- Open Access
Uniqueness of an inverse problem for an integro-differential equation related to the Basset problem
© Wu and Yu; licensee Springer. 2014
- Received: 8 June 2014
- Accepted: 1 October 2014
- Published: 15 October 2014
This paper concerns an inverse problem for an integro-differential equation related to the Basset problem. The inverse problem aims to determine a weakly singular term from the time trace at a fixed point . We use the maximum principle for an integro-differential operator to derive the uniqueness of the inverse problem. Additionally, we prove the existence and uniqueness of the direct Basset problem with a general kernel function.
MSC: 35L05, 35L10, 35R09, 35R30.
- inverse problems
- Basset problem
For details on the fractional derivative, we refer the readers to  or . The system (1.1) and (1.2) is called the Basset problem, which describes a classical problem in fluid dynamics with the unsteady motion of a particle accelerates in a viscous fluid due to the force of gravity . For the recent results on the Basset problem, we refer the readers to –. In Appendix A, we will prove the existence and uniqueness of the direct problem (1.1) and (1.2) for general k, which extends the results in  and . (References  and  were concerned the direct Basset problem with the fractional order , i.e. the problem (1.4) and (1.2).) For the other models related to the integro-differential equation, we refer the readers to –.
with a known function ϕ. Comparing with these papers, our study only needs the measurements at a fixed point . Another difference is that the kernel function discussed in our current paper has power singularity . Finally, our method in discussing the uniqueness of our inverse problem is different from those inverse memory kernel problems, in which the methods on the basis of the analytic semigroup theory were used. It is worth noting that in  the Fourier method was applied to prove the existence and uniqueness of determining a weakly singular kernel in a linear heat conduction model.
Now we state our main result in this paper.
Since k is continuous with power singularity , we will discuss the uniqueness of k in the sense of for all , when .
In this section, the notations , and are similar to , and in Section 1, namely, , and in . In order to prove Theorem 1.1, we first give the following lemmas.
Letting , we get the desired conclusion and the proof is complete. □
By using Lemma 2.2, we can prove the following lemma.
This contradicts with in . The proof of Lemma 2.3 is complete. □
Now we prove Theorem 1.1.
Proof of Theorem 1.1
Indeed, if (2.13) holds, then we have for , i.e.. Therefore, by Theorem 1.1, we obtain for .
In this paper we study an inverse weakly singular memory kernel problem for an integro-differential equation related to the Basset problem. In order to determine the weakly singular term, we only use the measurement data at a fixed point , rather than the usual measurement data on the whole or part of Ω in previous studies of inverse kernel problems. The uniqueness of our inverse problem is shown by using a maximum principle related to an integro-differential operator. In addition, the existence and uniqueness of the direct Basset problem with general kernel function are also given, which extends the results in  and .
We can prove the following.
whereis the Mittag-Leffler function defined by.
This is a time fractional parabolic equation to describe the Basset problem .
is well defined.
The first author is supported by NSFC (No. 11201238). This work has been completed while the first author visited Department of Mathematics and Statistics at The University of Vermont and he acknowledges the hospitality and support of the Department and the University.
