DSpace at VNU: Holder stability for a class of initial inverse nonlinear heat problem in multiple dimension

Received Date: 21 June 2011Revised Date: 7 October 2014 Accepted Date: 21 October 2014 Please cite this article as: Tuan, N.H., Duy, B.T., Minh, N.D., Khoa, V.A., Hölder stability for a

Received Date: 21 June 2011

Revised Date: 7 October 2014

Accepted Date: 21 October 2014

Please cite this article as: Tuan, N.H., Duy, B.T., Minh, N.D., Khoa, V.A., Hölder stability for a class of initial

inverse nonlinear heat problem in multiple dimension, Communications in Nonlinear Science and Numerical Simulation (2014), doi: http://dx.doi.org/10.1016/j.cnsns.2014.10.027

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heat problem in multiple dimension

Nguyen Huy Tuan 1, Bui Thanh Duy 2, Nguyen Dang Minh1 and Vo Anh Khoa 1

1

Faculty of Mathematics and Computer Science, University of Science,Vietnam National University, HoChiMinh City, Viet Nam

2

Faculty of Fundamental Science, Ho Chi Minh City Architecture University, Viet Nam

Abstract We consider an inverse heat conduction problem for the non-linear heat equation which appears

in some applied subjects The problem is severely ill-posed Using some modified regularization method, weestablish a regularized solution which gives error estimate of H¨older type for all t ∈ [0, T ] Some numerical

examples show that the computational effect of these methods are all satisfactory

Keywords and phrases: Backward heat problem, Nonlinearly Ill-posed problem, Quasi-boundary value

methods, Quasi-Reversibility methods

Mathematics subject Classification 2000: 35K05, 35K99, 47J06, 47H10.

From φ ε , we have to find a function v ε (x, t) which approximates stability the solution u of (1) The problem is

severely ill-posed in the sense of Hadamard In fact, for a given final data, we are not sure that a solution ofthe problem exists In case a solution exists, it may not depend continuously on the final data The problemhas many various application, for example in glass production [32], polymer processing

The linear homogeneous case of (1) have been studied by many various methods We can notably mentionthe quasi-solution method (QS-method)of Tikhonov [44, 45],the quasi-reversibility method (QR method) ofLattes and Lions [11, 19], the quasi boundary value method (Q.B.V method)[5, 7, 33, 34] and the C-regularizedsemigroups method [1, 2, 15, 16, 25], the numerical method [14, 20, 28, 18]

Previously, reconstruction of the initial temperature was considered only for the linear heat equation withsource terms independent of the temperature To the author’s knowledge, there are not many papers on theinitial inverse heat problem with linear and nonlinear source terms In 2006, Trong and Tuan proposed thequasi-reversibility regularization method for the nonhomogeneous backward heat, however, the retrieved time

T = 0.5 is still small After that, Trong and Tuan also proposed many regularization method to improve previous

stability results, for example [31, 36, 38, 39] In addition, Nam (2010) used the truncation method to studied thenonlinear backward heat and get the H¨older error estimate However, a numerical simulation for 2-D and 3-Dcases is not considered in this paper Recently, Li, Jiang and Hon (2010) [21] have proposed a meshless methodbased on RBFs method to solve 2-D and 3-D nonhomogeneous backward heat, but this algorithm was complexand hard to implement Without using a priori regularization, Chang and Liu [3] utilized the backward grouppreserving scheme (BGPS) to tackle those multi-dimensional nonlinear and nonhomogeneous backward heatand obtained good results Very recently, Chang [4] proposed a Fictitious Time Integration Method (FTIM) for

By employing the F T IM, Chang computed the solution and retrieve the initial data very well with a high order

accuracy and showed that the efficiency of the computations for three-dimensional backward heat

Motivated by these above reasons, in this paper, we propose a new modified quasi-boundary regularizationmethod to establish the H¨older estimates, i.e, the error is of order ε k (0 < k < 1) The method of quasi-boudary

value method was first introduced by Showalter [33] and then fast developed by many authors, for example Clarkand Oppenheimer [5], Denche and Bessila [7], C-W Chang, C-S Liu and J-R Chang [4], etc The method of

quasi-boudary value method is also called the perturbation method, whereby we modified the source term f and the final data φ The main idea of the quasi-boundary-value method is going to replace the boundary value

problem with an approximate well-posed one, then to construct approximate solutions of the given boundaryvalue problem

