Identifying a time dependent zeroth order coefficient in a time fractional diffusion wave equation by using the measured data at a boundary point 2

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Applicable Analysis

An International Journal

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Identifying a time-dependent zeroth-order

coefficient in a time-fractional diffusion-wave

equation by using the measured data at a

boundary point

Ting Wei & Kaifang Liao

To cite this article: Ting Wei & Kaifang Liao (2021): Identifying a time-dependent zeroth-ordercoefficient in a time-fractional diffusion-wave equation by using the measured data at a boundarypoint, Applicable Analysis, DOI: 10.1080/00036811.2021.1932834

To link to this article: https://doi.org/10.1080/00036811.2021.1932834

Published online: 27 May 2021.

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Identifying a time-dependent zeroth-order coefficient in a

time-fractional diffusion-wave equation by using the measured data at a boundary point

School of Mathematics and Statistics, Lanzhou University, Lanzhou, People’s Republic of China

ABSTRACT

In this paper, we investigate a nonlinear inverse problem of identifying

a time-dependent zeroth-order coefficient in a time-fractional

diffusion-wave equation by using the measured data at a boundary point We firstly

prove the existence, uniqueness and regularity of the solution for the

corre-sponding direct problem by using the contraction mapping principle Then

we try to give a conditional stability estimate for the inverse zeroth-order

coefficient problem and propose a simple condition for the initial value and

zeroth-order coefficient such that the uniqueness of the inverse coefficient

problem is obtained The Levenberg–Marquardt regularization method is

applied to obtain a regularized solution Based on the piecewise linear finite

elements approximation, we find an approximate minimizer at each

itera-tion by solving a linear system of algebraic equaitera-tions in which the Fréchet

derivative is obtained by solving a sensitive problem Two numerical

exam-ples in one-dimensional case and two examexam-ples in two-dimensional case

are provided to show the effectiveness of the proposed method.

ARTICLE HISTORY

Received 5 June 2020 Accepted 12 May 2021

COMMUNICATED BY

D XU

KEYWORDS

Time-fractional diffusion-wave equation; time-dependent zeroth-order coefficient; uniqueness; conditional stability;

Levenberg–Marquardt regularization method

2010 MATHEMATICS SUBJECT

in which(·) is the Gamma function, and the second-order elliptic operator Ais defined by

for the unit outward normal vectorν(x) = (ν1(x), , ν d (x)) at x ∈ ∂.

If all functions except u in problem (1) are given, it is a well-posed direct problem This model can

be used to formulate the anomalous superdiffusion phenomena of particles in heterogenous porousmedia, see [3–5] for some application backgrounds However, sometimes the time-dependent zeroth-

order coefficient p (t) may not know, for example the diffusion of pollutants in underground sandy

soil, if the particles of pollutants are absorbed on the surface of sand stone such that the particles donot have random walks or have a chemical reaction such that the pollutants are degraded, then the

zeroth-order term in mathematical equation is appeared where the coefficient p (t) is called absorbtion

coefficient or reaction coefficient which describes the absorbtion rate or reaction rate of pollutants, i.ethe amount of absorbed or degraded pollutants in unit volume at unit time Generally, it is difficult to

know the exact coefficient p (t) We want to identify it based on an additional condition In this paper,

we consider an inverse problem for identifying the zeroth-order coefficient p (t) in problem (1) The

additional condition is

Direct problems for time-fractional diffusion-wave equations have been investigated widely invarious aspects, for examples, the existence and uniqueness of weak solutions [6–8], numericalmethods [9–13]

For inverse problems of time-fractional diffusion-wave equations, there are not so many ences In [14], the authors consider a backward problem for a time-fractional diffusion-wave equationand use the Tikhonov regularization method to solve it Siskova and Slodicka in [15] investigate aninverse time-dependent source problem by an additional integral condition In [16], Liao et al con-sider an inverse time-dependent source problem by an additional boundary condition All the studiesmentioned before focused on the linear inverse problems for the time-fractional diffusion-waveequations

refer-In this paper, we try to solve a nonlinear inverse problem of recovering the time-dependent

coef-ficient p (t) We firstly give the existence, uniqueness and regularity result for the corresponding

direct problem by using a fixed point theorem and then prove a conditional stability estimate forthe inverse problem A numerical algorithm is provided to give a regularized approximate solution.For a similar problem on a time-fractional diffusion equation, Fujishiro et al in [17] identified azeroth-order coefficient from the measured data at an interior or a boundary point, and obtained

a stability result, however, no numerical method is provided Zhang in [18] determined a dependent diffusion coefficient from the Neumann data at a boundary data in a time-fractionaldiffusion equation and gave the uniqueness of inverse problem and an efficient algorithm In [19], Sun

