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BIT Numerical Mathematics
Erschienen in: BIT Numerical Mathematics 1/2019

06.10.2018

The time-fractional diffusion inverse problem subject to an extra measurement by a local discontinuous Galerkin method

verfasst von: Samaneh Qasemi, Davood Rostamy, Nazdar Abdollahi

Erschienen in: BIT Numerical Mathematics | Ausgabe 1/2019

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Abstract

This paper deals with an inverse problem of identifying a space dependent coefficient in a time-fractional diffusion equation on a finite domain with final observation. The existence and uniqueness of this inverse problem are proved. A numerical scheme is proposed to solve the problem. The main idea of the proposed scheme is approximating the time fractional derivative by Diethelm’s quadrature formula and use the local discontinuous Galerkin method in space variable. Also, an error estimate for this problem is presented. Finally, two numerical example is studied to demonstrate the accuracy and efficiency of the proposed method.
Vorheriger Artikel Stable application of Filon–Clenshaw–Curtis rules to singular oscillatory integrals by exponential transformations
Nächster Artikel Stability and convergence of a conservative finite difference scheme for the modified Hunter–Saxton equation

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Literatur
1.
Al-Refai, M., Luchko, Y.: Maximum principles for the fractional diffusion equations with the Riemann–Liouville fractional derivative and their applications. Fract. Calc. Appl. Anal. 17, 483–498 (2014)MathSciNetCrossRefMATH
2.
Arnold, D.N., Brezzi, F., Cockburn, B., Marini, D.: Discontinuous Galerkin methods for elliptic problems. In: Discontinuous Galerkin Methods: Theory, Computation and Applications, vol. 11, pp. 89–101 (2000)
3.
Asadzadeh, M., Schatz, A., Wendland, W.: Asymptotic error expansions for the finite element method for second order elliptic problems in \({ R} _N, N>=2\), I: local interior expansions. SIAM J. Numer. Anal. 48, 2000–2017 (2010)MathSciNetCrossRefMATH
4.
Asadzadeh, M., Beilina, L.: A posteriori error analysis in a globally convergent numerical method for a hyperbolic coefficient inverse. Inverse Probl. 26, 115–127 (2010)CrossRefMATH
5.
Bagley, R., Torvik, P.: A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheol. 27, 201–210 (1983)CrossRefMATH
6.
Bassi, F., Rebay, S.: A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier–Stokes equations. J. Comput. Phys. 131, 267–279 (1997)MathSciNetCrossRefMATH
7.
Brunner, H., Han, H., Yin, D.: The maximum principle for time-fractional diffusion equations and its application. Numer. Func. Anal. Optim. 36, 1307–1321 (2015)MathSciNetCrossRefMATH
8.
Cannon, J.R., Yin, H.-M.: A class of nonlinear non-classical parabolic equations. J. Differ. Equ. 79, 266–288 (1989)CrossRefMATH
9.
Cannon, J.R., Yin, H.-M.: Numerical solutions of some parabolic inverse problems. Numer. Methods Partial Differ. Equ. 6, 177–191 (1990)MathSciNetCrossRefMATH
10.
Castillo, P.: An optimal estimate for the local discontinuous Galerkin method. In: Discontinuous Galerkin Methods: Theory, Computation and Applications. Lecture Notes in Computational Science and Engineering, vol. 11, pp. 285–290 (2000)
11.
Chen, Q., Liu, J.J.: Solving an inverse parabolic problem by optimization from final measurement data. J. Comput. Appl. Math. 193, 183–203 (2006)MathSciNetCrossRefMATH
12.
Cheng, J., Nakagawa, J., Yamamoto, M., Yamazaki, T.: Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation. Inverse Probl. 25(11), 551–562 (2009)MathSciNetCrossRefMATH
13.
Cockburn, B., Shu, C.W.: The local discontinuous Galerkin method for time dependent converction-diffusion systems. SIAM J. Numer. Anal. 35, 2440–246 (1998)MathSciNetCrossRefMATH
14.
Cockburn, B., Kanschat, G., Perugia, I., Schötzau, D.: Superconvergence of the local discontinuous Galerkin method for elliptic problems on cartesian grids. SIAM J. Numer. Anal. 39(1), 264–285 (2001)MathSciNetCrossRefMATH
15.
16.
Diethelm, K.: Numerical Approximation for Finite-Part Integrals with Generalized Compound Quadrature Formulae. Hildesheimer Informatikberichte (1995)
17.
Diethelm, K.: An algorithm for the numerical solution of differential equations of fractional order. Electron. Trans. Numer. Anal. 5, 1–6 (1997)MathSciNetMATH
18.
Douglas, J., Dupont, J.T.: Interior penalty procedures for elliptic and parabolic Galerkin methods. Lecture Notes in Physics, vol. 58. Springer, Berlin (1976)
19.
Furati, KhM, Iyiola, O., Kirane, M.: An inverse problem for a generalized fractional diffusion. Appl. Math. Comput. 249, 24–31 (2014)MathSciNetMATH
20.
Georgoulis, E.H.: hp-version interior penalty discontinuous Galerkin finite element methods on anisotropic meshes. Int. J. Numer. Anal. Model. 3, 52–79 (2006)MathSciNetMATH
21.
