WAVELET SOLUTIONS AND STABILITY ANALYSIS OF
PARABOLIC EQUATIONS
Vinod Mishra and Sabina
Department of Mathematics,
Sant Longowal Institute of Engineering & Technology,
Longowal 148 106, Punjab, India, vinodmishrasliet@rediffmail.com
Abstract. In this paper wavelet solutions of extended Sideways and non stan-dard parabolic equations have been analyzed along with stabilization and errors estimation.
Key words and Phrases: Ill-posed Problem; Regularization; Meyer Wavelet; Inverse Heat Conduction; Sideways Heat Equation; Wavelet-Galerkin Method.
Abstrak. Di dalam paper ini dianalisa kestabilan dan estimasi kesalahan untuk penyelesaian wavelet dari persamaan parabolik non standar yang diperluas satu sisi. Kata kunci: Masalahill-posed; Regularisasi; Wavelet Meyer; Konduksi panas invers; Persamaan panas satu sisi; Metode Wavelet-Galerkin.
1. Sideways Heat Equation: An Introduction
Ill-posed problems have always been in the focus of industrial applications. An inverse ill-posed problem is one for which a small perturbation on the boundary specification (g) can amount to a big alteration on its solution, if it exists. That is, if the solution exists, it does not depend continuously on data (g). Meyer multiresolution analysis plays a key role in the solution of parabolic heat conduction problems.
Organization of paper is as follows. Wavelet regularized and Galerkin solu-tions of Standard Sideways heat equation (SHE) along with stability and errors estimation has been reviewed as first part. In the second part, wavelet regulariza-tion of extended SHE has been obtained and errors have been estimated. Third part introduces inequality based wavelet-Galerkin solution of extended SHE. In it
2000 Mathematics Subject Classification: 65T60 Received: 09-02-2010, accepted: 25-03-2010.
numerical solution of non standard parabolic equations of extended SHE has been subjected to stability consideration. Fourth part is devoted to test problem while fifth part is the conclusion. At the end set of references are listed.
Consider the following one dimensional parabolic heat conduction problem in quarter plane (x≥0, t≥0), assuming that body is large,
uxx=ut, x≥0, t≥0;
u(x,0) = 0, x≥0;
u(1, t) =g(t), t≥0, u|x→∞bounded.
(1)
Assumptionu|x→∞bounded guarantee the uniqueness of solution. A solution
u(x, t) ∈ L2(0,∞) for x ≥ 0 can be obtained from initial temperature data g. This equation called Sideways heat equation, is an inverse ill posed problem. This equation is a model of a situation where one wants to determine the surface heat flux (temperature) on the both sides of a heat conducting body from measured transient temperature at a fixed location inside the body. The simplest example can of lacquer coating to be applied on one side of a particle board. The surface under coating is inaccessible to measurement due to extreme heat condition. Inside temperature gε is measured atx= 1 indirectly through a thermocouple inserted
into plate through other side. Based on internal measurements, surface (x= 0) temperature is computed.
But even with precautions, measurement errors are bound to occur ing. Let
gε∈L2(R) be the perturbed data such that data error
kg−gεk ≤ε (2)
for some bound ε >0. Imposea priori bound on the solution atx= 0, i.e.,
ku(0, t)k ≤M. (3)
The problem 1 can now be moduled as
uxx=ut, x≥0, t≥0;
u(x,0) = 0, x≥0;
ku(1, t)−gεk ≤ε
ku(0, t)k ≤M
(4)
The problem now becomes well posed, that is, stable in the sense that for any two solutions u1 andu2
ku1(x, t)−u2(x, t)k ≤2M1−xεx,0≤x <1.
This was proved by Levine in 1983. The problem 4 was approximated for the first time by [18] and [11] by multiscale analysis and wavelet techniques of measured data. In frequency space,u(x, t), g(t), gε(t) extended to whole of t-axis by defining
Although we intend to recoverx >0 for 0≤x <1, the problem specification includes the heat equation for x > 1 together with boundedness at infinity. To obtain u(x, t) for x > 1, ux(1, t) is determined also. Equation 4 together with
ux(1, t) is a Cauchy problem.
