Two new regularization methods for solving sideways heat equation

R E S E A R C H

Open Access

Two new regularization methods for solving

sideways heat equation

Huy Tuan Nguyen

_{and Vu Cam Hoan Luu}

*_{Correspondence:} [email protected] 1_{Applied Analysis Research Group,} Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam Full list of author information is available at the end of the article

Abstract

We consider a non-standard inverse heat conduction problem in a bounded domain which appears in some applied subjects. We want to know the surface temperature in a body from a measured temperature history at a fixed location inside the body. This is an exponentially ill-posed problem in the sense that the solution (if it exists) does not depend continuously on the data. In this paper, we introduce the two new classes of quasi-type methods and iteration methods to solve the problem and prove that our methods are stable under botha priorianda posterioriparameter choice rules. An appropriate selection of a parameter in the scheme will get a satisfactory approximate solution. Furthermore, if we use the discrepancy principle we can avoid the selection of thea prioribound.

MSC: 35K05; 35K99; 47J06; 47H10

Keywords: Cauchy problem; sideways heat equation; ill-posed problem; error estimates

1 Introduction

In this paper, we want to determine the surface temperatureu(x,t) for  <x<Lfrom well-known temperature measurementsu(L,t) =g(t) whenu(x,t) satisfies the following system:

⎧ ⎪ ⎨ ⎪ ⎩

ut=uxx,  <x<L, ≤t≤π, u(L,t) =g(t), ≤t≤π,

u(x, ) = , ≤t≤π.

()

In order to guarantee the uniqueness of the solution, here we assume thatube bounded as

x→+∞. The functionsg,hgiven inL_{(, π) are to be noised. Physically, we only obtain} the measured Cauchy data with measurement errors. This is a severely ill-posed problem: any small perturbation in the observation data can cause large errors in the solutionu(x,t) forx∈[,L). Therefore, most classical numerical methods often fail to give an acceptable approximation of the solution. Thus it requires regularization techniques to stabilize the numerical computations [].

The problem has a long history and has attracted to many authors. In several engineering contexts, it is sometimes necessary to determine the surface temperature and heat flux in a body from a measured temperature history at a fixed location inside the body [, ]. The physical situation at the surface may be unsuitable for attaching a sensor, or the accuracy

of a surface measurement may be seriously impaired by the presence of the sensor. The special case of estimating a surface condition from interior measurements has come to be known as the inverse heat conduction problem (IHCP).

In recent years, IHCP have been researched by many authors and some valuable method are proposed, such as the difference regularization method [], the Fourier method [], quasi-reversibility method [, ], wavelet and wavelet-Galerkin method [], a spectral reg-ularization method [, ],etc.

Most of the papers involving regularization methods for sideways heat are devoted to giving a convergence analysis with the regularization parameter chosen by ana priori

choice rule, under which somea prioriinformation on the unknown exact solution which is to be simulated is often used. However, in general, thea prioriboundMcannot be known exactly in practice, and working with a wrong constantMmay lead to the bad regularized solution. In the present paper, we proposed two new regularization method: a modified quasi-boundary value method and an iteration method for solving Problem (). This method has been used by Deng and Liu to deal with the sideways parabolic equation [], but here we give another way to choose the regularization parameter by ana posteriori

rule. Moreover, we shall give a criterion for making ana posteriorichoice of the regular-ization parameter. To the authors’ knowledge, there are very few papers for choosing the regularization parameter by thea posteriorirule for this sideways heat.

This paper is constructed as follows: in Section , we propose two new regularization method. Convergence estimates under priori and posteriori assumptions are given in Sec-tion  and SecSec-tion . Finally, we draw a conclusion to our method.

2 Formulation of the problem and a quasi-boundary value regularization method

Supposeg _{is for measured data inL}_{(, π) and satisfies}

g–g_L_(,π)≤, ()

where>  represents a noisy level. Forf ∈L_{(, π), we have the Fourier series}

f(t) =

k∈Z

fkeikt, ()

where the Fourier coefficient is

fk=

 π

π



f(t)e–iktdt. ()

The norm off inLis

f_L_(,π)= π

k∈Z

|fk|. ()

The principal value of√ikis

√

ik=

⎧ ⎨ ⎩

( +i) |k_|, k≥, ( –i) |k_|, k< .

