Simultaneous identification of three parameters in a time-fractional diffusion-wave equation by a part of boundary Cauchy data

https://doi.org/10.1016/j.amc.2020.125382Get rights and content

Highlights

  • Find a new result for Mittag-Leffler functions.

  • Prove the uniqueness of the inverse problem.

  • Use an effective and stable algorithm to solve the inverse problem numerically.

Abstract

This paper is devoted to determine the fractional order, the initial flux speed and the boundary Neumann data simultaneously in a one-dimensional time-fractional diffusion-wave equation from part boundary Cauchy observation data. We prove the uniqueness result for this inverse problem by using a new result for the Mittag-Leffler function and Laplace transform combining with analytic continuation. Then we use the iterative regularizing ensemble Kalman method in Bayesian framework to solve the inverse problem numerically. And four numerical examples are provided to show the effectiveness and stability of the proposed algorithm.

Introduction

In recent years, time-fractional diffusion or diffusion-wave equations have attracted the attention of many researchers because the fractional derivatives have advantages in describing sub-diffusion and super-diffusion phenomena [31], [39]. At the same time, fractional calculus has been widely used in many fields, such as biology, physics, hydrology, chemistry and biochemistry, medicine and finance [13], [29], [30], [35], [48], [49].

In this paper, we consider the following time-fractional diffusion-wave equation with Neumann boundary condition and initial value condition{0+αu(x,t)=Δu(x,t),x(0,l),t>0,ux(0,t)=a(t),ux(l,t)=b(t),t>0,u(x,0)=φ(x),x(0,l),ut(x,0)=ψ(x),x(0,l),where 0+α denotes the Caputo fractional left-sided derivative of order α defined by0+αu(x,t)=1Γ(2α)0t2u(x,s)s2ds(ts)α1,t>0,1<α<2,in which Γ( · ) is the Gamma function.

If the fractional order α and the functions a(t), b(t), φ(x), ψ(x) are all given, then (1.1) is the direct problem for a time-fractional diffusion-wave equation. Direct problems for time-fractional diffusion (0 < α < 1) or diffusion-wave (1 < α < 2) equations have been studied extensively in recent years [1], [3], [7], [18], [19], [24], [28], [30], [32], [37], [44].

Inverse problems for time-fractional diffusion or diffusion-wave equations are to recover initial data or source function or diffusion coefficient and so on by some additional data. The inverse problem we consider in this paper is to recover (α, b(t), ψ(x)) simultaneously from the additional observation dataf(t)=u(0,t),t>0.

There are many works for inverse problems of the time-fractional diffusion equations (0 < α < 1). In [25], Liu et al. used a quasi-reversibility regularization method to solve the backward problem for a time-fractional diffusion equation in one-dimensional case. In [43], Wei et al. used a boundary element method to solve a Cauchy problem for a time-fractional diffusion equation. In [22], Li et al. considered an inverse problem for determining the space-dependent diffusion coefficient and the fractional order, and gave a uniqueness result. In [26], Liu et al. proved the uniqueness result of determining the initial status and heat flux on the boundary simultaneously from the heat measurement data given on the other boundary in a time-fractional diffusion equation, and gave a regularizing scheme for the inverse problem. In [36], Ruan et al. considered a simultaneous identification problem of the space-dependent source term and the fractional order for a time-fractional diffusion equation, and proved the uniqueness result of the inverse problem by the Laplace transform method and analytic continuation technique. In [27], Liu et al. considered an inverse problem of recovering a time-dependent factor of an unknown source on some sub-boundary for a time-fractional diffusion equation by nonlocal measurement data, and proposed two regularizing methods for the inverse problem.

For the inverse problems of time-fractional diffusion-wave equations (1 < α < 2), to our knowledge, there are few papers to consider them. In [38], Šišková et al. provided a numerical algorithm based on Rothe’s method to deal with an inverse time-dependent source problem for a time-fractional diffusion-wave equation, and gave a priori estimate and the convergence of iterated solution. In [42], Wei et al. used the Tikhonov regularization method to deal with the inverse initial value problem for a time-fractional diffusion-wave equation. In [45], Xian et al. used a variational method to solve a backward problem for a time-fractional diffusion-wave equation. In [23], Liao et al. identified the fractional order and the space source term simultaneously in a multi-dimensional time-fractional diffusion-wave equation, and provided the uniqueness and stability of the inverse problem. In [10], Gong et al. used a generalized Tikhonov regularization method to solve an inverse time-dependent source problem for a time-fractional diffusion-wave equation.

