Numerical Inversion for the Multiple Fractional Orders in the Multiterm TFDE

Hindawi Advances in Mathematical Physics Volume 2017, Article ID 3204959, 7 pages https://doi.org/10.1155/2017/3204959 Research Article Numerical Inversion for the Multiple Fractional Orders in the Multiterm TFDE Chunlong Sun,1,2 Gongsheng Li,1 and Xianzheng Jia1 1 School of Science, Shandong University of Technology, Zibo 255049, China Department of Mathematics, Southeast University, Nanjing 210096, China 2 Correspondence should be addressed to Gongsheng Li; Received 9 February 2017; Accepted 2 July 2017; Published 17 August 2017 Academic Editor: Ming Mei Copyright © 2017 Chunlong Sun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The fractional order in a fractional diffusion model is a key parameter which characterizes the anomalous diffusion behaviors. This paper deals with an inverse problem of determining the multiple fractional orders in the multiterm time-fractional diffusion equation (TFDE for short) from numerics. The homotopy regularization algorithm is applied to solve the inversion problem using the finite data at one interior point in the space domain. The inversion fractional orders with random noisy data give good approximations to the exact order demonstrating the efficiency of the inversion algorithm and numerical stability of the inversion problem. 1. Introduction The partial differential equations of fractional order have played an important role in modeling of the anomalous phenomena and in the theory of the complex systems during the last two decades; see, for example, [1–8]. The so-called time-fractional diffusion equation (TFDE) that is obtained from the classical diffusion equation by replacing the firstorder time derivative by a fractional derivative of order 𝛼 with 0 < 𝛼 < 1 has to be especially mentioned. On the other hand, by the attempts to describe some real processes with the equations of the fractional order, several researches were confronted with the situation that the order 𝛼 of the timefractional derivative from the corresponding models did not remain constant and changed, say, in the interval from 0 to 1, from 1 to 2, or even from 0 to 2. To manage these phenomena, several approaches were suggested. One of them introduces the fractional derivatives of the variable order, that is, the derivatives with the order that can change with the time or/and depending on the spatial coordinates [9–11], and the other way is to employ the distributed order TFDE, or the multiterm TFDE in discretization. Let Ω be a bounded domain in Rd (𝑑 ≥ 1) with smooth boundary 𝜕Ω, and let 𝑇 > 0; the multiterm homogeneous TFDE with variable coefficient in Ω is given as 𝑆 𝜕𝛽𝑠 𝑢 𝜕𝛼 𝑢 + 𝑟 = ∇ ⋅ (𝐷∇𝑢) , 𝑥 ∈ Ω, 0 < 𝑡 ≤ 𝑇, ∑ 𝑠 𝜕𝑡𝛼 𝑠=1 𝜕𝑡𝛽𝑠 (1) where 𝑢 = 𝑢(𝑥, 𝑡) denotes the state variable at space point 𝑥 and time 𝑡, 𝛼 denotes the principal fractional order, and 𝛽1 , 𝛽2 , . . . , 𝛽𝑆 are the multiterm fractional orders of the time derivatives, which satisfy the condition 0 < 𝛽𝑆 < 𝛽𝑆−1 < ⋅ ⋅ ⋅ < 𝛽1 < 𝛼 < 1, (2) and 𝑟1 , 𝑟2 , . . . , 𝑟𝑆 are positive constants, and 𝐷(𝑥) > 0 is the smooth diffusion coefficient tensor. All of the above timefractional derivatives are defined in the sense of Caputo; for example, the fractional derivative of the order 𝛽 ∈ (0, 1) is given by 𝑡 𝜕𝑢 (𝑥, 𝑠) 𝑑𝑠 1 𝜕𝛽 𝑢 = . ∫ 𝛽 𝜕𝑠 (𝑡 − 𝑠)𝛽 Γ (1 − 𝛽) 0 𝜕𝑡 (3) See, for example, Podlubny [12] and Kilbas et al. [13] for the definition and properties of Caputo’s derivative. There are still a few research works reported on the multiterm TFDE like (1). On theoretical analysis and analytical methods for the forward problem, we refer to DaftardarGejji and Bhalekar [14], Luchko [15, 16], Jiang et al. [17], 2 Ding et al. [18, 19], and Li et al. [20], and for numerical methods and simulations we refer to [21–23], and so on. However, for real problems, the fractional orders, the initial distribution, the diffusion coefficient, or the source term cannot be obtained directly and we have to determine them by some additional measurements, which contributes to inverse problems arising in the fractional diffusion models. There are still some researches on inverse problems for the one-term TFDE; see, for example, Murio [24], Liu et al. [25, 26], Sakamoto and Yamamoto [27], Tuan [28], Chi et al. [29], Yamamoto and Zhang [30], Luchko et al. [31], Wei et al. [32, 33], and Liu et al. [34]; also see Jin and Rundell [35] for a tutorial review on inverse problems for anomalous diffusion processes. It is noted that the research works stated above are almost related to coefficient identification problems in the one-term time/space fractional diffusion equations. However, it is also important to deal with inverse problems of determining the fractional orders in the fractional differential equations since the fractional order is an essential index characterizing the anomalous diffusion. As for inverse problems of determining fractional orders in the single-term time/space fractional diffusion models, we refer to [36–42], and so on. On the other hand, there are few literatures concerned with the inverse problems in the multiterm TFDEs to our knowledge. Li and Yamamoto [39] studied an inverse problem of identifying the multiple fractional orders in the multiterm TFDE, and they gave the uniqueness result using Laplace transform and analytical method, and later they considered the similar model [42], and also the uniqueness of determining the fractional orders, the number of the fractional terms, and the spatially varying coefficient simultaneously is proved. Recently, Sun et al. [43] considered a simultaneous inversion problem for determining the space-dependent diffusion and source coefficients in the multiterm TFDE using the optimal perturbation regularization algorithm, and quite a few numerical inversions are presented. Based on the above analysis, we are to deal with the inverse problem of determining the multiple fractional orders in the multiterm TFDE with the additional measurements at the interior point from numerics. The uniqueness results for such kind of inverse problems have been obtained (c.f. [39, 42], e.g.), but numerical inversions are still open to be implemented. Based on the difference solution to the forward problem, we perform numerical inversions by utilizing the homotopy regularization algorithm not only with the accurate data but also with random noisy data. The inversion fractional orders approximate to the exact orders as the noise level gets smaller demonstrating a numerical stability of the inverse problem here. The rest of the paper is organized as follows. In Section 2, an implicit finite difference solution to the forward problem is given and the inverse problem of determining the fractional orders is formulated. In Section 3, the homotopy regularization algorithm is introduced to solve the inversion problem and num (...truncated)