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Uniqueness of an inverse problem for an integro-differential equation related to the Basset problem
Boundary Value Problems volume 2014, Article number: 229 (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.
with the initial and boundary conditions:
Here the operator A is uniformly elliptic on , defined by
with satisfying for , the function k with some power singularity is unknown and has to be determined by the following measurement data at a fixed point :
In this case, (1.1) can be written as a fractional parabolic equation
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 –.
As for the inverse kernel problems related to the integro-differential equation, in – efficient strategies to prove the existence and uniqueness of inverse memory kernel problems were given. In particular, Colombo and Guidetti  showed that a semilinear integro-differential parabolic inverse problem had a unique solution global in time under suitable growth conditions for the nonlinearity involved in the evolution equation. Lorenzi and Rocca  studied an inverse two memory kernels problem in a hyperbolic phase-field model. But the additional measurements used in these papers are imposed on the whole or the part of Ω, which can be expressed in the following integral form:
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.
For some , we use the notation to denote the following Banach space: , endowed with the norm . Furthermore, we introduce
We assume that , and satisfy
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 .
2 Proof of Theorem 1.1
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.
Letand. If there existandsuch that u attains the minimum value aton, then we have
Here we borrow the ideas used in dealing with the extremum principle of the Caputo derivative in . Integration by parts yields
Since and , we have
Letandsatisfyin. Then we have
Next we consider the general case of in . Let with some . Then we have
According to the proved conclusion, it follows that
Letting , we get the desired conclusion and the proof is complete. □
By using Lemma 2.2, we can prove the following lemma.
By the equation of u and (1.6), we have
Letting and differentiating the equation in (1.1) with respect to t, we find that
where we have used
According to Theorem A.1 in Appendix A, we have under and . In addition, by (1.6) we have in and . Then applying Lemma 2.2, we obtain
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 .
Since , there exists sufficiently small such that for . Without loss of generality, we can assume that for . Next we prove
By Lemma 2.3, we have for . Therefore, in . Applying Lemma 2.2, we have in . Furthermore, we can obtain
Otherwise, there exists such that is the minimum value of on . Then we have and . Additionally, Lemma 2.1 gives and we find that
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 .
Here, we study the existence and uniqueness of the following direct problem:
We can prove the following.
Letwith some, , and. Then the direct problem (A.1) has a unique solutionsuch that
whereis the Mittag-Leffler function defined by.
When with , the equation of u can be rewritten as
This is a time fractional parabolic equation to describe the Basset problem .
We will use a fixed point argument to prove this theorem. To do this, we set
with some , which will be specified below. For given , we consider
Therefore, the following mapping:
is well defined.
We want to choose T small enough to prove Φ is a contraction on , which implies that Φ has a unique fixed point u in . By , we have
Then we can choose to satisfy
for all , and from which it follows that . On the other hand, given , satisfies
Therefore, applying (A.5) we have
for all . Then there exists sufficiently small such that
In order to obtain the global existence, it is sufficient to prove that the solution u of the problem (A.1) satisfies
for any T. Indeed, if (A.14) holds, then we can extend the local solution repeatedly to the whole interval by the above fixed point arguments. By the estimate of Schauder type for parabolic equation , we find that for any ,
Ashyralyev A: Well-posedness of fractional parabolic equations. Bound. Value Probl. 2013., 2013: 10.1186/1687-2770-2013-31
Ashyralyev A: Well-posedness of the Basset problem in spaces of smooth functions. Appl. Math. Lett. 2011, 24: 1176-1180. 10.1016/j.aml.2011.02.002
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.2012.02.057
Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.
Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.
Basset AB: On the descent of a sphere in a viscous liquid. Q. J. Math. 1910, 42: 369-381.
Cakir Z: Stability of difference schemes for fractional parabolic PDE with the Dirichlet-Neumann conditions. Abstr. Appl. Anal. 2012., 2012: 10.1155/2012/463746
Ashyralyev A, Cakir Z: On the numerical solution of fractional parabolic partial differential equations with the Dirichlet condition. Discrete Dyn. Nat. Soc. 2012., 2012:
Ashyralyev A: Well-posedness of parabolic differential and difference equations with the fractional differential operator. Malaysian J. Math. Sci. 2012, 6: 73-89. suppl.
Ashyralyev A, Cakir Z: FDM for fractional parabolic equations with the Neumann condition. Adv. Differ. Equ. 2013., 2013: 10.1186/1687-1847-2013-120
Ashyralyev A, Cakir Z: r -Modified Crank-Nicholson difference scheme for fractional parabolic PDE. Bound. Value Probl. 2014., 2014: 10.1186/1687-2770-2014-76
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: 10.1186/1687-1847-2014-97
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/s12043-013-0583-7
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.2013.10.014
Wheeler MF:A priori error estimates for Galerkin approximations to parabolic partial differential equations. SIAM J. Numer. Anal. 1973, 10: 723-759. 10.1137/0710062
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/S0218202507002017
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.2009.8.601
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.
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.2010.07.028
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.19970770403
Luchko Y: Maximum principle for the generalized time-fractional diffusion equation. J. Math. Anal. Appl. 2009, 351: 218-223. 10.1016/j.jmaa.2008.10.018
Wu Z, Yin J, Wang C: Elliptic and Parabolic Equations. World Scientific, Singapore; 2006.
Friedman A: Partial Differential Equations of Parabolic Type. Holt, Rinehart & Winston, New York; 1964.
Dixo J, Mckee S: Weakly singular discrete Gronwall inequalities. Z. Angew. Math. Mech. 1986, 66: 535-544. 10.1002/zamm.19860661107
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.
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.