Mattos and Lopes [14] gave another version of 1
k(x)uxx(x, t) =ut(x, t), t≥0,0≤x <1 [0< α≤k(x)<∞]
u(0, t) =g(t)
ux(0, t) = 0
wherek(x) is smooth. While Elden et al. [4, 3] gave yet another type:
(k(x)ux(x, t))x=ut(x, t), t≥0,0≤x <1 [0< α≤k(x)<∞];
u(0, t) =g(t)
ux(0, t) = 0
There are quite other methods available for solving various parabolic heat conduction equations as difference approximation, optimal filtering, optimal ap-proximation, method of lines, dual least square, singular value analysis and spectral and Tikhonov regularizations. See, for details, [22], [13], [5, 6], [20], [8, 10] and [24]. Taking Fourier transform (FT) on both sides of 1 w.r.t. t, the frequency space solution ˆu(x, ω)∈L2(R) is
ˆ
u(x, ω) =e(1−x)√iωgˆ(ω) (5)
Also ˆf(ω) = ˆu(0, ω) = e√iωgˆ(ω). Since √iω tends to infinity as |ω| → ∞, the
problem thus is ill-posed. Further, the existence of the solution inL2(R) depends on fast decay of ˆgεat high frequencies. The solutionu(x, t) to 1 is
u(x, t) =
Z ∞
−∞
eiωte(1−x)√iωˆg(ω)dω. (6)
By Parseval formula,
ku(x, t)k2=kuˆ(x, ω)k2=
Z ∞
−∞
e(1−x)√2|ω||gˆ(ω)|2dω,
where
√
|ω|
2 is real part of √
iω. This shows that ˆg(ω) has to decay rapidly as
ω→ ∞. If the initial datagis noisy, the ˆg(ω) will have high frequency components and are to be cut by the Meyer multiresolution analysis.
˜
u(x, t) =
Z ∞
−∞
eiωte(1−x)√iωˆgε(ω)χmaxdω, (7)
whereχmax is the characteristic function of the interval [−ωmax, ωmax]. Here we present some error estimates from [2].
Theorem 1.1. Suppose that we have two regularized solutions u˜1 andu˜2 defined by 7 with datag1andg2satisfyingkg1−g2k< ε. If we selectωmax= 2
¡
log¡M ε
¢¢2 , then we get the error bound
ku˜1(x, t)−u˜2(x, t)k< M1−xεx.
Theorem 1.2. Letuandu˜defined be the solutions of 6 and 7 with the same exact datag and letωmax= 2
¡
log¡M
ε
¢¢2 . Then
ku(x, t)−u˜(x, t)k< M1−xεx.
Theorem 1.3. Suppose thatuis given by 6 with exact datagand that u˜given by 7 with measured data gε. Selectωmax= 2¡log¡Mε¢¢
2
, then we get the error bound
ku(x, t)−u˜(x, t)k<2M1−xεx.
Meyer Multiresolution Analysis and Wavelet Regularization (MRA)
Letαj = 2jα0, whereα0 = 23π, j∈Z. The FT of Meyer scaling is given by [1]
ˆ
ϕ(ω) =
1, |ω| ≤α0;
coshπ2ν³|2ωα|−1´i, α0≤ |ω| ≤α1;
0, otherwise.
whereν isCk differentiable function (0≤k≤ ∞) satisfying
ν(x) =
(
1, x≤0; 0, x≥0.
with additional conditionν(x) +ν(1−x) = 1.