- Ashyralyev A: Well-posedness of fractional parabolic equations. Bound. Value Probl. 2013., 2013: 10.1186/1687-2770-2013-31Google Scholar
- Ashyralyev A: Well-posedness of the Basset problem in spaces of smooth functions. Appl. Math. Lett. 2011, 24: 1176-1180. 10.1016/j.aml.2011.02.002MathSciNetView ArticleGoogle Scholar
- Li F, Liang J, Xu HK: Existence of mild solutions for fractional integrodifferential equations of Sobolev type with nonlocal conditions. J. Math. Anal. Appl. 2012, 391: 510-525. 10.1016/j.jmaa.2012.02.057View ArticleGoogle Scholar
- Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.Google Scholar
- Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.Google Scholar
- Basset AB: On the descent of a sphere in a viscous liquid. Q. J. Math. 1910, 42: 369-381.Google Scholar
- Cakir Z: Stability of difference schemes for fractional parabolic PDE with the Dirichlet-Neumann conditions. Abstr. Appl. Anal. 2012., 2012: 10.1155/2012/463746Google Scholar
- Ashyralyev A, Cakir Z: On the numerical solution of fractional parabolic partial differential equations with the Dirichlet condition. Discrete Dyn. Nat. Soc. 2012., 2012:Google Scholar
- Ashyralyev A: Well-posedness of parabolic differential and difference equations with the fractional differential operator. Malaysian J. Math. Sci. 2012, 6: 73-89. suppl.MathSciNetGoogle Scholar
- Ashyralyev A, Cakir Z: FDM for fractional parabolic equations with the Neumann condition. Adv. Differ. Equ. 2013., 2013: 10.1186/1687-1847-2013-120Google Scholar
- Ashyralyev A, Cakir Z: r -Modified Crank-Nicholson difference scheme for fractional parabolic PDE. Bound. Value Probl. 2014., 2014: 10.1186/1687-2770-2014-76Google Scholar
- Ashyralyev A, Emirov N, Cakir Z: Well-posedness of fractional parabolic differential and difference equations with Dirichlet-Neumann conditions. Electron. J. Differ. Equ. 2014., 2014: 10.1186/1687-1847-2014-97Google Scholar
- Liu W, Chen K: The functional variable method for finding exact solutions of some nonlinear time-fractional differential equations. Pramana 2013, 81: 377-384. 10.1007/s12043-013-0583-7View ArticleGoogle Scholar
- Saedpanah F: Well-posedness of an integro-differential equation with positive type kernels modeling fractional order viscoelasticity. Eur. J. Mech. A, Solids 2014, 44: 201-211. 10.1016/j.euromechsol.2013.10.014MathSciNetView ArticleGoogle Scholar
- Wheeler MF:A priori error estimates for Galerkin approximations to parabolic partial differential equations. SIAM J. Numer. Anal. 1973, 10: 723-759. 10.1137/0710062MathSciNetView ArticleGoogle Scholar
- Colombo F, Guidetti D: A global in time existence and uniqueness result for a semilinear integrodifferential parabolic inverse problem in Sobolev spaces. Math. Models Methods Appl. Sci. 2007, 17: 537-565. 10.1142/S0218202507002017MathSciNetView ArticleGoogle Scholar
- Colombo F, Guidetti D: Identification of the memory kernel in the strongly damped wave equation by a flux condition. Commun. Pure Appl. Anal. 2009, 8: 601-620. 10.3934/cpaa.2009.8.601MathSciNetView ArticleGoogle Scholar
- Lorenzi A, Rocca E: Identification of two memory kernels in a fully hyperbolic phase-field system. J. Inverse Ill-Posed Probl. 2008, 16: 147-174.MathSciNetGoogle Scholar
- Wu B, Liu J: A global in time existence and uniqueness result for an integrodifferential hyperbolic inverse problem with memory effect. J. Math. Anal. Appl. 2011, 373: 585-604. 10.1016/j.jmaa.2010.07.028MathSciNetView ArticleGoogle Scholar
- Janno J, Von Wolfersdorf L: Identification of weakly singular memory kernels in heat conduction. Z. Angew. Math. Mech. 1997, 77: 243-257. 10.1002/zamm.19970770403MathSciNetView ArticleGoogle Scholar
- Luchko Y: Maximum principle for the generalized time-fractional diffusion equation. J. Math. Anal. Appl. 2009, 351: 218-223. 10.1016/j.jmaa.2008.10.018MathSciNetView ArticleGoogle Scholar
- Wu Z, Yin J, Wang C: Elliptic and Parabolic Equations. World Scientific, Singapore; 2006.View ArticleGoogle Scholar
- Friedman A: Partial Differential Equations of Parabolic Type. Holt, Rinehart & Winston, New York; 1964.Google Scholar
- Dixo J, Mckee S: Weakly singular discrete Gronwall inequalities. Z. Angew. Math. Mech. 1986, 66: 535-544. 10.1002/zamm.19860661107MathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd.Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.