Three landweber iterative methods for solving the initial value problem of time fractional diffusion wave equation on spherically symmetric domain

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Three Landweber iterative methods for solving the initial value problem of time-fractional diffusion- wave equation on spherically symmetric domain

Fan Yang, Qiao-Xi Sun & Xiao-Xiao Li

To cite this article: Fan Yang, Qiao-Xi Sun & Xiao-Xiao Li (2021): Three Landweber iterativemethods for solving the initial value problem of time-fractional diffusion-wave equation

on spherically symmetric domain, Inverse Problems in Science and Engineering, DOI:

10.1080/17415977.2021.1914603

To link to this article: https://doi.org/10.1080/17415977.2021.1914603

Published online: 17 Apr 2021.

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Three Landweber iterative methods for solving the initial value problem of time-fractional diffusion-wave equation on spherically symmetric domain

Fan Yang, Qiao-Xi Sun and Xiao-Xiao Li

School of Science, Lanzhou University of Technology, Lanzhou, People’s Republic of China

ABSTRACT

In this paper, the inverse problem for identifying the initial value

of time-fractional diffusion wave equation on spherically symmetric

region is considered The exact solution of this problem is obtained

by using the method of separating variables and the property the

Mittag–Leffler functions This problem is ill-posed, i.e the solution(if

exists) does not depend on the measurable data Three different

kinds landweber iterative methods are used to solve this problem.

Under the priori and the posteriori regularization parameters choice

rules, the error estimates between the exact solution and the

regular-ization solutions are obtained Several numerical examples are given

to prove the effectiveness of these regularization methods.

ARTICLE HISTORY

Received 6 April 2020 Accepted 28 March 2021

KEYWORDS

Time-fractional diffusion wave equation; spherically symmetric; Ill-posed problem; fractional Landweber method; inverse problem

2010 MATHEMATICS SUBJECT

prac-or part of the boundary value [13,14] is unknown, and it is necessary to invert themthrough some measurement data, which puts forward the inverse problem of fractionaldiffusion equation The study of the fractional diffusion equation is still on an initial stage,the direct problem of it has been studied in [15–17]

For the inverse problem of time-fractional diffusion equation as 0< α < 1, there are a

lot of research results For identifying the unknown source, one can see [18–26] Aboutbackward heat conduction problem, one can see [27–34] About identifying the initialvalue problem, one can see [35–37] For an inverse unknown coefficient problem of atime-fractional equation, one can see [38,39] About identifying the source term and ini-tial data simultaneous of time-fractional diffusion equation, one can see [40,41] Aboutidentifying some unknown parameters in time-fractional diffusion equation, one can see

CONTACT Fan Yang yfggd114@163.com School of Science, Lanzhou University of Technology, Lanzhou, Gansu

730050, People’s Republic of China

[42] About the inverse problem for the heat equation on a columnar axis-symmetric area,one can see [43–48] In [43–45], Landweber regularization method, a simplified Tikhonovregularization method and a spectral method are used to identify source term on a colum-nar axis-symmetric area In [46–48], the authors considered a backward problem on acolumnar axis-symmetric domain In [46], Yang et al used the quasi-boundary valueregularization method to solve the inverse problem for determining the initial value ofheat equation with inhomogeneous source on a columnar symmetric domain The errorestimate between the regular solution and the exact solution under the corresponding reg-ularization parameter selection rules is obtained Finally, numerical example is given toverify that the regularization method is very effective for solving this inverse problem.

In [47], Cheng et al used the modified Tikhonov regularization method to do with theinverse time problem for an axisymmetric heat equation Finally, Hölder type error esti-mate between the approximate solution and exact solution is obtained In [48], Djerrar

et al used standard Tikhonov regularization method to deal with an axisymmetric inverse

problem for the heat equation inside the cylinder a ≤ r ≤ b, and numerical examples is

used to show that this method is effective and stable About the inverse problem for thetime-fractional diffusion equation as 0< α < 1 on a columnar and spherical symmetric

areas, one can see [49,50] In [49], Xiong proposed a backward problem model of a fractional diffusion-heat equation on a columnar axis-symmetric domain Yang et al in[50] used Landweber iterative regularization method to solve identifying the initial value

time-of time-fractional diffusion equation on spherically symmetric domain Compare with theinverse problem of time-fractional diffusion equation as 0< α < 1, there are little research

result for the inverse problem of time-fractional diffusion wave equation as 1< α < 2.