Clearly ˆϕis a Ck function. Corresponding wavelet is given by
ˆ
ψ(ω) =
eiω/2sinhπ
2ν ³
|ω| 2α −1
´i
, α0≤ |ω|;
eiω/2coshπ
2ν ³
|ω| 2α −1
´i
, α0≤ |ω| ≤α1;
0, otherwise.
suppϕˆ= [−α1, α1]
suppψˆ= [−α2,−α0]∪[α0, α2]
Vj={ϕj,k, k∈Z}; ϕj,k= 2j/2ϕ(2jx−k), j, k∈Z
ˆ
ϕj,k=
Z
R
ϕj,ke−ixωdx= 2−j/2e−ik2
−jω ˆ
ϕ(2−jω).
Alsoψj,k(x) = 2j/2ψ(2jx−k), j, k ∈Z constitutes an orthonormal basis ofWj ∈
L2(R)(V
j+1=Vj⊕Wj).
ˆ
ψj,k= 2−j/2e−ik2
−jω ˆ
ψ(2−jω) suppϕˆj,k= [−αj+1, αj+1], k∈Z
suppψˆj,k= [−αj+2,−αj]∪[αj, αj+2], k∈Z
Let Πj andPj (fixedj∈N) be the orthogonal projections ofL2(R) ontoVj
andWj respectively. Then forh, w∈L2(R)
h= Πjh(t) =
X
k∈Z
hh, ϕlkiϕlk(t), l≤j
w=Pjw(t) =
X
k∈Z
hh, ψlkiψlk(t), l≤j
The corresponding orthogonal projections in frequency space follow as:
ˆ
Πj:L2(R)→Vˆj =span{ϕˆjk}k∈Z
ˆ
Pj:L2(R)→Wˆj=span{ψˆjk}k∈Z
According to 5, for any functiong∈L2(R) such that its FT ˆgbelongs toV
j,
there exists a solution ˆuwith boundary condition ˆu(1, ω) = ˆg(ω)(ˆg(ω) = ˆh(ω)). The whole mechanism suggests for a Fourier regularization process involving a family of problems in the frequency space parameterized byj∈Z defined by
½
ˆ
uxx(x, ω) =iωuˆ(x, ω), ω∈R,0≤x <∞;
ˆ
u(1, ω) = Πjgˆε(ω), t≥0,uˆ|x→∞ bounded. (8)
This has a unique solution since support of Πjˆgεis compact. The solution is
ˆ
u(x, ω) =e(1−x)√iωΠjgˆε(ω). (9)
For wavelet regularization,
½
ˆ
uxx(x, ω) =iωuˆ(x, ω), ω∈R,0≤x <∞;
ˆ
u(1, ω) =Pjˆgε(ω), t≥0,uˆ|x→∞ bounded. (10)
The unique solution ˆudoes not have any high frequency components as support of
ˆ
u(x, ω) =e(1−x)√ωPjgˆε(ω). (11)
Notice that
ˆ
g= ˆΠjgˆ+ (1−Πˆj)ˆg= ˆΠjgˆ+ ˆPjˆg=
X
k∈Z
hˆg,ϕˆlkiϕˆlk+
X
l≥j
X
k∈Z
hˆg,ψˆlkiψˆlk
This implies
ˆ
Πjgˆ= ˆg for|ω| ≤αj or |ω| ≥αj+2 since ˆψj,k(ω) = 0∀l≤j
ˆ
Pjgˆ= ˆg for|ω| ≥αj+1 ˆ
Pjgˆ= 0 forl > j and|ω| ≤αj+1
Pjcan be considered as a low pass filter as frequencies higher thanαj+1are filtered away.