Suppose that the solution of the problem is represented as a Fourier series

u(x,t) =

k∈Z

uk(x)eikt, ()

with

uk(x) =

 π

π



u(x,t)e–iktdt. ()

LetF(x,t) =_k∈ZFk(x)eiktwith

Fk(x) =

 π

π



F(x,t)e–iktdt. ()

Then by taking the partial derivative with respect totandx, we obtain

ut(x,t) =

k∈Z

uk(x)ikeikt,

uxx(x,t) =

k∈Z d

dxuk(x)e

ikt_.

()

The main equationut=uxxleads to

k∈Z

uk(x)ikeikt=

k∈Z

d

dxuk(x)e

ikt_.

Combining the Cauchy datag,h, we have the following systems of second ordinary equa-tions:

d

dxuk(x) –ikuk(x) = , uk(L) =gk, ()

where

gk=

g(t),e–ikt=  π

π



g(t)e–iktdt.

The solution of () is

uk(x) =Ckexp(

√

ikx) +Dkexp(–

√

ikx). ()

Sinceu(x,t) is bounded whenx→+∞, we conclude thatCk= . It follows fromuk(L) = Dkexp(–

√

ikL) =gkthat

Dk=gkexp(

√

ikL).

This leads to the formula ofuk:

uk(x) =exp

√

The exact form ofuis

u(x,t) =

k∈Z

exp√ik(L–x)g(t),e–ikteikt, ≤x≤L. ()

Lemma  For y∈R,we have

e

√

iky_=e |k|y_{. ()}

Proof We can rewrite the term|e

√

iky_|by

e

√

iky_=e(+i) |k_|y

=e |k|y

cos

|k|y



+sin

|k|y



=e |k|y_{. ()}

3 Quasi-boundary value method for a sideways heat equation

Define the Sobolev spaceHs_{(, π) as}

u(,t)_Hs_(,π)=

k∈Z

 +ksu(,t),e–ikt.

In this section, we present a QBV regularization problem as follows:

⎧ ⎪ ⎨ ⎪ ⎩

uβt()=u β()

xx ,  <x<L, ≤t≤π, uβ()_{(L,t) =Bg(t), ≤t≤π,} uβ()_{(x, ) = , ≤t≤π,}

()

with the additive information thatu_{(x,t) bounded whenx→+∞. AndBis defined as}

Bg(t) =

k∈Z

exp(– |k_|L)

β() +exp(– |k_|L)

g(t),e–ikteikt. ()

3.1 A priori parameter choice rule

Theorem  Let u be an exact solution to Problem().Let gbe as in().

() Ifu(,t)L_(,π)≤Mthen byβ() =

M,we have

_uβ()_{(x,·) –u(x,·)}

L_(,π)≤M–

x

L

x L.

() Ifu(,t)Hs_(,π)≤M_then byβ() = k

M, <k< ,we have

uβ()_{(x,·) –u(x,·)}

L_(,π)



M

x L

kxL–k_+M

A(s)

L

ln(M/)

s

,

where

Proof

Proof of Part. The solution of this regularized problem is

uβ()_{(x,t) =}

k∈Z

exp(– |k_|L)exp(√ik(L–x))

β() +exp(– |k_|L)

g_(t),e–ikt_eikt_{. ()}

Letvβ()_obey

vβ()(x,t) =

k∈Z

exp(– |k_|L)exp(√ik(L–x))

β() +exp(– |k_|L)

g(t),e–ikteikt.

Step. Estimateuβ()_–vβ()

L_(,π).

