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An inverse problem for a nonlinear diffusion equation with time-fractional derivative

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Published/Copyright: March 22, 2016
Journal of Inverse and Ill-posed Problems
From the journal Journal of Inverse and Ill-posed Problems Volume 25 Issue 2

Abstract

A nonlinear time-fractional inverse coefficient problem is considered. The unknown coefficient depends on the solution. It is proved that the direct problem has a unique solution. Afterwards the continuous dependence of the solution of the corresponding direct problem on the coefficient is proved. Then the existence of a quasi-solution of the inverse problem is obtained in the appropriate class of admissible coefficients.

Keywords: Fractional diffusion; inverse problem; quasi-solution; class of admissible coefficients
MSC 2010: 34A12; 35R11; 35R30; 65F22

Funding statement: This research has been supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK) through the project No. 113F373, also by the Zirve University Research Fund.

A Appendix

In this Appendix, for the convenience of readers, we provide some basic results and theorems used in the paper.

First, the example in [32, p. 845, (a) of Section 32.1b] states that if the operator A:X→X* is monotone and hemicontinuous, then A is a maximal monotone operator, and [32, p. 821, Proposition 31.5] implies that A is an m-accretive operator.

Proposition 1

Proposition 1 ([32, p. 586, Proposition 27.6 (a)])

Let A:X→X* be an operator on a real reflexive Banach space X. Then if A is monotone and hemicontinuous, then A is pseudomonotone.

Theorem 2

Theorem 2 ([32, p. 819, Theorem 31.A])

Let A:X→X* be an m-accretive operator. Then the following problem has a unique continuous solution:

(A.1){u′⁢(t)+A⁢u⁢(t)=0,0<t<∞,u⁢(0)=u0.
Proposition 3

Proposition 3 ([32, p. 821, Proposition 31.5])

Let A:D⁢(A)⊆H→H be an operator on a real Hilbert space H. Then the following properties of A are mutually equivalent:

  1. A is monotone and R⁢(I+A)=H.

  2. A is m-accretive.

  3. A is maximal monotone.

Lemma 4

Lemma 4 (Alikhanov inequality, [1, p. 661])

For any function v⁢(t) absolutely continuous on [0,T], one has the inequality

(A.2)v⁢(t)⁢∂β∂⁡tβ⁢v⁢(t)≥12⁢∂β∂⁡tβ⁢v2⁢(t).

References

[1] Alikhanov A. A., A priori estimates for solutions of boundary value problems for fractional order equations, Differ. Equ. 46 (2010), no. 5, 660–666. 10.1134/S0012266110050058Search in Google Scholar

[2] Bazhlekova E., Existence and uniqueness results for a fractional evolution equation in Hilbert space, Fract. Calc. Appl. Anal. 15 (2012), no. 2, 232–243. 10.2478/s13540-012-0017-0Search in Google Scholar

[3] Cheng J., Nakagawa J., Yamamoto M. and Yamazaki T., Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation, Inverse Problems 25 (2009), 115–131. 10.1088/0266-5611/25/11/115002Search in Google Scholar

[4] Del-Castillo-Negrete D., Carreras B. A. and Lynch E., Front dynamics in reaction-diffusion systems with Lévy flights: A fractional diffusion approach, Phys. Rev. Lett. 91 (2003), no. 4, Article ID 018302. 10.1103/PhysRevLett.91.018302Search in Google Scholar PubMed

[5] Demengel F. and Demengel G., Functional Spaces for the Theory of Elliptic Partial Differential Equations, Springer, London, 2012. 10.1007/978-1-4471-2807-6Search in Google Scholar

[6] de Ramos E. N., de Delgado P. L., de Souza R. E., Engelsberg M., Miranda M. and Silva M. A., Concentration-dependent diffusivity and anomalous diffusion: A magnetic resonance imaging study of water ingress in porous zeolite, Phys. Rev. E (3) 73 (2006), Article ID 011204. 10.1103/PhysRevE.73.011204Search in Google Scholar PubMed