Theorem 1.4. [18] Let gε be the measured data satisfying 2. Let uεj(x, t)denote
the inverse Fourier transform of the solution of 8 withg =gε. Ifj =j(ε)is such
that
εe√αj+1≤M andj(ε)→ ∞ whenε→0,
then for0≤x≤1
ku(x, t)−uεj(x, t)k →0 whenε→0,
and, moreover,
kuεj(x, t)−u(x, t)k2≤M2(1−x)£ε2x+ε2jx
¤
, whereεj=M e−
√
αj+1 (12)
Theorem 1.5. [18] Let gε be the measured data satisfying 2. Let vεj(x, t)denote
the inverse Fourier transform of the solution of 12 withg=gε. Ifj=j(ε)is such
that
εe√αj+1≤M andj(ε)→ ∞ whenε→0,
then forx > √√2−1 2
kvεj(x, t)−v(x, t)k →0 whenε→0,
and, the following inequality holds:
ku(x, t)−vεj(x, t)k2≤2M2(1−x)
£
2ε2x+ε2jx
¤
+ 2CjM2(1−
√ 2/2−x)
ε2(x+
√ 2/2−1)
j ,
whereεj =M e−
√
αj+1 and{C
j} is a certain sequence converging to 0 as j→ ∞.
Theorem 1.6. [18] Let gε be the measured data satisfying 2. Let {xk} be the
decreasing sequence of knots that holds:
1> x0> ³
1−2−12
´
x1>· · ·> xk−1> xk >
³
1−2−k+12
´
xk−1, k= 1,2,· · ·
ˆ
vεj(x, ω) =
(
e(1−x)√iωPjˆgε(ω), x∈[x0,1); e(xk−1−x)
√
iωP
j−kvˆεj(xk−1, ω), x∈[xk, xk−1), k= 1,2,· · ·.
Then for any x∈(0,1)
kvεj(x, t)−u(x, t)k →0when ε→0.
Remarks. According to [9] interval (0,1) is to be replaced by (e∗,1), where
e∗ = limk→∞ek, ek =
³
1−2−1 2
´ ³
1−2−2 2
´
· · ·³1−2−k
2
´
and 0.037513 < e∗ <
0.037514.
Galerkin Solution of 1 in Scaling Spaces Vj
Approximating solution of 1 in scaling spaces
huxx−ut, ϕjki = 0,
hu(0, t), ϕjki = hPjg, ϕjki, (13)
hux(0, t), ϕjki = h0, ϕjki, k∈Z.
where ϕjk is the orthonormal basis ofVj given by the scaling functionϕ. Letting
the approximate solutionuj(x, t)∈Vj be in the form of variable separable
uj(x, t) =
X
l∈Z
Wl(x)ϕjl(t).
The equation 13 reduces to infinite dimensional differential equation?? d2W
dx2 =Dj(x)W withW(1) =γ, W′(1) =ν, (14) where γ = Pjg = Pl∈Zγlϕjl = Pl∈Zhg, ϕjliϕjl. ν = Pjh = Pl∈Zνlϕjl =
P
l∈Zhν, ϕjliϕjl. Here kDj(x)k ≤ π2−j. Solution W = γe((1−x)
√
Dj) for exact
dataγ.
Theorem 1.7. [19]Let uj andu˜j be solutions in Vj of the approximating problem
1 withg=Pjg for the boundary specificationsg andg˜respectively. Ifkg−˜gk< ε,
then
kuj(x, t)−u˜j(x, t)k ≤e
³
(1−x)√1 22−jπ
´ . Forj=j(ε)such that2−j ≤ 2
π
¡
logMε¢
, we have
kuj(x, t)−u˜j(x, t)k ≤εxM1−x.
Theorem 1.8. [19] If uis a solution of problem 1, then
ku(x, t)−Pju(x, t)k ≤M e
j
−x√1 3π2−j
k ,
Forj such that 2−j≤ 3
πlogε−
1, we have
ku(x, t)−Pju(x, t)k ≤M ε−x 2
2. Wavelet Regularization of Extended Sideways Heat Equation
Now we find wavelet regularization of the other version of sideways heat conduction problem, the extended SHE. Consider the following heat conduction problem [Mattos et al.]:
k(x)uxx(x, t) =ut(x, t), [0< α≤k(x)<∞];
u(0, t) =g(t), ux(0, t) = 0,
(15)
whereg∈L2(R) is such that the measured datag
ε satisfieskg−gεk< εfor some
constant ε >0, and 0< α≤k(x)<∞, k continuous. To findu∈L2(R) subject toa priori boundkFk=ku(1, t)k ≤M, andu|x→∞bounded.