We have

uβ()–vβ()_L_(,π) = π

k∈Z

exp(– |k|L) β() +exp(– |k_|L)

exp√ik(L–x) 

×g(t) –g(t),e–ikt

= π

k∈Z

exp(–√|k|x)

[β() +exp(– |k_|L)]

g(t) –g(t),e–ikt

≤β()Lx–_{g(t) –g(t)}

L_(,π)≤β() x

L–_{. ()}

This implies that

uβ()–vβ()_L_(,π)≤β()

x

L–_=M–xLLx_{. ()}

Step. Estimatevβ()_–u

L_(,π). Using u(,t),e–ikt=exp(

√

ikL)g(t),e–ikt, we have

vβ()–u_L_(,π) = π

k∈Z

β()exp( |

k| L)

 +βexp( |k_|L)

exp√ik(L–x)g(t),e–ikt

= π

k∈Z

β_|exp(–√_ikx)|

[β() +exp(– |k_|L)]

u(,t),e–ikt

=β()π

k∈Z

_exp(– |k|

x)

β() +exp(– |k_|L)



u(,t),e–ikt

≤β()β()Lx–_u(,t)

L_(,π)≤β() x

L_M_{. ()}

Fromβ() =

M, we obtain vβ()–u_L_(,π)≤ x

This implies that

vβ()–u_L_(,π)≤M–

x

LxL_{. ()}

Combining () and (), we obtain

_uβ()_–u

L_(,π)≤uβ()–vβ()_L_(,π)+vβ()–u_L_(,π)≤M–

x

L

x L_.

Proof of Part.

Lemma  Let s> ,X≥.Then for <β< ,we have

β

( + |k_|)s_(β+exp(– |k|

L))

≤sse–s +L–s L

ln(/β)

s

.

Proof

Case. |k_|∈[,_L]. It is clear to see that

β

( + |k_|)s_(β+exp(– |k|

L))

≤ β

( + |k_|)s_exp(– |k|

L)

≤βexp

_|

k|

 L

≤eβ.

From the inequalityβ≤(s_e)s₍ 

ln(/β))s, we get

β

( + |k_|)s_(β+exp(– |k|

L))

≤sse–s +L–s L

ln(/)

s

. ()

Case. |k_|>_L. SetY=exp(– |k|

L)

β . Then we obtain

β

( + |k_|)s_(β+exp(– |k|

L))

= β

β+βY

L L–ln(βY)

s

=   +Y

L L–ln(βY)

s

=   +Y

L

ln(/β)

s

–ln(β)

L–ln(βY)

s

=

L

ln(/β)

s

  +Y

–ln(β)

L–ln(βY)

s

. ()

We continue to estimate the term _+_Y(_L–_–ln(ln(_ββ_Y)₎)s_.

If  <Y≤ then  < –ln(β) < –ln(βY), thus

  +Y

–ln(β)

L–ln(βY)

s

else ifY>  thenlnY>  andln(βY) = –L |_k|< – due to the assumption |k_|∈(_L,∞). ThereforelnY( +ln(βY))≤. This implies that

 < –lnβ

L–ln(βY)< –lnβ

–ln(βY)<  +lnY.

Hence, in this case, we get

  +Y

–ln(β)

L–ln(βY)

s

<( +lnY)

s

Y = ( +lnY) s_Y–_.

Setg(Y) = ( +lnY)s_Y–_forY>e–_{. Taking the derivative of this function, we get}

g(Y) = ( +lnY)s–Y–(s–  –lnY).

The function g has a maximum at the point Y such thatg(Y) = . This implies that

Y=es–. And therefore

sup

Y≥

( +lnY)sY–≤g(Y) =sse–s. ()

Since () and () hold, we have

  +Y

–ln(β)

L–ln(βY)

s

≤sse–s.

From (), we get

β

( + |k_|)s_(β+exp(– |k|

L)) ≤sse–s

T

ln(/β)

s

≤sse–s +T–s T

ln(/β)

s

. ()

We have

v–u_L_(,π)

= π

k∈Z

_βexp(– |k|

x)

β+exp(– |k_|L)



u(,t),e–ikt

≤π

k∈Z

β

( + |k_|)s_(β+exp(– |k| L))

 +

|k|



s

u(,t),e–ikt

≤(s)se–s +L–s

L

ln(/β)

s

π

k∈Z

 +

|k|



s

u(,t),e–ikt

≤s(s)se–s +L–s

L

ln(/β)

s

π

k∈Z

 +ksu(,t),e–ikt

≤A(s)

L

ln(/β)

s

where

A(s) = s(s)se–s +L–s.

