Simultaneous determination of a source term and diffusion concentration for a multi-term space-time fractional diffusion equation
Abstract
An inverse problem of determining a time dependent source term along with diffusion/temperature concentration from a non-local over-specified condition for a space-time fractional diffusion equation is considered. The space-time fractional diffusion equation involve Caputo fractional derivative in space and Hilfer fractional derivatives in time of different orders between 0 and 1. Under certain conditions on the given data we proved that the inverse problem is locally well-posed in the sense of Hadamard. Our method of proof based on eigenfunction expansion for which the eigenfunctions (which are Mittag-Leffler functions) of fractional order spectral problem and its adjoint problem are considered. Several properties of multinomial Mittag-Leffler functions are proved.
Keyword : inverse problem, fractional derivative, Bi-orthogonal system of functions, multinomial Mittag-Leffler function
How to Cite
Malik, S. A., Ilyas, A., & Samreen, A. (2021). Simultaneous determination of a source term and diffusion concentration for a multi-term space-time fractional diffusion equation. Mathematical Modelling and Analysis, 26(3), 411-431. https://doi.org/10.3846/mma.2021.11911
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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A. Mohebbi and M. Abbasi. A fourth-order compact difference scheme for the parabolic inverse problem with an overspecification at a point. Inverse Probl. Sci. Eng., 23:457–478, 2015. https://doi.org/10.1080/17415977.2014.922075
M. Di Paola. Complex Fractional Moments and Their Use in Earthquake Engineering: in Encyclopedia of Earthquake Engineering. Springer Berlin Heidelberg, 2014.
Y. Povstenko. Linear fractional diffusion-wave equation for scientists and engineers. Springer Internatinal Publishing Switzerland, 2015. https://doi.org/10.1007/978-3-319-17954-4
G.S. Samko, A.A. Kilbas and D.I. Marichev. Fractional Integrals and Derivatives: Theory and applications. Gordon and Breach Science Publishers, 1993.
H. Sun, Y. Zhang, D. Baleanu, W. Chen and Y. Chen. A new collection of real world applications of fractional calculus in science and engineering. Commun Nonlinear Sci Numer Simul, 64:213–231, 2018. https://doi.org/10.1016/j.cnsns.2018.04.019
Z. Tomovski, R. Hilfer and H.M. Srivastava. Fractional and operational calculus with generalized fractional derivative operators and MittagLeffler type functions. Integral Transforms and Special Functions, 21:797–814, 2010. https://doi.org/10.1080/10652461003675737
P.J. Torvik and R.L. Bagley. On the appearance of the fractional derivative in the behavior of real materials. Journal of Applied Mechanics, 51:294–298, 1984. https://doi.org/10.1115/1.3167615
K. Šišková and M. Slodička. Recognition of a time-dependent source in a time-fractional wave equation. Applied Numerical Mathematics, 121:1–17, 2017. https://doi.org/10.1016/j.apnum.2017.06.005
T. Wei, L. Sun and Y. Li. Uniqueness for an inverse space-dependent source term in a multi-dimensional time-fractional diffusion equation. Applied Mathematics Letters, 61:108–113, 2016. https://doi.org/10.1016/j.aml.2016.05.004
G.H. Weiss. Aspects and Applications of the Random Walk. North-Holland, Amsterdam, 1994.
T.S. Aleroev, M. Kirane and Y.F. Tang. The boundary-value problem for a differential operator of fractional order. J. Math. Sci., 194:499–512, 2013. https://doi.org/10.1007/s10958-013-1543-y
M. Ali, S. Aziz and S.A. Malik. Inverse source problem for a space-time fractional diffusion equation. Fract. Calc. Appl. Anal., 21:844–863, 2018. https://doi.org/10.1515/fca-2018-0045
M. Ali, S. Aziz and S.A. Malik. Inverse source problem for a spacetime fractional diffusion equation: Application of fractional Sturm-Liouville operator. Mathematical Methods for applied Sciences, 41:2733–2744, 2018. https://doi.org/10.1002/mma.4776
M. Ali, S. Aziz and S.A. Malik. Inverse source problems for a space-time fractional diffusion equation. Inverse Problems in Science and Engineering, 122:1– 22, 2019. https://doi.org/10.1080/17415977.2019.1597079
M. Ali and S.A. Malik. An inverse problem for a family of time fractional diffusion equations. Inverse Problems in Science and Engineering, 25:1299–1322, 2017. https://doi.org/10.1080/17415977.2016.1255738
S. Aziz and S.A. Malik. Identification of an unknown source term for a time fractional fourth-order parabolic equation. Electronic Journal of Differential Equations, 293:1–28, 2016.
M. Caputo, J.M. Carcione and M.A.B. Botelho. Modeling extreme-event precursors with the fractional diffusion equation. Fract. Calc. Appl. Anal., 18:208–222, 2015. https://doi.org/10.1515/fca-2015-0014
M.M. Dzhrbashyan and A.B. Nersesyan. Expansions in certain bi-orthogonal systems and boundary-value problems for differential equations of fractional order. Tr. Mosk. Mat. Obs., 10:89–179, 1961.