time-et al considered to seek a time-dependent potential coefficient in a multi-term time-fractional sion equation and used the Levenberg–Marquardt regularization method to solve the corresponding

diffu-inverse problem numerically In [20], Sun et al investigated a time-dependent convection coefficient

in a time-fractional diffusion equation

There are four different points compared with the references For the direct problem, the Fouriermethod is fail to give an explicit expression of the solution for problem (1) since the zeroth-order

coefficient is concerned with variable t Secondly, we obtain a higher regularity of solution for the

direct problem compared with ones in [17,19] such that the Caputo derivative is well defined in apointwise meaning, in fact the solutions for the integral equations in [19] may not be the solutions

of the corresponding direct problems On the other hand, we give a conditional stability estimate forthis inverse zeroth-order coefficient problem which is a new issue as we know Finally, we propose anumerical method combined with a finite element approximation to solve this inverse problem.Throughout this paper, if unspecified, we always use the following assumptions

to find an approximate time-dependent zeroth-order coefficient Numerical results for four examples

in one- and two-dimensional cases are provided to illustrate the efficiency of our used method inSection6 Finally, we give a brief conclusion in Section7

2 Preliminaries

In this paper, the space AC[0, T] is the space of absolutely continuous functions on [0, T] And define

AC n [0, T] : = {z(t)|z ∈ C n−1[0, T], z (n−1) (t) ∈ AC[0, T]}, n ≥ 2.

Denote the norms in L2() and L∞(0, T) as  ·  =  ·  L2(), · ∞=  · L∞(0,T), and the inner

product in L2() as (·, ·) H s (), s ∈Ris the standard Sobolev space (see Adams [21])

We define the operator A=A + 1 in D(A) := {u ∈ H2()|∂ ν u = 0on∂}, then by the standard theorems on second-order elliptic equations we know A is a self-adjoint and positive operator Let {λ k,φ k}∞

k=1be an eigensystem of A in D (A), then we have 0 < λ1 < λ2 ≤ λ3≤ · · · , limk→∞λ k=

∞, Aφ k = λ k φ k, and suppose{φ k}∞k=1⊂ D(A) be an orthonormal basis of L2().

We can define the Hilbert scale space D (A γ ) for γ ≥ 0 (see, e.g [22]) by

4.

Definition 2.1 ([ 1]): Let f (t) ∈ AC[0, T] for α ∈ (0, 1) and f (t) ∈ AC2[0, T] for α ∈ (1, 2) The

Caputo left-sided fractional derivative∂0+α f is defined by

Lemma 2.1 ([ 25]): Let f ∈ L p (0, T) and g ∈ L q (0, T) with 1 ≤ p, q ≤ ∞ and 1/p + 1/q = 1 Then the function f ∗ g defined by f ∗ g(t) = t

0f (t − s)g(s) ds belongs to C[0, T] and satisfies

|f ∗ g(t)| ≤ f  L p (0,t) g L q (0,t), t ∈ [0, T].

Lemma 2.2 ([ 25]): Let u, v ∈ H2() and d ≤ 3 Then uv ∈ H2() with the estimate

with C > 0 depending on u H2() .

Lemma 2.3 ([ 17]): Let C, α > 0 and u, d ∈ L1(0, T) be nonnegative functions satisfying

where C0 > 0 is a constant depending on b, α, T.

Definition 2.2 ([ 1 ]): The Mittag-Leffler function is

whereα > 0 and β ∈Rare arbitrary constants

Proposition 2.1 ([ 1]): Let 0 < α < 2 and β ∈Rbe 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

| E α,β (z) |≤ 1+ | z |C , μ ≤| arg(z) |≤ π.

Proposition 2.2 (See [ 8]): Let α > 0, λ > 0, then we have

If E1 = 0, we know f (t) = 0 for t ∈ [0, T] almost every, which deduces easily the result (10) In

the following, we suppose E1> 0.

Denote t0= f  C[0,T] /E1 We have to consider two cases

(1) If t0≥ T, then by (11), we can obtain

|∂0α+f (t)| ≤ 1

(2 − α) fL∞(0,T) T1−α

 t

t −t0

f (s) (t − s) α ds

(1 − α) 0)t0−α − f (0)t −α − α t −t0

0

f (s) (t − s) α+1 ds+ t

t −t0

f (s) (t − s) α ds

Putting the value of t0into the inequality above and combining with (12), we get the estimate (10)

By the interpolation theorems in Adames [21] ( See Theorem 5.8 in page 140, Theorem 5.2 in page

135), we have two estimates for f (t) as

in which we usef  L2(0,T) ≤ T12f  L∞(0,T) Putting (18) into (15), we have

∂0+α fC[0,T] ≤ C2(α, T)E2+α4

2 f 2−α4



3 Existence, uniqueness and regularity of solution for the direct problem

In this section, by the fixed point theorem, we can obtain the following existence, uniqueness and

regularity results for problem (1) Throughout this paper, the notation C means a generic constant independent of u which may take a different value appearing everywhere.