This leads to

_vβ()_–u

L_(,π)≤A(s)

L

ln(/β)

s

_u(,t) Hs_(,π)

≤MA(s)

L

ln(/β)

s

. ()

Combining () and (), we obtain

uβ()–u_L_(,π)≤u

β()_–vβ()

L_(,π)+v

β()_–u

L_(,π)

≤β()xL–_+MA(s)

L

ln(/β)

s

.

3.2 A posteriori parameter choice rule

Theorem  Suppose that <<g

L_(,π).Choose m> such that

 <m<g L_(,π).

Then there exists a unique numberβ()such that

uβ()(L,t) –g_L_(,π)=m. ()

Furthermore,ifu(,t)L_(,π)≤M then byβ() =

M,we have the following estimate:

uβ()(x,·) –u(x,·)_L_(,π)≤

m m– M

–_Lx

(m+ )

x L_.

Proof The following results are straightforward.

Lemma  Setρ(β) =uβ()_{(L,t) –g}

L_(,π).If <<g_L_(,π)thenρis a continuous

strictly increasing function and satisfies

(a) limβ→ρ(β) = ,

(b) limβ→+∞ρ(β) =gL_(,π).

From this lemma, it follows that there exists a unique numberβ() satisfying (). Setzβ()_{(x,t) =u(x,t) –u}β()_{(x,t). Thenz}β()_{satisfies the heat equation}

ztβ()=zβxx(). ()

From the formula ofuanduβ()_{, we get}

zβ()(x,t) =uβ()(x,t) –u(x,t) =

k∈Z

exp√ik(L–x)Rk

where

Rk

β,g,g=

exp(– |k_|L)

β() +exp(– |k_|L)

g(t),e–ikt–g(t),e–ikt.

Then using the Hölder inequality, we get

zβ()(x,t)_L_(,π) =

k∈Z

exp|k|L

 –x

L

Rk

β,g,g–Lx_R k

β,g,gLx

≤

k∈Z

exp|k|L

 –x

L

Rk

β,g,g–Lx



–x_L–

x L

×

k∈Z _Rk

β,g,g

x

L



x L

x L

=

k∈Z

exp|k|LRk

β,g,g

–x_L

k∈Z

Rk

β,g,g x

L

=zβ()(,t)–

x L L_(,π)z

β()_(L,t)Lx L_(,π).

This implies that

zβ()(x,t)_L_(,π)≤z

β()_(,t)–xL L_(,π)z

β()_(L,t)xL

L_(,π). ()

Proof of Part(a). It is easy to see that

_zβ()_(L,t)

L_(,π) =uβ()(L,t) –u(L,t)_L_(,π)

≤uβ()(L,t) –g_L_(,π)+g–g

L_(,π)≤(m+ ) ()

and

zβ()_(,t)

L_(,π)≤u(,t)_L_(,π)+uβ()(,t)_L_(,π). ()

We have

m =uβ()(L,t) –g(t)_L_(,π)

=

k∈Z

β()exp(– |k_|L)

 +β()exp(– |k_|L)

g(t),e–ikteikt L_(,π)

≤

k∈Z

β()exp(– |k_|L)

 +β()exp(– |k_|L)

g_{(t) –g(t),e}–ikt_eikt L_(,π)

+

k∈Z

β()

β() +exp(– |k_|L)

≤

k∈Z

g(t) –g(t),e–ikteikt L_(,π)

+

k∈Z

β()exp

_|

k|

 L

g(t),e–ikteikt L_(,π)

=g_–g

L_(,π)+β()u(,t)_L_(,π)

≤+β()u(,t)_L_(,π).