[7] Engl H. W., Kunisch K. and Neubauer A., Convergence rates for Tikhonov regularisation of non-linear ill-posed problems, Inverse Problems 5 (1989), 523–540. 10.1088/0266-5611/5/4/007Search in Google Scholar

[8] Fomin S., Chugunov V. and Hashida T., Application of fractional differential equations for modeling the anomalous diffusion of contaminant from fracture into porous rock matrix with bordering alteration zone, Transp. Porous Media 81 (2010), 187–205. 10.1007/s11242-009-9393-2Search in Google Scholar

[9] Fomin S., Chugunov V. and Hashida T., Mathematical modeling of anomalous diffusion in porous media, Fract. Differ. Calc. 1 (2011), no. 1, 1–28. 10.7153/fdc-01-01Search in Google Scholar

[10] Kilbas A. A., Srivastava H. M. and Trujillo J. J., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. Search in Google Scholar

[11] Li G., Zhang D., Jia X. and Yamamato M., Simultaneous inversion for the space-dependent diffusion coefficient and the fractional order in the time fractional diffusion equation, Inverse Problems 29 (2013), Article ID 065014. 10.1088/0266-5611/29/6/065014Search in Google Scholar

[12] Metzler R. and Klafter J., The random walks guide to anomalous diffusion: A fractional dynamics approach, Physics Rep. 339 (2000), 1–77. 10.1016/S0370-1573(00)00070-3Search in Google Scholar

[13] Nigmatulin R. R., To the theoretical explanation of the universal response, Phys. Stat. Sol. (B) 123 (1984), 739–745. 10.1515/9783112495506-040Search in Google Scholar

[14] Nigmatulin R. R., The realization of the generalized transfer equation in a medium with fractal geometry, Phys. Stat. Sol. (B) 133 (1986), 425–430. 10.1515/9783112495483-049Search in Google Scholar

[15] Ou Y. H., Hasanov A. and Liu H. L., Inverse coefficient problems for nonlinear parabolic differential equations, Acta Math. Sin. (Engl. Ser.) 24 (2008), no. 10, 1617–1624. 10.1007/s10114-008-6384-0Search in Google Scholar

[16] Pachepskya Y., Timlinb D. and Rawlsc W., Generalized Richards’ equation to simulate water transport in unsaturated soils, J. Hydrol. 272 (2003), 3–13. 10.1016/S0022-1694(02)00251-2Search in Google Scholar

[17] Plociniczak L., Approximation of the Erdélyi–Kober operator with application to the time-fractional porous medium equation, Siam J. Appl. Math. 74 (2014), no. 4, 1219–1237. 10.1137/130942450Search in Google Scholar

[18] Plociniczak L., Analytical studies of a time-fractional porous medium equation. Derivation, approximation and applications, Commun. Nonlinear Sci. Numer. Simul. 24 (2015), 169–183. 10.1016/j.cnsns.2015.01.005Search in Google Scholar

[19] Plociniczak L. and Okrasińska H., Approximate self-similar solutions to a nonlinear diffusion equation with time-fractional derivative, Phys. D 261 (2013), 85–91. 10.1016/j.physd.2013.07.006Search in Google Scholar

[20] Podlubny I., Fractional Differential Equations, Academic Press, San Diego, 1999. Search in Google Scholar

[21] Ramos N. M. M., Delgado J. M. P. Q. and de Freitas V. P., Anomalous diffusion during water absorption in porous building materials – Experimental evidence, Defect Diffusion Forum Vols. 273–276 (2008), 156–161. 10.4028/3-908451-51-5.156Search in Google Scholar

[22] Richards L. A., Capillary conduction of liquids through porous mediums, Physics 1 (1931), no. 5, 318–333. 10.1063/1.1745010Search in Google Scholar