The rest of the paper is organized as follows In Section 2, we state the linear case of the inhomogeneous

backward problem We also give the estimates which are of order ε k In Section 3, we extend our consideration

to the nonlinear Cauchy problem, given by (1) Numerical examples are tested in Section 4 to verify the efficacy

of the our method Finally, proofs of main results will be give in Appendix The proof for the m-dimensionalcase is similar to the one for the one dimensional case Hence, for simplicity, in Appendix we will give the proof

in the one dimensional case

In this section, we consider the linear inhomogeneous ill-posed problem as follows

∂u

∂t − ∆u = f(x, t), (x, t) ∈ Ω × (0, T ), (2)

u | ∂Ω = 0, t ∈ [0, T ], (3)

u(x, T ) = φ(x), x∈ Ω. (4)Based on the ideas mentioned in paper [37], we consider the following approximation problem

π

)n∫Ω

f (x, t)ϕm(x)dx,

φm =

(2

π

)n∫Ω

e (s −T )|m|2

fm(s)ds

)

ϕm(x), 0≤ t ≤ T. (8)

)n∫Ω

φ ε (x)ϕm(x)dx.

Throughout this section, we suppose that f ∈ L2((0, T ); L2(Ω)) and g ∈ L2(Ω) Now we state main results

of Secion 2 In Proposition 2.1, we have the neccessary and sufficient conditions for the existence of solution

of Problem 2 – 4 Proposition 2.2 is devoted to the existence and the stability of regularization u ε,a FinallyTheorem 2.1 gives rate convergence of error estimated

Proposition 2.1 The problem (2)–(4) has a unique solution u if and only if

Proposition 2.2 Let f ∈ L2((0, T ); L2(Ω)) and g ∈ L2(Ω) The problem (5)–(7) has a unique weak solution

u ε,a ∈ C([0, T ]; L2(Ω))∩ L2((0, T ); H01(Ω))∩ C1([0, T ]; H01(Ω)) satisfying (8) The solution of problem (5)–(7)

depends continuously on g in L2(Ω).

Theorem 2.1 Let v ε,a (., t), defined in (9), be the unique solution of Problem (5)–(7)corresponding to the noisy

data g ε Suppose that the problem (2)–(4) has a unique solution u(., t) ∈ W Let f, g be the functions satisfying the condition (10).

a) Assume that the function f satisfying the condition

From the above Theorem, we have some remarks

1 In t = 0, the error in (11) does not tend to zero when ε → 0 Its order is also the same one as in [5, 43].

The error (12) becomes

∥u(., 0) − v ε,a (., 0) ∥ ≤ C2

(ln( aT

Notice that the error (13) (k > 0) is of order of Holder type for all t ∈ [0, T ] It is easy to see that the

convergence rate ε p , (0 < p) is more quickly than the logarithmic order

(ln(1

ε)

)−q

(q > 0) when ε → 0 This

proves that our method is effective

In this section, we analyse the ill-posedness of the nonlinear backward heat problem in frequency space andexplain why we introduce the approximation problem

Consider the following heat equation system (1) with T = 1, f ∈ L ∞(Ω× [0, 1] × R) The inverse problem is to

determine the value of u(x, t) for 0 ≤ t < 1 from the data φ ε (x) If the solution exists, then the problem has a

unique solution ([36], Theorem 3.1, p 239) For system (1) we do not guarantee that the solutions exist In [36],

we present an simple way to check the existence of system (1) (See Theorem 3.2a, page 239) The main purpose

of this section is to find a computation method for the exact solution when it exists Hence, the regularizationtechniques are required

Informally, Problem (1) can be transformed the integral equation

e −(T −s)|m|2

fp(u)(s)ds

)

ϕm(x) (14)

fm(u)(t)ϕm(x) are the expansion of φ and f (u) respectively.

Since t < T , we know from (14) that, when m become large, e (T −t)|m|2

increase rather quickly Thus, the

term e −(t−T )|m|2

is the unstability cause Hence, to regularize the problem, we have to replace the term by

different better term Let a ≥ 1 be the fixed number We shall replace e −(t−T )|m|2

by the new better term

π

)n

< f (., t, u(., t)), ϕm(.) >=

(2

π

)n∫Ω

f (x, t, u(x, t))ϕm(x)dx, (16)

φm=

(2

π

)n

< φ(.), ϕm(.) >=

(2

π

)n∫Ω

φ(x)ϕm(x)dx (17)

and < , > is the inner product in L2(Ω).

