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.
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 .
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.
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.
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