An inverse fractional diffusion problem of source identification type. (English) Zbl 1543.65146
Summary: One of the major objectives in the field of inverse problems is to construct a space-dependent term of an unknown source in a stable manner. Many different fields of science have used this source term, especially when the extra condition is accompanied by noise. We focus on a one-dimensional situation in a fractional diffusion problem to recover the source term that is unknown. In order to accomplish this, the major problem was transformed into an equation of operator form in a way that allowed the unique solvability of this equation to be established. The Ritz-Galerkin method is then used to implement a numerical solution to the inverse problem. In conjunction with the Galerkin method, shifted Bernoulli wavelets (BWs) are used as basis functions to reduce the main problem to an algebraic equation. It is essential to include some kind of regularization method within the numerical algorithm to obtain a stable solution to the resulting linear system. We concluded by giving numerical examples that demonstrate the proposed algorithm’s validity and efficiency in the presence of noise.
MSC:
| 65M32 | Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs |
| 65M30 | Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65T60 | Numerical methods for wavelets |
| 65J20 | Numerical solutions of ill-posed problems in abstract spaces; regularization |
| 65F22 | Ill-posedness and regularization problems in numerical linear algebra |
| 35R30 | Inverse problems for PDEs |
| 35R25 | Ill-posed problems for PDEs |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 26A33 | Fractional derivatives and integrals |
| 35R11 | Fractional partial differential equations |
Keywords:
Caputo derivative; ill-posed problem; regularization approach; Ritz-Galerkin procedure; time-fractional inverse problemSoftware:
Regularization toolsReferences:
| [1] | Adams, RA; Fournier, JJ, Sobolev Spaces, 2013, Amsterdam: Elsevier, Amsterdam |
| [2] | Baleanu, D.; Machado, J.; Luo, A., Fractional Dynamics and Control, 2011, Cham: Springer Science & Business Media, Cham |
| [3] | Changpin, L.; Yujiang, W.; Ruisong, Y., Recent Advances in Applied Nonlinear Dynamics with Numerical Analysis: Fractional Dynamics, Network Dynamics, Classical Dynamics and Fractal Dynamics with their Numerical Simulations, 2013, Singapore: World Scientific, Singapore · Zbl 1261.65005 |
| [4] | Chen, A.; Li, C., Numerical solution of fractional diffusion-wave equation, Numer. Funct. Anal. Optim., 37, 1, 19-39, 2016 · Zbl 1382.65236 · doi:10.1080/01630563.2015.1078815 |
| [5] | Deng, W.; Hesthaven, JS, Local discontinuous Galerkin methods for fractional diffusion equations, ESAIM Math. Model. Numer. Anal., 47, 6, 1845-1864, 2013 · Zbl 1282.35400 · doi:10.1051/m2an/2013091 |
| [6] | Dou, F.; Hon, Y., Kernel-based approximation for Cauchy problem of the time-fractional diffusion equation, Eng. Anal. Bound. Elem., 36, 9, 1344-1352, 2012 · Zbl 1352.65309 · doi:10.1016/j.enganabound.2012.03.003 |
| [7] | Ebel, A.; Davitashvili, T., Air, Water and Soil Quality Modelling for Risk and Impact Assessment, 2007, Cham: Springer, Cham · Zbl 1187.35261 · doi:10.1007/978-1-4020-5877-6 |
| [8] | Engl, HW; Hanke, M.; Neubauer, A., Regularization of Inverse Problems, 1996, Cham: Springer Science & Business Media, Cham · Zbl 0859.65054 · doi:10.1007/978-94-009-1740-8 |
| [9] | Hansen, PC, Regularization tools version 4.0 for Matlab 7.3, Numer. Algorithms, 46, 2, 189-194, 2007 · Zbl 1128.65029 · doi:10.1007/s11075-007-9136-9 |
| [10] | Hansen, PC; O’Leary, DP, The use of the L-curve in the regularization of discrete ill-posed problems, SIAM J. Sci. Comput., 14, 6, 1487-1503, 1993 · Zbl 0789.65030 · doi:10.1137/0914086 |
| [11] | Jin, B.; Rundell, W., An inverse problem for a one-dimensional time-fractional diffusion problem, Inverse Probl., 28, 7, 2012 · Zbl 1247.35203 · doi:10.1088/0266-5611/28/7/075010 |
| [12] | Li, C.; Zhao, Z.; Chen, Y., Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion, Comput. Math. Appl., 62, 3, 855-875, 2011 · Zbl 1228.65190 · doi:10.1016/j.camwa.2011.02.045 |
| [13] | Li, G.; Zhang, D.; Jia, X.; Yamamoto, M., Simultaneous inversion for the space-dependent diffusion coefficient and the fractional order in the time-fractional diffusion equation, Inverse Probl., 29, 6, 2013 · Zbl 1281.65125 · doi:10.1088/0266-5611/29/6/065014 |
| [14] | Lin, Y.; Xu, C., Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225, 2, 1533-1552, 2007 · Zbl 1126.65121 · doi:10.1016/j.jcp.2007.02.001 |
| [15] | Ma, YK; Prakash, P.; Deiveegan, A., Generalized Tikhonov methods for an inverse source problem of the time-fractional diffusion equation, Chaos Solitons Fractals, 108, 39-48, 2018 · Zbl 1390.35425 · doi:10.1016/j.chaos.2018.01.003 |
