Determine a Space-Dependent Source Term in a Time Fractional Diffusion-Wave Equation
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Determine a Space-Dependent Source Term in a Time Fractional Diffusion-Wave Equation X.B. Yan1
· T. Wei1
Received: 31 October 2017 / Accepted: 25 February 2019 © Springer Nature B.V. 2019
Abstract This paper is devoted to identify a space-dependent source term in a multidimensional time fractional diffusion-wave equation from a part of noisy boundary data. Based on the series expression of solution for the direct problem, we improve the regularity of the weak solution for the direct problem under strong conditions. And we obtain the uniqueness of inverse space-dependent source term problem by the Titchmarsh convolution theorem and the Duhamel principle. Further, we use a non-stationary iterative Tikhonov regularization method combined with a finite dimensional approximation to find a stable source term. Numerical examples are provided to show the effectiveness of the proposed method. Keywords Inverse spatial source problem · Uniqueness · Non-stationary iterative Tikhonov regularization
1 Introduction Time or space fractional equations have been studied extensively in recent years, which have preferably advantages for describing anomalous diffusion phenomena due to nonlocal property of fractional order derivatives. Especially during the past decades, fractional order partial differential equations have been successfully used for modeling many processes and systems. For example, anomalous diffusion is often observed in materials with memory, e.g., viscoelastic materials, and heterogeneous media, such as soil, heterogeneous aquifer, and underground fluid flow. The literatures on the theories and applications of fractional order differential equations are very extensive, one can consult books [13, 20]. The direct problems for fractional equations have been studied in recent years [2, 4, 5, 12, 18, 19, 21, 31]. The inverse problems for time and space fractional equations are to recover unknown data by some additional
B X.B. Yan
[email protected] T. Wei [email protected]
1
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730030, P.R. China
X.B. Yan, T. Wei
data and also have been investigated recently, refer to [1, 6, 15, 17, 21, 23, 26, 28, 29, 33]. However, to our best knowledge, there is rare work on inverse problems for time fractional diffusion-wave equations. In this paper, we study an inverse space-dependent source problem for a time fractional diffusion-wave equation in a bounded domain. Let Ω be a bounded domain in Rd with sufficiently smooth boundary ∂Ω. We consider the following time fractional diffusion-wave problem ⎧ α ∂0+ u(x, t) + Lu(x, t) = f (x)g(t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ u(x, 0) = a(x), x ∈ Ω, ¯
x ∈ Ω, 0 < t ≤ T , (1.1)
⎪ ¯ ⎪ ∂t u(x, 0) = b(x), x ∈ Ω, ⎪ ⎪ ⎪ ⎩ ∂ν u(x, t) = 0, x ∈ ∂Ω, 0 < t ≤ T ,
α where 1 < α < 2 and ∂0+ u(x, t) is the Caputo left-sided fractional derivative defined by α u(x, t) = ∂0+
1 Γ (2 − α)
t
(t − s)1−α 0
∂ 2u (x, s)ds, ∂s 2
t > 0,
and L is a symmetric uniformly elliptic operator given by Lu(x, t) = −
d ∂ ∂ u(x, t) + c(x)u(x, t), aij (x) ∂xj ∂xi
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