Theorem 3.1: Let conditions (3)–(6) hold Then the integral Equation (23) has a unique solution u ∈

C ([0, T]; D(A)) ∩ C1([0, T]; L2()) satisfying

u C([0,T];D(A)) + u C1([0,T];L2())+ AuC([0,T];L2())

where C > 0 is depending on α, T, p∞ and q L∞(0,T;D(A))

Moreover, if f ∈ AC([0, T]; L2()), p ∈ AC[0, T], q ∈ AC([0, T]; D(A)), then there is a unique solution to the direct problem (1) satisfying u ∈ AC2([0, T]; L2()) ∩ C([0, T]; D(A)) and ∂ α

where C > 0 is a constant depending on α.

Since the Mittag-Leffler function E α,1 (−λ n t α ) is continuous over t ≥ 0, then by the uniform

convergence theorem, we know S1(t)ϕ ∈ C([0, T]; D(A)) By Lemma 2.2, we have

where C is depending on α Thus we know ∂ t (S1(t)ϕ) ∈ C([0, T]; L2()) and further S1(t)ϕ ∈

C1([0, T]; L2()) The following estimate holds

S1(t)ϕC1([0,T];L2()) + S1(t)ϕC([0,T];D(A)) ≤ Cϕ D(A), (27)

where C is depending on α.

Similarly, we have

S2(t)ψ D(A) ≤ Cψ D (A1−1/α), (28)and S2(t)ϕ ∈ C([0, T]; D(A)) By Lemma 2.3, we have

where we use λ n = O(n2/d ) and ∞n=11/λ2

n < ∞ for d = 1, 2, 3 It yields that ∂ t u3 ∈

C ([0, T]; L2()) by the uniform convergence theorem and

Since p (t) ∈ L∞(0, T) and q ∈ L∞(0, T; D(A)), by Lemma 2.2, we have

(bu)(·, t) D(A) ≤ u(·, t) D(A) + |p(t)|C(q(·, t) D(A) )u(·, t) D(A) ≤ C p,q u(·, t) D(A), (36)

where C p,q = C(p∞,q L∞(0,T;D(A) ) > 0 By taking f = bu in (35), we know Q b u ∈ X and

Q b uX ≤ Cu L∞(0,T;D(A))

Denote F = S1(t)ϕ + S2(t)ψ + t

0S3(t − τ)f (·, τ) dτ, G(u) = F + Q b u, then G is an affine

map-ping from X into X By induction, we have

we need to prove the operator G m is a contraction mapping for sufficiently large m By the generalized

Minkowski inequality and Lemma 2.1, estimate (36), we have

(Q b v )(·, t) D(A) ≤ C

 t

0 (t − τ) α−1 (bv)(·, τ) D(A)dτ

≤ C

 t

0 (t − τ) α−1 v(·, τ) D(A)dτ ≤ C/αt α v L∞(0,T;D(A)), (40)and

constant C is depending only on E p

whereρ m2=  m−1(mα) (α)(α−1) and C > 0 depending on α, T, A, p∞,q L∞(0,T;D(A), ifp∞≤ E p, the

constant C is depending only on E p Therefore, we have

Q m

It is easy to verifyρ m = (ρ m1 + ρ m2 )T mα → 0 as m → ∞ Therefore, the operator G mis a

contrac-tion mapping from X into itself for sufficiently large m∈N Hence the mapping Gmhas a unique

fixed point denoted by u ∈ X, that is, G m (u) = u Then we know the equation u = Q b u + F has a unique solution u in X.

thus, note that 1< α < 2, we have ∂ tt (S1(t)ϕ) ∈ L1(0, T; L2()) and S1(t)ϕ ∈ AC2([0, T]; L2()).

From Lemma 2.2, we know the second derivative for S2(t)ψ satisfy

∂ tt (S2(t)ϕ) =∞

n=1

(ψ, φ n )(−λ n t α−1 )E α,α (−λ n t α )φ n (x), (51)then by Lemma 2.1, we have

Therefore u = F + Q b u ∈ AC2([0, T]; L2()) and estimate (21) is easy to obtain. 