Throught out this section, we assume φ ∈ L2(Ω), α > 0 and f ∈ L ∞(Ω× [0, T ] × R) satisfy

Theorem 3.1 Assume that there exists a positive number P1 such that

This order error is the same in some paper [5, 36]

2 The assumption of the nonlinear term f (u) in (20) is introduced in [36] ( Theorem 3.2, p.239 and Remark

3.3, page 242) Then, the assumption is acceptable

The approximation error depends continuously on the measurement error for fixed 0 < t ≤ T However, as

t → 0, the accuracy of regularized solution becomes progressively lower This is a common property in the

theory of ill-posed problems, if we do not have additional conditions on the smoothness of the solution To

retain the continuous dependence of the solution at t = 0, we introduce a stronger a priori assumption and get

Theorem 3.2 Assume that there exists a positive number P2 such that

The order of convergence of this error is ε a−1 a By choosing a suitable constant a, we have a better error.

2 The estimate (25) is of order of Holder type for all t ∈ [0, T ] As known, the convergence rate of ε a−1 a ismore quickly than the logarithmic order

(ln

(1

In this subsection, we will implement proposed regularization methods by mainly giving two simple examples

(with choosing T = 1) We divide the section into three subsections The first one is to consider the main

numerical example More precisely, we take a type of the Ginzburg-Landau equation as an example of theconsidered problem Then, the example in [42] is taken again in the second subsection in order to compare

the stability with the present paper in the case of exact data (noise amplitude ε = 0) Finally, the last one is

showing our comments and discussions

Figure 1: The exact solution u (x, y, t) = y (π − y) e − t

2sin x at t = 0.9 (left) and 0.6 (right).

and

g (x, y) = y (π − y) e −1

The exact solution to problem (26) is u (x, y, t) = y (π − y) e − t

2sin x The aim of the numerical examples is to observe ε = c · 10 −r where c ∈ R is a positive number and r ∈ N will be given later The measured data g εis

defined by two ways below Clearly speaking, we will take perturbation, intended to define as ϵ · rand where

each random term rand will be determined on [−1, 1] uniformly, in exact data g In particular, it can be chosen

in one of the following

Besides, the regularized one is expected to be closed to the exact under a proper discretization The l2-norm

and l ∞-norm errors and the relative root mean square (RRMS) error will be considered Simultaneously, 2-Dand 3-D graphs are applied and analysed The regularized solution in the problem (15) with γ = 2, a = 1 is

where M, N are the truncation numbers.

In order to control the nonlinear term, choose v ε L

1(x, y) ≡ v ε (x, y, t L1) when dividing the time t i = i∆t, ∆t =

In general, the whole process of computation is summarized in the following steps

Step 1 Choose L1 and L2 and L3 to have

Step 3 For j = 0, L2 and l = 0, L3, put v i (x j , y l ) = v i,j,l and u (x j , y l , t i ) = u i,j,l We construct two

matrixes containing all discrete values of v ε and u at fixed time t i , denoted by A i and B i, respectively

u i,0,0 u i,0,1 · · · u i,0,L3

u i,1,0 u i,1,1 · · · u i,1,L3

E3(t i) =

√∑L2j=0

∑L3l=0 |v ε (x j , y l , t i)− u (x j , y l , t i)|2

√∑L2

j=0

∑L3

l=0 |u (x j , y l , t i)|2 . (47)

Remark 4.1 In order to calculate the first term in the right-hand side of (38), we use Gauss-Legendre

quadra-ture method (see in [47]) to estimate the term as follows.

E1(0.9) 4.71220069E-01 1.12516916E-01 3.22133054E-02 2.86983883E-02 2.34940542E-02

E1(0.6) 6.49468644E-01 2.15469544E-01 4.99372084E-02 5.03003853E-02 1.56110998E-01

E1(0.2) 1.05361303E+00 6.97860068E-01 2.15092411E-01 1.26670646E+00 diverged

E2(0.9) 9.24424713E-01 1.90953498E-01 7.77483355E-02 6.50941685E-02 5.33289371E-02

E2(0.6) 1.30362385E+00 3.88595089E-01 1.11034701E-01 1.01245647E-01 4.49331253E-01

E2(0.2) 2.14672372E+00 1.41073544E+00 3.87588157E-01 4.37498467E+00 diverged

E3(0.9) 6.01148142E-01 1.43622910E-01 4.16326776E-02 3.70915938E-02 3.03638626E-02

E3(0.6) 7.22458773E-01 2.39684955E-01 5.55493705E-02 5.59533628E-02 1.73655436E-01

E3(0.2) 9.59571010E-01 6.35571381E-01 1.95893972E-01 1.15364442E+00 divergedTable 1: Error estimates defined by (45)-(47) between the exact solution and regularized solution in Example 1

at t = 0.9; 0.6; 0.2.