Šišková et al in [51] used the regularization method to solve the inverse source lem of time-fractional diffusion wave equation Liao et al in [52] used conjugate gradientmethod combined with Morozovs discrepancy principle to identify the unknown sourcefor the time-fractional diffusion wave equation Šišková et al in [53] used the regulariza-tion method to deal with an inverse source problem for a time-fractional wave equation.Gong et al in [54] used a generalized Tikhonov to identify the time-dependent sourceterm in a time-fractional diffusion-wave equation In recent years, in physical oceanogra-phy and global meteorology, the inversion of initial boundary value problems has alwaysbeen a hot issue In order to increase the accuracy of numerical weather prediction, usu-ally by the model combined with the observation data of the inversion of initial boundaryvalue problems, and in numerical weather prediction model, provide a reasonable initialfield At present, many domestic and foreign ocean circulation model, atmospheric gen-eral circulation model, numerical weather prediction model and torrential rain forecastingmodel belongs to the inversion of initial boundary value problems during initialization,

prob-so such problems of scientific research application prospect is very broad Yang et al in[55] used the truncated regularization method to solve the inverse initial value problem

of the time-fractional inhomogeneous diffusion wave equation Yang et al in [56] usedthe Landweber iterative regularization method to solve the inverse problem for identifyingthe initial value problem of a space time-fractional diffusion wave equation Wei et al in[57] used the Tikhonov regularization method to solve the inverse initial value problem

of time-fractional diffusion wave equation Wei et al in [58] used the conjugate gradientalgorithm combined with Tikhonov regularization method to identify the initial value oftime-fractional diffusion wave equation Until now, we find that there are few papers for the

inverse problem of time-fractional diffusion-wave equation on a columnar axis-symmetricdomain and spherically symmetric domain In [59], the authors used the Landweber itera-tive method to solve an inverse source problem of time-fractional diffusion-wave equation

on spherically symmetric domain It is assumed that the grain is of a spherically symmetricdomain diffusion geometry as illustrated in Figure (a-b), which is actually consistent withlaboratory measurements of helium diffusion from a physical point of view from apatite As

a consequence of radiogenic production and diffusive loss, u (r, t) which only depends on

the spherically radius r and t denotes the concentration of helium For the inverse problem

of inversion initial value in spherically symmetric region, there are few research results atpresent Whereupon, in this paper, we consider the inverse problem to identify the initialvalue of time-fractional diffusion-wave equation on spherically symmetric region and givethree regularization methods to deal with this inverse problem in order to find a effectiveregular method

In this paper, we consider the following problem:

The existence and uniqueness of the direct problem solution has been proved in the [60]

The inverse problem is to use the measurement data g (r) and the known function f (r)

to identify the unknown initial dataϕ(r), ψ(r) The inverse initial value problem can be

transformed into two cases:

(IVP1): Assumingψ(r) is known, we use the final value data g(r) and the known function

f (r) to invert the initial value ϕ(r).

(IVP2): Assumingϕ(r) is known, we use the final value data g(r) and the known function

f (r) to invert the initial value ψ(r).

Because the measurements are error-prone, we remark the measurements with error as

f δ and g δand satisfy

whereδ > 0 In this paper, L2[0, r0; r2] represents the Hilbert space with weight r2on the

interval [0, r0] of the Lebesgue measurable function.(·, ·) and  ·  represent the inner

product and norm of the space of [0, r0; r2], respectively. ·  is defined as follows:

This paper is organized as follows In Section2, we recall and state some preliminary oretical results In Section3, we analyse the ill-posedness of the problem (IVP1) and theproblem (IVP2), and give the conditional stability result In Section4, we give the corre-sponding a priori error estimates and posteriori error estimates for three regularizationmethods In Section5, we conduct some numerical tests to show the validity of the pro-posed regularization methods Since most of the solutions of fractional partial differentialequations contain special functions (Mittag–Leffler functions), and the calculation of thesefunctions is quite difficult In this paper, the difficulties are overcome through [61,62].Finally, we give some concluding remarks

the-2 Preliminary results

In this section, we give some important Lemmas

Lemma 2.1 ([ 57]): If 0 < α < 2, and β ∈R be arbitrary Suppose μ satisfy πα2 < μ < min {π, πα} Then there exists a constant C1= C(α, β, μ) > 0 such that

where the constant c(a1, p ) is given by c(a1, p ) = ( p

4a1) p4 and c(a2, p ) is given by c(a2, p ) = ( p

4a2) p4.