This implies that

β()≤



m– u(,t)L_(,π). ()

It follows from () that

uβ()_(,t)

L_(,π) =

k∈Z

exp(– |k_|L)

β() +exp(– |k_|L)

exp(√ikL)g_(t),e–ikt_eikt L_(,π)

≤

k∈Z

exp(– |k_|L)

β() +exp(– |k_|L)

exp(√ikL)g_{(t) –g(t),e}–ikt_eikt L_(,π)

+

k∈Z

exp(– |k_|L)

β() +exp(– |k_|L)

exp(√ikL)g(t),e–ikteikt L_(,π)

≤

k∈Z  β()

g_{(t) –g(t),e}–ikt_eikt L_(,π)

+

k∈Z

exp(√ikL)g(t),e–ikteikt L_(,π)

≤  β()g

_–

g_L_(,π)+u(,t)_L_(,π)

≤

β()+u(,t)L_(,π). ()

Combining () and (), we obtain

uβ()(,t)_L_(,π)≤



m– + 

u(,t)_L_(,π)=

m

m– u(,t)L_(,π). ()

From () and (), we obtain

zβ()(,t)_L_(,π)≤

 + m

m– 

u(,t)_L_(,π)=

m– 

m–  u(,t)L_(,π). ()

This together with () and () leads to

zβ()(x,t)_L_(,π)≤zβ()(,t)

–x_L

L_(,π)z

β()_(L,t)xL L_(,π)

≤

m– 

m– u(,t)L_(,π)

–x_L

Hence

u(x,t) –uβ()_(x,t)

L_(,π)≤

(m– )M m– 

–x_L

(m+ )LxxL_.

4 An iteration regularization method 4.1 A priori parameter choice

To the authors’ knowledge, there are few papers for choosing the regularization parameter bya posteriorirule for the sideways heat equation. In this section, a new regularization method of iteration type for solving this problem will be given. We will use this method solving Problem (), ana priorianda posteriorirule for choosing regularization param-eters and the Hölder-type error estimates are given. To solve the problem, we introduce the following iteration scheme:

u_n(x,t),e–ikt= ( –h)u_n–(x,t),e–ikt+hexp√ik(L–x)g(t),e–ikt ()

with initial guess u

(x,t),e–iktand  <h=e–L |k|

 _{< , which plays an important role in}

the convergence proof. From the formula, it is easy to see that

u

n(x,t),e

–ikt_{= ( –h)}n_u

(x,t),e–ikt

+

n–

i=

( –h)i_hexp√_{ik(L–x)g(t),e}–ikt

= ( –h)nu_(x,t),e–ikt+ – ( –h)nexp√ik(L–x)g(t),e–ikt. ()

Therefore, the approximate solution of Problem () has the following form:

u n(x,t) =

k∈Z

u

n(x,t),e–ikt

eikt. ()

We have the main theorem as follows.

Theorem  Let u(x,t)be the exact solution of Problem(),and u

nbe its regularization approximation defined by()with u

(x,t),e–ikt= .Assume thatu(,t)L_(,π)≤M

for M> and take n= [M],where[k]denotes the largest integer not exceeding k,then we have the estimate

u_n(x,·) –u(x,·)_L_(,π)≤M–

x LLx_.

Lemma  For≤h≤and n≥,the following inequalities hold:

( –h)nh≤  n+ ,  – ( –h)n

h ≤n.

Lemma  For≤h≤, ≤α≤,and n≥,the following inequalities hold:

( –h)nhα≤ 

 – ( –h)n

hα ≤n

α

.

From u

(x,t),e–ikt= , we see that

u_n(x,t) =

k∈Z

 – ( –h)nexp√ik(L–x)g(t),e–ikteikt.

Letv_{(x,t) be given by}

v n(x,t) =

k∈Z

 – ( –h)nexp√ik(L–x)g(t),e–ikteikt.

We divide the argument into two steps.

Step. Estimateu

n(x,·) –vn(x,·)L_(,π). We have

u_n(x,·) –v_n(x,·)_L_(,π)

= π

k∈Z

 – ( –h)n_exp√_{ik(L–x)g–g(t),e}–ikt

≤πsup

k∈Z

 – ( –h)nexp√ik(L–x)

k∈Z

g–g(t),e–ikt

≤πsup

k∈Z

 – ( –h)nexp|k|(L–x)g–g_L_(,π).