Figure 2: The regularized solution at t = 0.9 (top left and right) and t = 0.6 (bottom left and right) with

ε = 10 −1 (left) and 10−3 (right).

G (x, y, t) = 3e t sin x sin y − e 4tsin4x sin4y, g (x, y) = e sin x sin y, (53)

and the exact solution is u (x, y, t) = e t sin x sin y As a result, we define regularized solutions in [42] respectively)

E11,ε (0.9) 6.53556866E-01 3.30804984E-01 2.83283777E-01 3.67386177E-01 6.71467859E-01

E12,ε (0.9) 1.10140054E+00 7.60878356E-01 3.70407512E-01 2.86192683E-01 2.76786902E-01

E11,ε (0.3) 2.81765079E-01 1.55555915E+00 diverged diverged diverged

E12,ε (0.3) 3.27159097E-01 2.66759844E-01 9.68434841E-03 5.73627429E-02 1.37365647E-01

E21,ε (0.9) 1.37396007E+00 7.07800698E-01 7.01846261E-01 1.29988472E+00 2.40033189E+00

E22,ε (0.9) 2.31294123E+00 1.59784557E+00 7.77865981E-01 6.01106684E-01 5.82272167E-01

E21,ε (0.3) 4.52457191E-01 4.67752062E+00 diverged diverged diverged

E22,ε (0.3) 6.87056458E-01 5.59943221E-01 2.38268082E-02 1.55947626E-01 4.69132845E-01

E31,ε (0.9) 5.58004424E-01 2.82440067E-01 2.41866636E-01 3.13672953E-01 5.73296764E-01

E32,ε (0.9) 9.40371689E-01 6.49635114E-01 3.16252558E-01 2.44350249E-01 2.36319629E-01

E31,ε (0.3) 4.38347079E-01 2.42001177E+00 diverged diverged diverged

E32,ε (0.3) 5.08967383E-01 4.15003162E-01 1.50661177E-02 8.81047035E-02 2.13778309E-01

Table 2: Error estimates between the exact solution and regularized solution u 1,ε , u 2,ε in Example 2 at t = 0.9 and t = 0.3.

The regularized solution in this example is established from (15) with γ = 8 and a = 1 as follows

All remaining steps are similar to the ones in the first example, from forming the solution and linearization

of the nonlinearity in (36)-(38) to four general steps of computation in (39)-(47) However, we will rewrite

Figure 6: The regularized solutions u 1,ε (top left and right) and u 2,ε (bottom left and right) at t = 0.9 for

ε = 10 −3 and ε = 10 −5.

γ = 10

1.0E-03 1.65233157E-01 2.91088082E-01 2.13548366E-011.0E-04 3.61894296E-02 8.53471471E-02 4.67714453E-021.0E-05 2.95162893E-02 6.77239962E-02 3.81470371E-02

ε E1(0.6) E2(0.6) E3(0.6)

γ = 6

1.0E-03 1.13829997E-01 1.98224388E-01 1.26622710E-011.0E-04 3.85381043E-02 9.27254809E-02 4.28691851E-021.0E-05 3.62892084E-02 8.40959558E-02 4.03675484E-02

γ = 10

1.0E-03 2.95599617E-01 5.58023649E-01 3.28820395E-011.0E-04 6.14237214E-02 1.24456258E-01 6.83267880E-021.0E-05 3.57159042E-02 8.57040999E-02 3.97298138E-02

ε E1(0.2) E2(0.2) E3(0.2)

γ = 6

1.0E-03 4.53264182E-01 8.75094861E-01 4.12807316E-011.0E-04 1.17132588E-01 2.14771505E-01 1.06677719E-011.0E-05 1.56893146E-01 3.60611080E-01 1.42889382E-01

γ = 10

1.0E-03 8.36364766E-01 1.68266799E+00 7.61713606E-011.0E-04 2.54957571E-01 4.56377563E-01 2.32200899E-011.0E-05 8.46973156E-02 1.65865204E-01 7.71375126E-02

Table 3: Error estimates between the exact solution and regularized solution in Example 1 with choosing γ = 6 and γ = 10.

Figure 8: The exact solution u (x, y, t) = e t sin x sin y at t = 0.9 (top) and graphs of error estimates generated

by difference between the exact solution and regularized solutions u 1,ε , u 2,ε for ε = 10 −1 (bottom left) and

ε = 10 −3 (bottom right).