Defineu, g, F to the wholet-axis by defining them to be zero fort <0. Taking FT of 15 w.r.t. t,
k Z ∞
0
e−iωtuxxdt=
Z ∞
0
e−iωtutdtor ˆuxx=
iω k(x)u.ˆ This implies
ˆ
u(x, ω) =Aex√iω/k+Be−x√iω/k (16) But ˆu(0, ω) = ˆg(ω), and so
ˆ
g(ω) =A+B (17)
Differentiating 16 w.r.t. x,
ˆ
ux(x, ω) =
iω k
h
Aex√iω/k−Be−x√iω/ki. (18)
Using ˆux(0, ω) = 0 in 18,
0 =A−B. (19)
From 17 and 19,
A=B= 1 2gˆ(ω). So, the frequency space solution ˆu(x, ω)∈L2(R) is
ˆ
u(x, ω) = cosh
r iω
k xgˆ(ω) (20)
ˆ
F = ˆu(1, ω) = cosh
r iω
k ˆg(ω)
Parseval’s formula yields
kuk2=kuˆk2=
Z ∞
−∞
¯ ¯ ¯ ¯ ¯
cosh
r iω
k ˆg(ω) ¯ ¯ ¯ ¯ ¯ 2
dω
showing the rapid decay of ˆg(ω) at high frequencies. Fourier regularized solution
ˆ
u(x, ω) = cosh
r iω
Wavelet regularized solution of (22 is)
ˆ
u(x, ω) = cosh
r iω
k xPˆjgˆε(ω).
Stability and Error Estimation
We state and approve the following theorems:
Theorem 2.1. Let gbe the true data of the problem 15 andgε the noisy measured
data satisfyingkg−gεk ≤εfor some ε >0. Let there exist a priory bound kFk=
kcoshqiω
k ˆgk ≤ M. Further, assume that j =j(ε) be such εcosh
q
iαj+2
k ≥ M.
Then
kgˆ−Pˆjgεk=kg−Pjgεk ≤
M
cosh
q
iαj+2 k
+
Cj
ε2+ M 2
cosh
q
iαj+2 k
1 2
whereCj is a certain sequence converging to zero asj→ ∞.
Lemma 2.2. kˆg−Πjgˆεk2≤ M 2
cosh2qiαj+2
k
Proof.
kgˆ−Πjˆgεk2 =
Z ∞
−∞| ˆ
g−Πjgˆε|2dω
=
Z
|ω|≤αj+2
|gˆ−Πjˆgε|2dω+
Z
|ω|≥αj+2
|gˆ−Πjgˆε|2dω
=
Z
|ω|≥αj+2
|gˆ|2dω=
Z
|ω|≥αj+2
|Fˆ|2 cosh2qiω
k
dω≤ M
2
cosh2
q
iαj+2 k
.
¤
Lemma 2.3. kΠjgˆε−Pjˆgεk2≤Cj
Ã
ε2+ M2
cosh2qiαj+2
k
Proof.
using Lemma 2.2 and Lemma 2.3. ✷
Theorem 2.4. Let gbe the true data of the problem 15 andgε the noisy measured
data satisfying kg−gεk ≤ε for someε >0. Let there exist a priory boundkFk=
kcoshqiωk ˆgk ≤ M. Further, assume that j =j(ε) be such εcosh
q
iαj+2
using Lemma 2.3 ¤
Theorem 2.7. Let gbe the true data of the problem 15 andgε the noisy measured
data satisfying kg−gεk ≤ε for someε >0. Let there exist a priory boundkFk=
3. Inequality Based Wavelet-Galerkin Solutions
A. Wavelet-Galerkin Solution of Sideways Heat Equation
infinite dimensional second order initial value ordinary differential equation with variable coefficients.