R. Hilfer. Applications of Fractional Calculus in Physics. World Scientific, 2000. https://doi.org/10.1142/3779
F. Höfling and T. Franosch. Anomalous transport in the crowded world of biological cells. Rep. Prog. Phys., 76:046602, 2013. https://doi.org/10.1088/0034-4885/76/4/046602
D.B. Hughes. Random Walks and Random Environments, Volume I: Random Walks. Oxford University Press, 1995.
C. Ionescu, A. Lopes, D. Copot, J.A.T. Machado and J.H.T. Bates. The role of fractional calculus in modelling biological phenomena: A review. Commun. Nonlinear Sci. Numer. Simulat., 51:141–159, 2017. https://doi.org/10.1016/j.cnsns.2017.04.001
A.A. Kilbas, H.M. Srivastava and J.J. Trujillo. Theory and applications of fractional differential equations. Elsevier Science Limimited, 204, 2006.
N. Kinash and J. Janno. An inverse problem for a generalized fractional derivative with an application in reconstruction of time- and space-dependent sources in fractional diffusion and wave equations. Mathematics, 7(12):1138, 2019. https://doi.org/10.3390/math7121138
M. Kirane, S.A. Malik and M.A. Al-Gwaiz. An inverse source problem for a two dimensional time fractional diffusion equation with nonlocal boundary conditions. Math. Meth. Appl. Sci., 36:1056–1069, 2013. https://doi.org/10.1002/mma.2661
Z. Li, Y. Liu and M. Yamamoto. Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients. Applied Mathematics and Composition, 257:381–397, 2015. https://doi.org/10.1016/j.amc.2014.11.073
Z. Li and M. Yamamoto. Uniqueness for inverse problems of determining orders of multi-term time- fractional derivatives of diffusion equation. Applicable Analysis, 94:570–579, 2015. https://doi.org/10.1080/00036811.2014.926335
H. Lopushanska and V. Rapita. Inverse coefficient problem for the semi-linear fractional telegraph equation. Electronic Journal of Differential Equations, 153:1–13, 2015.
Y. Luchko. Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation. J. Math. Anal. Appl., 374:538–548, 2011. https://doi.org/10.1016/j.jmaa.2010.08.048
Y. Luchko and R. Gorenflo. An operational method for solving fractional differential equations with the Caputo derivatives. Acta Mathematica Vietnamica, 24:207–233, 1999.
J.A.T. Machado and A.M. Lopes. Relative fractional dynamics of stock markets. Nonlinear Dyn., 86:1613–1619, 2016. https://doi.org/10.1007/s11071-016-2980-1
F. Mainardi. Fractional calculus and waves in linear viscoelaticity. Imperial College Press, 2010. https://doi.org/10.1142/p614
A.K. Mani and M.D. Narayanan. Analytical and numerical solution of an n-term fractional nonlinear dynamic oscillator. Nonlinear Dynamics, 1000:999–1012, 2020. https://doi.org/10.1007/s11071-020-05539-0
R. Metzler, J.H. Jeon, A.G. Cherstvy and E. Barkai. Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking. Physical Chemistry Chemical Physics, 16:24128– 24164, 2014. https://doi.org/10.1039/C4CP03465A
A. Mohebbi and M. Abbasi. A fourth-order compact difference scheme for the parabolic inverse problem with an overspecification at a point. Inverse Probl. Sci. Eng., 23:457–478, 2015. https://doi.org/10.1080/17415977.2014.922075
M. Di Paola. Complex Fractional Moments and Their Use in Earthquake Engineering: in Encyclopedia of Earthquake Engineering. Springer Berlin Heidelberg, 2014.
Y. Povstenko. Linear fractional diffusion-wave equation for scientists and engineers. Springer Internatinal Publishing Switzerland, 2015. https://doi.org/10.1007/978-3-319-17954-4
G.S. Samko, A.A. Kilbas and D.I. Marichev. Fractional Integrals and Derivatives: Theory and applications. Gordon and Breach Science Publishers, 1993.
H. Sun, Y. Zhang, D. Baleanu, W. Chen and Y. Chen. A new collection of real world applications of fractional calculus in science and engineering. Commun Nonlinear Sci Numer Simul, 64:213–231, 2018. https://doi.org/10.1016/j.cnsns.2018.04.019
Z. Tomovski, R. Hilfer and H.M. Srivastava. Fractional and operational calculus with generalized fractional derivative operators and MittagLeffler type functions. Integral Transforms and Special Functions, 21:797–814, 2010. https://doi.org/10.1080/10652461003675737
P.J. Torvik and R.L. Bagley. On the appearance of the fractional derivative in the behavior of real materials. Journal of Applied Mechanics, 51:294–298, 1984. https://doi.org/10.1115/1.3167615
K. Šišková and M. Slodička. Recognition of a time-dependent source in a time-fractional wave equation. Applied Numerical Mathematics, 121:1–17, 2017. https://doi.org/10.1016/j.apnum.2017.06.005
T. Wei, L. Sun and Y. Li. Uniqueness for an inverse space-dependent source term in a multi-dimensional time-fractional diffusion equation. Applied Mathematics Letters, 61:108–113, 2016. https://doi.org/10.1016/j.aml.2016.05.004
G.H. Weiss. Aspects and Applications of the Random Walk. North-Holland, Amsterdam, 1994.