4 Conditional stability for the inverse zeroth-order coefficient problem

In this section, we give a stability result and a conditional stability estimate for the inverse zeroth-ordercoefficient problem in Theorem 4.1

Theorem 4.1: Let a ij (x) ∈ C2( ¯) for i, j = 1, 2, , d Assume conditions (5)–(6) hold and

f ∈ AC([0, T]; L2()), q ∈ AC([0, T]; D(A)) Let u i be the solution of (1) for p = p i ∈ AC[0, T] with

p iL∞(0,T) ≤ M (i = 1, 2) Assume that there exist x0∈ ∂ and c1 > 0 such that

Moreover, if u1(x0 , t ), u2(x0 , t ) ∈ U = {f (t) ∈ W2, ∞(0, T), f  W2,∞(0,T) ≤ E}, then we have a

condi-tional stability estimate

p1− p2C[0,T] ≤ ¯Cu1(x0, t ) − u2(x0 , t )2−α4

where the constant ¯C > 0 depends on M, T, α, c1 , E, and q L∞(0,T;D(A)

Proof: Let u i be the solutions to (1) corresponding to p = p i (i = 1, 2) We denote u = u1 − u2and

p = p2− p1 Then u satisfies the following problem

In the following, we give an estimate forAu(·, t) D(A γ )ford4 < γ < 1 such that Au(·, t) is

meaning-ful at a point x0 Denote b1(x, t) = 1 − p1(t)q(x, t), and the solution u(x, t) of problem (58) can be

In the above inequality (60), takingγ = 0, by (36) and

(pqu2)(·, t)D(A) ≤ |p(t)|C(q L∞(0,T;D(A)) )u2(·, t) D(A),

where C is depending on the norms for given functions ϕ, ψ, f and M.

Substituting the above inequality into (60), we have

≤ C

 t

in which we use

q (x0 , t )u2 (x0 , t )[∂0α+u (x0 , t ) + Au(x0 , t ) − (1 − p1(t)q(x0 , t ))u(x0 , t )], (63)

by assumption (55) and the embedding theorem, noting thatd4 < γ < 1, we have

|p(t)| ≤ 1/c1[|∂α

0 +u(x0 , t )| + |Au(x0 , t )| + |(1 − p1 (t)q(x0 , t ))u(x0 , t )|],

≤ C[|∂0+α u (x0 , t )| + Au(·, t) D (A γ ) + u(·, t) D (A γ ) + |p1(t)|q(·, t))D (A) u(·, t) D (A)],

| p(t) |≤ C ∂ α

0+u (x0 , t )

L∞(0,T), t ∈ (0, T).

By Lemma 2.8, the conditional stability estimate (57) is easily obtained The proof is completed 

Remark 4.1: From Theorem 4.1, we know the solution p(t) in the inverse zeroth-order

prob-lem (1)–(2) is unique Since we use the additional condition u (x0 , t ) = g(t) to identify the coefficient

p (t), although we just know the perturbed data of g, it is rational to suppose u(x0 , t ) be known roughly,

thus the condition (55) can be verified approximately

Remark 4.2: In Theorem 4.1, if we know ϕ(x0) = 0, |q(x0 , t )| > 0 for t ∈ [0, T], by Theorem 3.1,

we know the solutions for the direct problem satisfy u1, u2 ∈ C([0, T]; D(A)), according to the embedding theorem, we know u1, u2∈ C([0, T] × ¯) Thus there is T0 ∈ (0, T] such that

|u1(x0, t )|, |u2(x0 , t )| > 0 for t ∈ [0, T0 ], under this case, we know the estimate (56) is true for t∈

[0, T0] Further, if u1(x0, t ) = u2 (x0 , t ) = g(t) for t ∈ [0, T], we know p1(t) = p2(t) for t ∈ [0, T0] If

p1 , p2are real analytic functions over [0, T], then we have p1(t) = p2(t) for t ∈ [0, T] That means the

uniqueness for the inverse zeroth-order coefficient is true under the simple conditionsϕ(x0) = 0 and

p is an analytic function.

5 Levenberg–Marquardt method and finite element approximation

In this section, we recover numerically the time-dependent zeroth-order function p (t) with the

additional boundary condition (2) by using the Levenberg–Marquardt regularization method

It is convenient to consider an inverse problem in a Hilbert space Since H1(0, T) ⊂ AC[0, T], thus

in the following, we constraint p (t) ∈ H1(0, T).

Define a forward operator

F : p (t) ∈ H1(0, T) → u(x0 , t; p ) ∈ L2(0, T), (66)

Putting the value of t0into the inequality above and combining with ( 12) , we get the estimate (10)

By the interpolation theorems in. ..