| [16] | Metzler, R.; Glöckle, WG; Nonnenmacher, TF, Fractional model equation for anomalous diffusion, Phys. A Stat. Mech. Appl., 211, 1, 13-24, 1994 · doi:10.1016/0378-4371(94)90064-7 |
| [17] | Murio, DA, Stable numerical solution of a fractional-diffusion inverse heat conduction problem, Comput. Math. Appl., 53, 10, 1492-1501, 2007 · Zbl 1152.65463 · doi:10.1016/j.camwa.2006.05.027 |
| [18] | Murio, DA, Time fractional IHCP with Caputo fractional derivatives, Comput. Math. Appl., 56, 9, 2371-2381, 2008 · Zbl 1165.65386 · doi:10.1016/j.camwa.2008.05.015 |
| [19] | Oldham, K.; Spanier, J., The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, 1974, Amsterdam: Elsevier, Amsterdam · Zbl 0292.26011 |
| [20] | Podlubny, I., Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to methods of their solution and some of their applications, 1998, Amsterdam: Elsevier, Amsterdam · Zbl 0922.45001 |
| [21] | Qian, Z., Optimal modified method for a fractional-diffusion inverse heat conduction problem, Inverse Probl. Sci. Eng., 18, 4, 521-533, 2010 · Zbl 1195.65125 · doi:10.1080/17415971003624348 |
| [22] | Sakamoto, K.; Yamamoto, M., Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382, 1, 426-447, 2011 · Zbl 1219.35367 · doi:10.1016/j.jmaa.2011.04.058 |
| [23] | Shahsahebi, F.S., Damirchi, J., Janmohammadi, A.: The Ritz-Galerkin procedure for an inverse space-dependent heat source problem. Jpn. J. Ind. Appl. Math. 1-19 (2021) · Zbl 07378253 |
| [24] | Tatar, S.; Ulusoy, S., An inverse source problem for a one-dimensional space-time fractional diffusion equation, Appl. Anal., 94, 11, 2233-2244, 2015 · Zbl 1327.35416 · doi:10.1080/00036811.2014.979808 |
| [25] | Tikhonov, A., Arsenin, V.Y.: Methods for Solving Ill-Posed Problems. New York, p. 12 (1977) · Zbl 0354.65028 |
| [26] | Tuan, NH; Ngoc, TB; Zhou, Y.; O’Regan, D., On existence and regularity of a terminal value problem for the time fractional diffusion equation, Inverse Probl., 36, 5, 2020 · Zbl 1469.35233 · doi:10.1088/1361-6420/ab730d |
| [27] | Tuan, NH; Huynh, LN; Ngoc, TB; Zhou, Y., On a backward problem for nonlinear fractional diffusion equations, Appl. Math. Lett., 92, 76-84, 2019 · Zbl 1524.35723 · doi:10.1016/j.aml.2018.11.015 |
| [28] | Wang, T.; Wang, YM, A modified compact ADI method and its extrapolation for two-dimensional fractional subdiffusion equations, J. Appl. Math. Comput., 52, 1-2, 439-476, 2016 · Zbl 1354.65173 · doi:10.1007/s12190-015-0949-8 |
| [29] | Wei, T.; Li, X.; Li, Y., An inverse time-dependent source problem for a time-fractional diffusion equation, Inverse Probl., 32, 8, 2016 · Zbl 1351.65072 · doi:10.1088/0266-5611/32/8/085003 |
| [30] | Wei, T.; Wang, J., A modified quasi-boundary value method for an inverse source problem of the time-fractional diffusion equation, Appl. Numer. Math., 78, 95-111, 2014 · Zbl 1282.65141 · doi:10.1016/j.apnum.2013.12.002 |
| [31] | Wei, T.; Zhang, Z., Reconstruction of a time-dependent source term in a time-fractional diffusion equation, Eng. Anal. Bound. Elem., 37, 1, 23-31, 2013 · Zbl 1351.35267 · doi:10.1016/j.enganabound.2012.08.003 |
| [32] | Wojtaszczyk, P., A Mathematical Introduction to Wavelets, 1997, Cambridge: Cambridge University Press, Cambridge · Zbl 0865.42026 · doi:10.1017/CBO9780511623790 |
| [33] | Xu, Q.; Hesthaven, JS, Discontinuous Galerkin method for fractional convection-diffusion equations, SIAM J. Numer. Anal., 52, 1, 405-423, 2014 · Zbl 1297.26018 · doi:10.1137/130918174 |
| [34] | Yang, F.; Fu, CL, The quasi-reversibility regularization method for identifying the unknown source for time-fractional diffusion equation, Appl. Math. Model., 39, 5-6, 1500-1512, 2015 · Zbl 1443.35199 · doi:10.1016/j.apm.2014.08.010 |
| [35] | Yang, F.; Fu, CL; Li, XX, The inverse source problem for time-fractional diffusion equation: stability analysis and regularization, Inverse Probl. Sci. Eng., 23, 6, 969-996, 2015 · Zbl 1329.35357 · doi:10.1080/17415977.2014.968148 |
| [36] | Yeganeh, S.; Mokhtari, R.; Hesthaven, JS, Space-dependent source determination in a time-fractional diffusion equation using a local Discontinuous Galerkin method, BIT Numer. Math., 57, 3, 685-707, 2017 · Zbl 1377.65124 · doi:10.1007/s10543-017-0648-y |
| [37] | Zhou, J.; Li, H.; Xu, Y., Ritz-Galerkin method for solving an inverse problem of parabolic equation with moving boundaries and integral condition, Appl. Anal., 98, 10, 1741-1755, 2019 · Zbl 1418.65121 · doi:10.1080/00036811.2018.1434512 |
| [38] | Zhou, Y.; Wang, J.; Zhang, L., Basic Theory of Fractional Differential Equations, 2016, Singapore: World Scientific, Singapore · Zbl 1360.34003 · doi:10.1142/10238 |
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