Proof: Refer to the appendix for the details of the proof. 

Lemma 2.5: For 1 < α < 2 and any fixed T > 0, there is at most a finite index set I1=

{n1, n2, , n N } such that E α,1 (−( n π

r0)2T α ) = 0 for n ∈ I1 and E α,1 (−( n π

r0)2T α ) = 0 for

n /∈ I1 Meanwhile there is at most a finite index set I2= {m1, m2, , m M } such that

E α,2 (−( nπ r0)2T α ) = 0 for n ∈ I2and E α,2 (−( nπ r0)2T α ) = 0 for n /∈ I2.

Proof: From Lemma 2.2, we know that there exists L0> 0 such that

Remark 2.1: The index sets I1and I2may be empty, that means the singular values for the

operators K1and K2are not zeros Here and below, all the results for I1=∅and I2=∅are regarded as the special cases

Lemma 2.6 ([ 57]): For 1 < α < 2 and λ n = ( nπ r0)2, there exists positive constants C, C depending on α, T such that

Proof: Refer to the appendix for the details of the proof. 

Lemma 2.10: For a1> 0, a2> 0, p > 0, m1> 0, m2> 0, we have

4 .

Proof: Refer to the appendix for the details of the proof. 

Lemma 2.11: For a1> 0, a2> 0, p > 0, m5≥ 1, m6≥ 1, we have

Proof: Refer to the appendix for the details of the proof. 

3 The ill-posedness and the conditional stability

Theorem 3.1: Let ϕ(r), ψ(r) ∈ L2[0, r0; r2], then there exists a unique weak solution and

the weak solution for (1) is given by

where (ϕ(r), R n (r)) and (ψ(r), R n (r)) are the Fourier coefficients.

and due to [59], we know the linear operators K1and K2are compact from L2[0, r0; r2] to

L2[0, r0; r2] The problem (IVP1) and the problem (IVP2) are ill-posed

Let K∗1be the adjoint of K1and K2∗be the adjoint of K2 Since R n (r) =

√

2n πsin( nπr r0 )

Therefore, the singular system of K1is(σ 1n (1) ; R n,ψ n (1) ).

By the similar verification, we know the singular system of K2is(σ 2n (2) ; R n,ψ n (2) ), where

It is not hard to prove that the kernel spaces of the operators K1and K2are

N(K1) = span {R n ; n ∈ I1} for I1 =∅; N(K1) = {0} for I1=∅,

N(K2) = span {R n ; n ∈ I2} for I2 =∅; N(K2) = {0} for I2=∅,

and the ranges of the operators K1and K2are

Therefore, we have the following existence of the solutions for the integral equations:

Theorem 3.2: If I1=∅, for any g1∈ R(K1), there exists a unique solution in L2[0, r0; r2]

for the integral Equation (28) given by

Proof: Suppose ϕ(ξ) =∞n=1ϕ n R n (ξ), put g1=∞n =1,n/∈I1g 1n R n (r) into (28), according

to the orthonormality of{R n}, it is not hard to obtain the results 

Theorem 3.3: If I2=∅, for any g2∈ R(K2), there exists a unique solution in L2[0, r0; r2]

for the integral equation (30) given by

Proof: The proof is similar to the Theorem 3.2. 

We have the following theorem on conditional stability:

Theorem 3.4: When ϕ(r) satisfies the a-priori bound condition

where E1and p are positive constants, we have

ϕ(r) ≤ C11E

2 +2

Applying Lemma 2.6 and (39), we obtain

where C12= ( TCr π22) p+2p is a constant.

Proof: The proof is similar to the Theorem 3.4, so it is omitted. 

4 Regularization method and error estimation

By referring to [59,64,65], we find a classical Landweber regularization method and twofractional Landweber iterative regularization methods, but it has not been explained which

of these methods is better So, in this section, we mainly make use of two kinds of fractionalLandweber regularization methods and the classical Landweber regularization method tosolve the problem (IVP1) and the problem (IVP2) The error estimates between the exactsolution and the corresponding regular solution are given, respectively By using threeregularization methods to solve the same problem, an optimal regularization method isobtained

By [64], the fractional Landweber regularization solution is given as follows:

4.1 The priori error estimate

Lemma 4.1: Suppose f (x) and g(x) be ∈ L2(0, r0; r2), there is a constantM1to make:

n2π2T We complete the proof of Lemma 4.1 

1+ 1)δ.