Guo, B., Pu, X., Huang, F.: Fractional Partial Differential Equations and Their Numerical Solutions. World Scientific, Singapore (2015)CrossRefMATH
22.
23.
Ismailov, M.I., Çiçek, M.: Inverse Source Problem for a time-fractional diffusion equation with nonlocal boundary conditions. Appl. Math. Model. 40, 4891–4899 (2016)MathSciNetCrossRef
24.
Jin, B., Rundell, W.: An inverse problem for a one-dimensional time-fractional diffusion problem. Inverse Probl. 28(7), 234–245 (2012)MathSciNetCrossRefMATH
25.
Larsson, S., Racheva, M., Saedpanah, F.: Discontinuous Galerkin method for an integro-differential equation modeling dynamic fractional order viscoelasticity. Comput. Methods Appl. Mech. Eng. 283, 196–209 (2015)MathSciNetCrossRefMATH
26.
Li, X.J., Xu, C.J.: Existence and uniqeness of the weak solution of the space–time fractional diffusion equation and a spectral method approximation. Commun. Comput. Phys. 8, 1016–1051 (2010)MathSciNetMATH
27.
28.
Liu, L.L., Yamamoto, M., Yan, L.: On the uniqueness and reconstruction for an inverse problem of the fractional diffusion process. Appl. Numer. Math. 87, 1–19 (2015)MathSciNetCrossRefMATH
29.
Luchko, Y.: Maximum principle for the generalized time-fractional diffusion equation. J. Math. Anal. Appl. 351, 218–223 (2009)MathSciNetCrossRefMATH
30.
Metzler, R., Klafter, J.: The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A Math. Gen. 37, 161–208 (2004)MathSciNetCrossRefMATH
31.
Metzler, R., Klafter, J.: The random walks guide to an omalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)CrossRefMATH
32.
Murio, D.A.: Implicit finite difference approximation for time fractional diffusion equations. Comput. Math. Appl. 56, 1138–1145 (2008)MathSciNetCrossRefMATH
33.
34.
Oldham, K.B., Spanier, J.: The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order. Academic Press, New York (1974)MATH
35.
Prilepko, A.I., Orlovsky, D.G., Vasian, I.A.: Methods for Solving Inverse Problems in Mathematical Physics. Marcel Dekker Inc., New York (2000)
36.
Podlubny, I.: Fractiorlal Differential Eqnations. Academic Press, New York (1999)
37.
Rostamy, D., Abdollahi, N.: Gegenbauer cardinal functions for the inverse source parabolic problem with a time-fractional diffusion equation. J. Comput. Theor. Transp. 46, 307–329 (2017)MathSciNetCrossRef
38.
Rostamy, D.: The new streamline diffusion for the 3D coupled Schrodinger equations with a cross-phase modulation. ANZIAM J. 55, 51–78 (2013)MathSciNetCrossRefMATH
39.
Rostamy, D., Qasemi, S.: Discontinuous Petrov-Galerkin and Bernstein-Legendre polynomials method for solving fractional damped heat- and wave-like equations. Transp. Theory Stat. Phys. 44, 1–23 (2015)MathSciNet
40.
Scalas, E., Gorenflo, R., Mainardi, F.: Fractional calculus and continuous-time finance. Phys. A Stat. Mech. Appl. 284, 376–84 (2000)MathSciNetCrossRef
41.
Shao, L., Feng, X., He, Y.: The local discontinuous Galerkin finite element method for Burgers equation. Math. Comput. Model. 54, 2943–2954 (2011)MathSciNetCrossRefMATH
42.
43.
Tatar, S., Ulusoy, S.: An inverse source problem for a one-dimensional space-time fractional diffusion equation. Appl. Anal. 94, 2233–2244 (2015)MathSciNetCrossRefMATH
44.
45.
Wei, T., Zhang, Z.Q.: Reconstruction of a time- dependent source term in a time-fractional diffusion equation. Eng. Anal. Bound. Elem. 37, 23–31 (2013)MathSciNetCrossRefMATH
46.
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)MathSciNetCrossRefMATH
47.
Wei, T., Li, X.L., Li, Y.S.: An inverse time-dependent source problem for a time-fractional diffusion equation. Ineverse Probl. 32, 085003 (2016)MathSciNetCrossRefMATH
48.
Wei, T., Zhang, Z.Q.: Robin coefficient identification for a time-fractional diffusion equation. Inverse Probl. Sci. Eng. 24, 647–666 (2016)MathSciNetCrossRefMATH
49.
Wu, B., Wu, S.: Existence and uniqueness of an inverse source problem for a fractional integrodifferential equation. Comput. Math. Appl. 68, 1123–1136 (2014)MathSciNetCrossRefMATH
50.
Xu, Y., Shu, C.W.: Local discontinuous Galerkin method for the Camassa–Holm equation. SIAM J. Numer. Anal. 46, 1998–2021 (2008)MathSciNetCrossRefMATH
51.
Yeganeh, S., Mokhtari, R., Hesthaven, J.S.: Space-dependent source determination in a time-fractional diffusion equation using a local discontinuous Galerkin method. BIT Numer. Math. 57, 685–707 (2017)MathSciNetCrossRefMATH
52.
Zhang, Y., Meerschaert, M.M., Baeumer, B.: Particle tracking for time-fractional diffusion. Phys. Rev. E. 78, 1–7 (2008)MATH
53.
Metadaten
Titel
The time-fractional diffusion inverse problem subject to an extra measurement by a local discontinuous Galerkin method
verfasst von
Samaneh Qasemi
Davood Rostamy
Nazdar Abdollahi
Publikationsdatum
06.10.2018
Verlag
Springer Netherlands
Erschienen in
BIT Numerical Mathematics / Ausgabe 1/2019
Print ISSN: 0006-3835
Elektronische ISSN: 1572-9125
DOI
https://doi.org/10.1007/s10543-018-0731-z

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