However, to our best knowledge, there are no works to restructure (α, b(t), ψ(x)) simultaneously from the part boundary Cauchy data f(t)=u(0,t) in a time-fractional diffusion-wave equation. In this paper, we consider the following two problems:

• Is it possible to identify (α, b(t), ψ(x)) uniquely from the additional data f(t)? How to prove the uniqueness result?

• Can we find an effective and stable algorithm to solve the inverse problem numerically?

For the first problem, we firstly prove a new result for the Mittag-Leffler function, then we obtain the uniqueness result for the inverse problem by using integral mean value theorem and Laplace transform combining with analytic continuation. For the second problem, we use the iterative regularizing ensemble Kalman method which is a derivative-free optimization method to solve the inverse problem numerically.

Our paper is organized as follows. In Section 2, we present some preliminary which will be used in this paper. The uniqueness result for our inverse problem is provided in Section 3. In Section 4, we present the iterative regularizing ensemble Kalman method. In Section 5, numerical results for four examples are investigated. Finally, we give a conclusion in Section 6.

Section snippets

Preliminaries

Definition 2.1

[21], [33] The Mittag-Leffler function isEα,β(z)=k=0zkΓ(αk+β),zC,where α > 0 and βR are arbitrary constants.

Proposition 2.2

[33] Let 0 < α < 2 and βR be arbitrary. We suppose that μ is such that πα/2 < μ < min{π, πα}. Then there exists a constant C=C(α,β,μ)>0 such that|Eα,β(z)|C1+|z|,μ|arg(z)|π.

Proposition 2.3

[21] Let 0 < α < 2, βR, λ > 0 and mN, thendmdtm(tβ1Eα,β(λtα))=tβm1Eα,βm(λtα),(βm)>0,t>0.

Proposition 2.4

[21] For α > 0, β > 0, we have0+ezttβ1Eα,β(±atα)dt=zαβzαa,(z)>|a|1α.

Proposition 2.5

Let 1 < α < 2 and t > 0, then Eα,α(t)

The uniqueness of the inverse problem

We denote the eigensystem of Δ with the homogeneous Neumann boundary condition as {λn,ϕn(x)}n=0 such that ϕn(x)=λnϕn(x) and ϕn(0)=ϕn(l)=0. In fact, it is easy to obtain thatλn=(nπl)2,ϕn(x)=cos(nπlx),n=0,1,2,···

By using the Sturm-Liouville theorem, we know that the sequence {ϕn(x)}n=0 is a complete orthogonal basis in L2(0, l), and for arbitrary f(x) ∈ L2(0, l), we havef(x)=n=0ρn(f,ϕn)ϕn(x),where ( · ,  · ) is the inner product in L2(0, l), andρ0=1l,ρn=2l,n=1,2,···Moreover, fL2(0,l)=(

Iterative regularizing ensemble Kalman method

In this section, we want to find an effective algorithm to obtain the numerical solution of the inverse problem. In section 3, we prove the uniqueness of the inverse problem by using the additional data f(t)=u(0,t) for all t(0,+), it is impossible in practical application because the data can only be measured in a finite time interval [0, T]. Thus, in this section, we reconstruct (α, b(t), ψ(x)) by the additional data f(t)=u(0,t) only in the finite time interval [0, T].

The inverse problem in

Numerical experiments

In this section, we present some numerical results for four examples to show the effectiveness of IREKM.

Without the loss of generality, we assume l=1 and T=1. We solve the direct problem (1.1) by using a finite element scheme given in [41], and we take the grid sizes for time and space variables in the finite element algorithm are Δt=150 and Δx=150 respectively.

In order to show the accuracy of numerical solution, we compute the approximate relative errors denoted byeαn=|αn¯α||α|,ebn=bn¯b2

Conclusion

In this paper, we consider an inverse problem for recovering three parameters simultaneously by part boundary Cauchy data in a time-fractional diffusion-wave equation in one-dimensional case. By using a new result for the Mittag-Leffler function we have proven in this paper, combining with the integral mean value theorem and the Laplace transform, we prove the uniqueness of the solution for the inverse problem. In order to obtain the numerical results of the inverse problem, we use the

Acknowledgments

This paper was supported by the NSF of China (11371181, 11771192) and the Fundamental Research Funds for the Central Universities, No. lzujbky-2020-12.

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