Define an orthogonal projection operatorQ1: L2[0, r0; r2]→ R(K1), combining

Equa-tions (3) and (4), we have:

Q1g1δ−Q1g1 ≤ g1δ − g1 = g δ − g ≤2(M2

1+ 1)δ.

Define an orthogonal projection operatorQ2: L2[0, r0; r2]→ R(K2), combining

Equa-tions (3) and (4), we have:

Q2g2δ−Q2g2 ≤ g2δ − g2 = g δ − g ≤2(M2

1+ 1)δ.

Theorem 4.1: Let m1= [( E1

δ ) +24 ] If the a-priori condition (42) and Lemma 4.1 hold, we

have the following convergence estimate

ϕ m1 ,δ (r) − ϕ(r) ≤ C13E

2 +2

δ ) p+24 ] If the a-priori condition (46) and Lemma 4.1 hold, we

have the following convergence estimate

δ ) p+24 ] If the a-priori condition (42) and Lemma 4.1 hold, we

have the following convergence estimate

ϕ m3 ,δ (r) − ϕ(r) ≤ C15E

2 +2

where C15=2a1(M2

1+ 1) + C3, and [x] denotes the largest integer smaller than or equal

to x.

Proof: By the triangle inequality, we have

B(n) ≤ C3(m3+ 1)−p

Theorem 4.4: Let m4= [( E2

δ ) +24 ] If the a-priori condition (46) and Lemma 4.1 hold, we

have the following convergence estimate

ψ m4 ,δ (r) − ψ(r) ≤ C16E

2 +2

δ )2(γ +1) p+2 ] If the a-priori condition (42) and Lemma 4.1 hold, we

have the following convergence estimate

ϕ m5 ,δ (r) − ϕ(r) ≤ C17E

2 +2

On the other hand, we have

δ )2(γ +1) p+2 ] If the a-priori condition (46) and Lemma 4.1 hold, we

have the following convergence estimate

ψ m6 ,δ (r) − ψ(r) ≤ C18E

2 +2

Proof: The proof process is similar to Theorem 4.5, so it is omitted. 

4.2 The posteriori error estimate

(c) limm1 →∞ρ(m1) = 0;

(d) ρ(m1) is strictly decreasing for any m1∈ (0, +∞).

Theorem 4.7: If formula (3) and (4) is true, then the regularization parameter

δ

 4 +2.The convergence result is given in the following theorem 

Theorem 4.8: Assuming that Lemma 4.1 and (39) are valid, and the regularization

param-eter are given by Equation (70), then

Combining (72), (73) and (74), we obtain the convergence estimate Assuming thatτ2>2(M2

1+ 1) is the given constant, the selection of m2= m2(δ) ∈

N0is that when m2satisfying

appears for the first time, the iteration stops, whereg δ

2(r) ≥ τ2δ.

(d) ρ(m2) is strictly decreasing for any m2∈ (0, +∞).

Theorem 4.9: If formula (3) and (4) is true, then the regularization parameter

τ2 −√

2(M2+1) ) p+24 .

Proof: The proof process is similar to Theorem 4.7, so it is omitted. 

Theorem 4.10: Assuming that (3), (4) and (41) are valid, and the regularization parameters

are given by (76), then we have

ψ m2 ,δ (r) − ψ(r) ≤ C22E

2 +2

1+ 1) is the given constant, the selection of m3= m3(δ) ∈

N0is that when m3satisfying

appears for the first time, the iteration stops, whereg1δ (r) ≥ τ3δ.

Lemma 4.4: Assuming ρ(m3) = K1ϕ m3 ,δ (r) − g1δ (r), then

(a) ρ(m3) is a continuous function;

(b) limm3→0ρ(m3) = g δ

1;

(c) limm3 →∞ρ(m3) = 0;

(d) ρ(m3) is strictly decreasing for any m3∈ (0, +∞).

Theorem 4.11: If formula (3) and (4) is true, then the regularization parameter

Proof: From (50), we have

So, we have H (n)E1≥ (τ3−2(M2

E1

δ

 4 +2

The convergence result is given in the following theorem

Theorem 4.12: Assuming that (3), (4) and (39) are valid, and the regularization parameters

are given by (79), then we hold

ϕ m3 ,δ (r) − ϕ(r) ≤ C24E

2 +2

δ.

(82)For the second part, we have