Lemma 3.1. If{ϕjk}k∈Z is the orthogonal basis of scaling spacesVj such that the
matrix
⌊(Dj)lk(x)⌋l,k∈Z =
¹
1
k(x)hϕ ′
jl, ϕjki
º
l,k∈Z
.
The matrixDj is skew symmetric and equal along diagonals. Moreover,kDj(x)k ≤ π2−j
k(x).
Theorem 3.2. [14]Let uandvbe positive continuous functions,x≥aandc >0. If
u(x) =c+
Z x
a
Z s
a
v(τ)u(τ)ds
then
u(x)≤ceRax
Rs
av(τ)u(τ)ds.
Solution of 15 in Frequency Domain
k(x)ˆuxx(x, ω) =iωuˆ(x, ω), ω∈R,0≤x <1;
ˆ
u(0, ω) = ˆg(ω),
ˆ
ux(0, ω) = 0.
(21)
ˆ
u(0, ω) = ˆg(ω) +
Z x
0 Z s
0 iω
k(τ)uˆ(τ, ω)dτ ds Using Theorem 3.2,
|uˆ(0, ω)| ≤ |gˆ|e|ω|R0x
Rs
0 1
k(τ)dτ ds
Galerkin Solution of 15 in Scaling Spaces Vj
Approximating solution of 21 in scaling spaces
hk(x)uxx−ut, ϕjki = 0
hu(0, t), ϕjki = hPjg, ϕjki, (22)
hux(0, t), ϕjki = h0, ϕjki, k∈Z.
whereϕjk is the orthonormal basis ofVj given by the scaling functionϕ.
Letting the approximate solutionuj(x, t)∈Vj be
uj(x, t) =
X
l∈Z
Wl(x)ϕjl(t).
The equation 22 reduces to infinite dimensional differential equation [14]
d2W
dx2 =Dj(x)W withW(0) =γ, W
′(0) = 0, (23)
where γ=Pjg =Pz∈Zγzϕjz =Pz∈Zhg, ϕjziϕjz. Solution of 23 is analogous to
W(x) =γ+
Theorem 3.3. [14]Let uj andu˜j be solutions in Vj of the approximating problem
15 withg=Pjg for the boundary specificationsgandg˜respectively. Ifkg−˜gk< ε,
B. Wavelet-Galerkin Solution of Non Standard Parabolic Equation
We have stated and proved the following Theorem and found the solution of 27 in frequency domain as well as in scaling spaces. For details, refer to authors’ paper [16].
Consider the following heat conduction problem
Solution of 27 in Frequency Domain
Using Theorem 3.5
Galerkin Solution of 27 in Scaling Spaces
kW(x)k ≤ kγke
Stability of Wavelet-Galerkin Method
We prove the following convergence theorems.
Theorem 3.6. Let uj and u˜j be solutions inVj of the approximating problem 27
Graph of norm error versus space axis is
Equation 31 yields
kuj(x, t)−Pju(x, t)k ≤M ε(1−x 2
)e−E2(1−x2) (33)
forj≥ log[(2π/3αlog 2)(1/logε−1)]
Graph of norm error error versus space axis is
In authors paper [16], inequality based wavelet-Galerkin solution of 27 has been found by taking k(x) equal to (x+a)2 and errors have been computed and compared at various values ofa.
5. Conclusion
References
[1] Daubechies, I.,Ten Lectures on Wavelets, SIAM Publications, Philadelphia, 1992.
[2] Elden, L., F. Berntsson and T. Regniska, ”Wavelet and Fourier Methods for Solving the Sideways Heat Equation”,SIAM J. Sci. Compu.21(2000), 2187-2205.
[3] Elden, L., Numerical Solution of the Sideways Heat Equation. In: Inverse Problems in Diffussion Process (ed. H. Engl and W. Rundell), SIAM, Philadelphia, 1995, 130-150. [4] Elden, L.,The Numerical Solution of a Non-characteristic Cauchy Problem for a Parabolic
Equation, in: Numerical Treatment of Inverse Problems in Differential and Integral Equa-tions (ed. P. Deuflhard and E. Hairer), Proceedings of an Internal Workshop, Heidelberg, 1982, Birkhauser, Boston, 1983, 246-268.