It follows fromexp(√|k|(L–x)) =hLx–andg–g

L_(,π)≤that

u_n(x,·) –v_n(x,·)_L_(,π)≤π sup

<h<

 – ( –h)nhLx–_.

Because of Lemma , we have

u_n(x,·) –v_n(x,·)_L_(,π)≤n– x

L_.

Therefore

u_n(x,·) –v_n(x,·)_L_(,π)≤n–

x

L_{. ()}

Step. Estimatev

n(x,·) –u(x,·)L_(,π). We have

v

n(x,·) –u(x,·)



L_(,π) = π

k∈Z

( –h)n_{exp|k|(L–x)g(t),e}–ikt

≤u(,t)_L_(,π) sup

<h<

( –h)nhLx

≤M(n+ )–Lx_.

This implies that

v_n(x,·) –u(x,·)_L_(,π)≤M(n+ )–

x

Combining () and (), we obtain

u_n(x,·) –u(x,·)_L_(,π)≤u

(x,·) –v(x,·)_L_(,π)+v

(x,·) –u(x,·)_L_(,π)

≤n–xL_{+M(n+ )}–Lx_.

Due ton= [M], we haven≤M andn+ ≥M, and therefore

u_n(x,·) –u(x,·)_L_(,π)≤M

M

–_Lx +

M

–x_L

= M–xLxL_.

4.2 The discrepancy principle

In this section, we discuss thea posterioristopping rule for iteration scheme () based on ‘discrepancy principle’ of Morozov in the following form:

g(t) –uβ(L,t)L_(,π)=m, ()

wherem>  is a constant andβdenotes the regularization parameter. If we take u

(x,t),e–ikt= , then () can be simplified to the form

g_{(t) –u}

β(L,t)L_(,π)=

k∈Z

g_(t),e–ikt_eikt_– k∈Z

 – ( –h)β_g(t),e–ikt_eikt L_(,π)

=

k∈Z

( –h)βg(t),e–ikteikt L_(,π)

=m. ()

We have the main theorem as follows.

Theorem  Let u(x,t)be the exact solution of Problem(),and u

nbe its regularization ap-proximation defined by()with u

(x,t),e–ikt= .If the a priori boundu(,t)L_(,π)≤

M is valid and the iteration()is stopped by the discrepancy principle(),then

u

β(x,·) –u(x,·)L_(,π)≤EM–

x LxL_,

where

E= (m+ )xL

m m– 

–x_L .

The following results are obvious.

Lemma  Set P(β) =g_{(t) –u}

β(L,t)L_(,π)=( –h)βg(t)_L_(,π).

Lemma  Setting

w β(x,t) =

k∈Z

exp√ik(L–x)g(t),e–ikt– – ( –h)n

the following inequality holds:

wβ(x,t)L_(,π)≤wβ(,t)

–x_L

L_(,π)w

β(L,t)

x L L_(,π).

Proof We have

w β(,t) =

k∈Z

exp(√ikL)g(t),e–ikt– – ( –h)nexp(√ikL)g_(t),e–ikt_eikt

and

wβ(L,t) =

k∈Z

g(t),e–ikt– – ( –h)ng(t),e–ikteikt.

We have

wβ(x,t)



L_(,π)=

k∈Z

exp|k|(L–x)g(t),e–ikt– – ( –h)ng(t),e–ikt.

For brevity in this case, we setQk(h,n) = g(t),e–ikt– [ – ( –h)n]g(t),e–ikt. Then using

the Hölder inequality, we get

wβ(x,t)



L_(,π) =

k∈Z

exp|k|L

 –x

L

Qk(h,n)

–_Lx

Qk(h,n)

x L

≤

k∈Z

exp|k|L

 –x

L

Qk(h,n)

–_Lx

 –x

L –x_L

×

k∈Z

Qk(h,n)

x

L



x L

x L

=

k∈Z

exp|k|L∗Qk(h,n)

–x_L

k∈Z

Qk(h,n) x

L

=w β(,t)

–_Lx

L_(,π)wβ(L,t)

x L L_(,π).

