second kind. Integr. Transf. and Spec. Funct. 5, (1997), 47–58.
E.F.D. Goufo, A biomathematical view on the fractional dynamics of cellulose degradation. Fract. Calc. Appl. Anal. 18, No 3 (2015), 554–564; DOI:10.1515/fca-2015-0034 https://www.degruyter.com/view/journals/fca/18/3/fca.18.issue-3.xml.
R. Hilfer, Applications of Fractional Calculus in Physics World Scientific Singapore, (2000).
R. Hilfer, Y. Luchko, Z. Tomovski, Operational method for the solution of fractional differential equations with generalized Riemann-Liouville fractional derivatives. Fract. Calc. Appl. Anal. 12, No 3 (2009), 299–318at http://www.math.bas.bg/complan/fcaa.
V.A. Il’in, How to express basis conditions and conditions for the equiconvergence with trigonometric series of expansions related to non-self-adjoint differential operators. Computers and Math. with Appl. 34, (1997), 641–647.
V.A. Il’in, L.A. Kritskov, Properties of spectral expansion corresponding to non-self-adjoint differential operators. J. of Math. Sci. 116, (2003), 3489–3550.
H. Jiang, F. Liu, I. Turner, K. Burrage, Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain. J. Math. Anal. Appl. 389, (2012), 1117–1127.
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).
M.H. Kim, G.C. Ri, O. Hyong-Chol, Operational method for solving multi-term fractional differential equations with the generalized fractional derivatives. Fract. Calc. Appl. Anal. 17, No 1 (2014), 79–9510.2478/s13540-014-0156-6 https://www.degruyter.com/view/journals/fca/17/1/fca.17.issue-1.xml.
R. Klages, G. Radons, I.M. Sokolov, Anomalous Transport Willey-VCH Verlag GmbH & Co. KGaA Weinheim, (2008).
Z. Li, Y. Liu, M. Yamamoto, Initial-boundary value problems for multi-term time-fractioanl diffusion equations with positive constnt coefficients. Appl. Math. and Comput. 257, (2015), 381–397.
Y. Liu, Strong maximum principle for multi-term time-fractional diffusion equations and its application to an inverse source problem. Computers and Math. with Appl. 73, (2017), 96–108.
C. Li, D. Qian, Y.Q. Chen, On Riemann-Liouville and Caputo derivatives. Discrete Dynam. in Nature and Society (2011), #562494, 15.
Y.F. Luchko, The exact solution of certain differential equations of fractional order by using operational calculus. Computers and Math. with Appl. 29, (1995), 73–85.
Y. Luchko, R. Gorenflo, An operational method for solving fractional differential equations with the Cpauto dervatives. Acta Mathematica 24, (1999), 207–233.
S.Y. Lukashchuk, Conservation laws for time-fractional subdiffusion and diffusion-wave equations. Nonlinear Dyn. 80, (2015), 791–802.
B.P. Moghaddam, J.A.T. Machado, A stable three-level explicit spline finite difference scheme for a class of nonlinear time variable order fractional partial differential equations. Computers and Math. with Appl. 73, (2017), 1262–1269.
A.Y. Mokin, On a family of Initial-boundary value problems for the heat equation. Diff. Equations 45, (2009), 126–141.
A.Y. Mokin, Applications of nonclassical separation of variables to a nonlocal heat problem. Diff. Equations 49, (2013), 59–67.
I. Podlubny, Fractional Differential Equations Academic Press San Diego, (1999).
M.S. Salakhitdinov, E.T. Karimov, Direct and inverse source problems for two-term time-fractional difusion equation with Hilfer derivative. Uzb. Math. J. 4, (2017), 140–149.
M.S. Salakhitdinov, E.T. Karimov, Corrigendum to “Direct and inverse source problems for two-term time-fractional difusion equation with Hilfer derivative”[Uzb. Math. J. 4 (2017), 140-149]. Uzb. Math. J. 3, (2018), 139.
S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications Gordon and Breach Science Publishers Amsterdam, (1993).
C. Sun, G. Li, X. Jia, Simultaneous inversion for the diffusion and source coefficients in the multi-term TFDE. Inverse Problems in Sci. and Engin. 336, (2018), 114–126.
H. Sun, Y. Zhang, D. Baleanu, W. Chen, Y. Chen, A new collection of real world applications of fractional calculus in science and engineering. Commun. Nonlinear Sci. Numer. Simul. 64, (2018), 213–231.
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Ali, M., Aziz, S. & Malik, S.A. Inverse Problem for a Multi-Term Fractional Differential Equation. Fract Calc Appl Anal 23, 799–821 (2020). https://doi.org/10.1515/fca-2020-0040
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DOI: https://doi.org/10.1515/fca-2020-0040