[5] Elden. L., ”Numerical Solution of the Sideways Heat Equation by Difference Approximation in Time”,Inverse Problems11(1995), 913-923.
[6] Elden. L., ”Solving an Inverse Conduction Problem by ’Method of Lines’”,Transaction of ASME J. of Heat Transfer 119(1997), 406-412.
[7] Engl, H.W. and P. Manselli, ”Stability Estimates and Regularization for an Inverse Heat Con-duction Problem in Semi-infinite and Finite Time Intervals”,Numer. Funct. Anal. Optimiz.
10(1989), 517-540.
[8] Fu, Chu-li, Xiang-Tuan Xiong, Hong-fong Li and You-bin Zhu, ”Wavelet and Spectral Reg-ularization Methods for a Sideways Parabolic Equation”,xxx(2004), pp.1-28.
[9] Fu, Chu-li, You-bin Zhu and Chun-yu Qiu, ”Wavelet Regularization for an Inverse Heat Conduction Problem”,Math. Analy. Appl.288(2003), 212-222.
[10] Fu, Peng, Chu-li Fu, Xiang-Tuan Xiong and Hong Feng, ”Regularization Methods and the Order Optimal Error Estimates for a Sideways Parabolic Equations”,Compu. Math. Appl.
49(2005), 777-788.
[11] Hao, D.N., A. Schneider and H.-J. Reinhardt, ”Regularization of a Non-characteristic Cauchy Problem for a Parabolic Equation”,Inverse Problems11(1995), 1247-1263.
[12] Knabner, P. and S. Vessella,Stability Estimates for Ill-posed Cauchy Problems for Parabolic Equations, Inverse and Ill-posed Probems (ed. H.W. Engl and C.W. Groetsch), St. Wolfgang: Academic Press, 1987, 351-368.
[13] Lamm, P.K. and Lars Elden, ”Numerical Solution of First-Kind Volterra Equations by Se-quential Tikhonov Regularization”,SIAM J. Numer. Anal.34(1997), 1433-1450.
[14] Mattos, Linhares de, Jose Roberto, and Ernesto Prado Lopes, ”A Wavelet-Galerkin Method Applied to Partial Differential Equations with Variable Coefficients”,Fifth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference10, 2003, 211-225.
[15] Mejia, C.E. and Murio, D.A., ”Numerical Solution of Generalized IHCP by Discrete Mollifi-cation”,Compu. Math. Appl.32(1996), 33-50.
[16] Mishra,Vinod and Sabina, ”Wavelet Solutions of Parabolic Equations”, Matematika 26
(2010), 61-69.
[17] Qiu, Chun-yu, Chu-li Fu and You-bin Zhu, ”Wavelets and Regularization of the Sideways Heat Equation”,Compu. Math. Appl.46(2003), 821-829.
[18] Reginska, T., ”Sideways Heat Equation and Wavelets”,J. Compu. Appl. Math.63(1995),
209-204.
[19] Regniska, T., and L. Elden, ”Solving the Sideways Heat Conduction by a Wavelet -Galerkin Method”,Inverse Problems 13(1997), 1093-1106.
[20] Regniska, T., ”Application of Wavelet Shrinkage to Solving the Sideways Heat Equation”, BIT Numer. Math.41(2001), 1101-1110.
[21] Regniska, T., ”Stability and Convergence of a Wavelet Galerkin -Method for the Sideways Heat Equation”,J. Inverse Ill Posed Probl.8(2000), 31-49.
[22] Seidman, T. and L. Elden, ”An Optimal Filtering Method by Sideways Heat Equation by Difference Approximation in Time”,Inverse Problems 11(1995), 913-923.