[23] Sakamoto K. and Yamamato M., Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl. 382 (2011), 426–447. 10.1016/j.jmaa.2011.04.058Search in Google Scholar

[24] Sakamoto K. and Yamamato M., Inverse source problem with a final overdetermination for a fractional diffusion equation, Math. Control Relat. Fields 4 (2011), 509–518. 10.3934/mcrf.2011.1.509Search in Google Scholar

[25] Sun H., Meerschaert M. M., Zhang Y., Zhu J. and Chen W., A fractal Richards’ equation to capture the non-Boltzmann scaling of water transport in unsaturated media, Adv. Water Resources 52 (2013), 292–295. 10.1016/j.advwatres.2012.11.005Search in Google Scholar PubMed PubMed Central

[26] Szymkiewicz A., Modeling Water Flow in Unsaturated Porous Media. Accounting for Nonlinear Permeability and Material Heterogeneity, Springer, Berlin, 2012. 10.1007/978-3-642-23559-7Search in Google Scholar

[27] Tatar S., Tinaztepe R. and Ulusoy S., Determination of an unknown source term in a space-time fractional diffusion equation, J. Fract. Calc. Appl. 6 (2015), no. 2, 94–101. Search in Google Scholar

[28] Tatar S., Tinaztepe R. and Ulusoy S., Simultaneous inversion for the exponents of the fractional time and space derivatives in the space-time fractional diffusion equation, Appl. Anal. 95 (2016), no. 1, 1–23. 10.1080/00036811.2014.984291Search in Google Scholar

[29] Tatar S. and Ulusoy S., A uniqueness result in an inverse problem for a space-time fractional diffusion equation, Electron. J. Differ. Equ. 2013 (2013), no. 258. Search in Google Scholar

[30] Tatar S. and Ulusoy S., An inverse source problem for a one-dimensional space-time fractional diffusion equation, Appl. Anal. 94 (2015), no. 11, 2233–2244. 10.1080/00036811.2014.979808Search in Google Scholar

[31] Tikhonov A. and Arsenin V., Solution of Ill-Posed Problems, Wiley, New York, 1977. Search in Google Scholar

[32] Zeidler E., Nonlinear Functional Analysis and its Applications. II/B: Nonlinear Monotone Operators, Springer, New York, 1990. 10.1007/978-1-4612-0981-2Search in Google Scholar

[33] Zhang Y. and Xu X., Inverse source problem for a fractional diffusion equation, Inverse Problems 27 (2011), Article ID 035010. 10.1088/0266-5611/27/3/035010Search in Google Scholar

Received: 2015-11-16
Revised: 2016-2-17
Accepted: 2016-2-21
Published Online: 2016-3-22
Published in Print: 2017-4-1

© 2017 by De Gruyter

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  4. Recovery of harmonic functions from partial boundary data respecting internal pointwise values
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https://doi.org/10.1515/jiip-2015-0100
Keywords for this article
Fractional diffusion; inverse problem; quasi-solution; class of admissible coefficients

Articles in the same Issue

  1. Frontmatter
  2. A functional Hodrick–Prescott filter
  3. On the peakon inverse problem for the Degasperis–Procesi equation
  4. Recovery of harmonic functions from partial boundary data respecting internal pointwise values
  5. Inverse spectral problems of transmission eigenvalue problem for anisotropic media with spherical symmetry assumptions
  6. An inverse problem for a nonlinear diffusion equation with time-fractional derivative
  7. Imaging of complex-valued tensors for two-dimensional Maxwell’s equations
  8. Direct and inverse source problems for a space fractional advection dispersion equation
  9. A conditional Lipschitz stability for determining a space-dependent source coefficient in the 2D/3D advection-dispersion equation
  10. Inverse transmission eigenvalue problems with the twin-dense nodal subset
  11. Stability of the inverse boundary value problem for the biharmonic operator: Logarithmic estimates
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