This completes the proof of the lemma.

Lemma  The following inequality holds:

β≤ M

m– .

Proof It follows from () that

m =

k∈Z

( –h)β_g(t),e–ikt_eikt

L_(,π)≤

k∈Z

( –h)β_{g(t) –g(t),e}–ikt_eikt L_(,π)

+

k∈Z

≤ sup <h<

( –h)β

k∈Z

g(t) –g(t),e–ikteikt L_(,π)

+sup <h<

( –h)β_g(t),e–ikt_eikt L_(,π).

Moreover, it is easy to see that

( –h)β≤  β

and

g(t),e–ikteikt_L_(,π)=exp(–

√

ikL)u(,t),e–ikteikt_L_(,π)≤u(,t)_L_(,π)≤M.

This leads to

β≤+M

m.

This completes the proof of the lemma.

Lemma  The following inequality holds:

w

β(L,·)L_(,π)≤(m+ ).

Proof Due to the triangle inequality, we have

wβ(L,·)_L_(,π) =

k∈Z

g(t),e–ikt– – ( –h)ng(t),e–ikteikt L_(,π)

≤

k∈Z

g(t) –g_(t),e–ikt_eikt L_(,π)

+

k∈Z

( –h)βg(t),e–ikteikt L_(,π)

≤g–g_L_(,π)+

k∈Z

( –h)βg(t),e–ikteikt L_(,π)

≤+m= (m+ ).

Lemma  The following inequality holds:

w

β(,·)L_(,π)≤

mM m– .

Proof Due to the triangle inequality andh=e–L |k|_{, we have}

w

β(,·)L_(,π)

=

k∈Z

≤

k∈Z

exp(√ikL)g(t),e–ikt– – ( –h)nexp(√ikL)g(t),e–ikteikt L_(,π)

+

k∈Z

 – ( –h)β_exp(√

ikL)g(t) –g(t),e–ikteikt L_(,π)

≤

k∈Z

( –h)βexp(√ikL)g(t),e–ikteikt L_(,π)

+

k∈Z

 – ( –h)β h

g(t) –g_(t),e–ikt_eikt L_(,π)

. ()

Moreover, we have

k∈Z

( –h)βexp(√ikL)g(t),e–ikteikt L_(,π)

≤

k∈Z

exp(√ikL)g(t),e–ikteikt L_(,π)

=u(,t)_L_(,π)≤M, ()

and using –(–_hh)β ≤β, we have the estimate

k∈Z

 – ( –h)β h

g(t) –g(t),e–ikteikt

L_(,π)≤

β

k∈Z

g(t) –g(t),e–ikteikt L_(,π)

≤βg–g_L_(,π)≤β. ()

Combining (), (), and (), we obtain

w

β(,·)L_(,π)≤M+β.

Lemma  leads to

wβ(,·)_L_(,π)≤M+

M m– =

Mm

m– .

Now, we return to the proof of the theorem.

Combining Lemma , Lemma , and Lemma , we obtain

u(x,·) –u(x,·)_L_(,π) =wβ(x,·)L_(,π)

≤w β(,t)

–x_L

L_(,π)wβ(L,t) x L L_(,π)

≤

Mm m– 

–_Lx

(m+ )xL

=EM–xLxL_,

where

E= (m+ )xL

m m– 

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

HTN and VCHL organized and wrote this manuscript. VCHL contributed to all the steps of the proofs in this research together. HTN participated in the discussion and corrected the main results. All authors read and approved the final manuscript.

Author details

1_{Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City,} Vietnam.2_{Faculty of Basic Science, Posts and Telecommunications Institute of Technology, Ho Chi Minh City, Vietnam.}

Acknowledgements

The authors wish to express their sincere thanks to the anonymous referees and the handling editor for many constructive comments, leading to the improved version of this paper. The second author is supported by Posts and Telecommunications Institute of Technology, Ho Chi Minh City, Vietnam.

Received: 22 September 2014 Accepted: 14 January 2015 References

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