We consider the following Cauchy problem for the elliptic equation with inhomogeneous source in a rectangular domain with Dirichlet boundary conditions at x=0 and x=π. The
problem is ill-posed. The main aim of this paper is to introduce a regularization method and use it to solve the problem. Some sharp error estimates between the exact solution and its regularization approximation are given and a numerical example shows that the method works effectively.
1. Introduction
The Cauchy problem for the elliptic equation has been extensively investigated in many practical areas. For example, some problems relating to geophysics [1], plasma physics [2], and bioelectric field problems [3] are equivalent to solving the Cauchy problem for the elliptic equation. In this paper, we consider the following Cauchy problem for elliptic equation with nonhomogeneous source:
(1)uxx+uyy=f(x,y),(x,y)∈(0,π)×(0,1),(2)u(0,y)=u(π,y)=0,(3)uy(x,0)=0,(4)u(x,0)=g(x),
where g∈L2(0,π), f∈L2(0,1;L2(0,π)) are given.
Problem (1)–(4) is well known to be ill-posed in the sense of Hadamard: a small perturbation in the data g may cause dramatically large errors in the solution u(x,y) for 0<y≤1. An explicit example to emphasize this fact is given in [4]. In the past, there were many studies on the homogeneous problem, that is f=0 in (1). Using the boundary element method, the homogeneous problems were considered in [5–7] and the references therein. Similarly, many authors have investigated the Cauchy problem for linear homogeneous elliptic equation, for example, the quasireversibility method [4, 8–10], fourth-order modified method [11, 12], Fourier truncation regularized method (or spectral regularized method) [13–15], the Backus-Gilbert algorithm [16] and so forth. Some other authors also considered the homogeneous problem such as Beskos [5], Eldén et al. [17, 18], Marin and Lesnic [19], Qin and Wei [20], Regińska and Tautenhahn [21], Tautenhahn [22].
Very recently, in 2009, Hào and his group [23] applied the nonlocal-boundary value method to regularize the abstract homogeneous elliptic problem. This method is also given in [24] for solving an elliptic problem with homogeneous source in a cylindrical domain. A mollification regularization method for the Cauchy problem in a multidimensional case has been considered in the recent paper of Cheng and his group [25].
Although there are many papers on the homogeneous elliptic equation, the result on the inhomogeneous case is very scarce, while the inhomogeneous case is, of course, more general and nearer to practical application than the homogeneous one. Shortly, it allows the occurrence of some elliptic source which is inevitable in nature. The main aim of this paper is to present a simple and effective regularization method and investigate the error estimate between the regularization solution and the exact solution. In a sense, this paper may be an extension of many previous results.
The remainder of the paper is divided into two sections. In Section 2, we will study the regularization of problem (1)–(4) and obtain convergence estimates. In Section 3, a numerical test case for inhomogeneous problems is given to describe the effectiveness of our method.
2. Regularization and Error Estimate
By the method of separation of variables, the solution of problem (1)–(4) is given by
(5)u(x,y)=∑n=1∞[(eny+e-ny2)gn+∫0y(en(y-s)-en(s-y)2n)fn(s)ds]sinnx,
where
(6)gn=2π∫0πg(x)sinnxdx,fn(s)=2π∫0πf(x,s)sinnxdx.
We can see that the instability is caused by the fast growth of eny, y>0 as n tends to infinity. Even though these exact Fourier coefficients gn,fn(s) may tend to zero rapidly, in practice, performing classical calculation is impossible because the given data is usually diffused by a variety of reasons such as round-off error and measurement error. A small perturbation in the data can arbitrarily deduce a large error in the solution. Therefore, some special regularization methods are required. From (5), we replace the term eny that causes dramatically the increasing of the right side by several bounded approximations. We assume that the exact data g(x) and the measured data gϵ(x) both belong to L2(0,π) and satisfy ∥gϵ-g∥2≤ϵ where ∥·∥2 is the norm on L2(0,π) and ϵ denotes the noise level, respectively.
In the paper, we will use a modification method to regularize our problem. The regularized solution is given as follows:
(7)uϵ(x,y)=∑n=1∞[∫0y12n(e-nsα+e-ny-en(s-y))fn(s)ds12(1α+e-ny+e-ny)gn+∫0y12n(e-nsα+e-ny-en(s-y))fn(s)ds]sinnx.
Here α∈(0,1) is a parameter regularization which depends on ϵ. The explicit error estimates including error estimates have been given according to some priori assumptions on the regularity of the exact solution.
Let vϵ be the solution of problem (7) corresponding to the measured data gϵ. Then, it is given by
(8)vϵ(x,y)=∑n=1∞[12(1α+e-ny+e-ny)gnϵ+∫0y12n(e-nsα+e-ny-en(s-y))fn(s)ds+∫0y12n(e-nsα+e-ny-en(s-y))fn(s)ds]sinnx.
We first have the following theorem.
Theorem 1.
Let g,gϵ∈L2(0,π) such that ∥gϵ-g∥2≤ϵ. Then one has
(9)∥vϵ(·,y)-uϵ(·,y)∥2≤α-1ϵ,
for all y∈[0,1].
Proof.
It follows from (7) and (8) that
(10)vϵ(·,y)-uϵ(·,y)=∑n=1∞[12(1α+e-ny+e-ny)(gnϵ-gn)]sinnx.
We have
(11)∥vϵ(·,y)-uϵ(·,y)∥22=π2∑n=1∞14(1α+e-ny+e-ny)2|gnϵ-gn|2≤π8∑n=1∞(1α+1)2|gnϵ-gn|2≤α-2∥gϵ-g∥22≤α-2ϵ2.
Therefore, we get
(12)∥vϵ(·,y)-uϵ(·,y)∥2≤α-1ϵ.
This completes the proof of Theorem 1.
Theorem 2.
Let g,gϵ be as in Theorem 1. Assume that ∫01∑n=1∞e2nfn2(s)ds<∞. If we select α=ϵ1/2, then for every y∈[0,1] one has
(13)∥vϵ(·,y)-u(·,y)∥2≤Mϵ(1-y)/2,
where
(14)M=1+32×∥u(·,1)∥22+∥uy(·,1)∥22+π2∫01∑n=1∞e2nfn2(s)ds.
Proof.
First, we have
(15)un(y)=(eny+e-ny2)gn+∫0y(en(y-s)-en(s-y)2n)fn(s)ds,(16)unϵ(y)=12(1α+e-ny+e-ny)gn+∫0y12n(e-nsα+e-ny-en(s-y))fn(s)ds.
Subtracting (16) to (15), we have
(17)unϵ(y)-un(y)=12(1α+e-ny-eny)gn+12n∫0y(e-nsα+e-ny-en(y-s))fn(s)ds=-12(αα+e-ny)[enygn+1n∫0yen(y-s)fn(s)ds].
We have
(18)uy(x,y)=∑n=1∞[n(eny-e-ny2)gn+∫0yn(en(y-s)+en(s-y)2n)fn(s)ds+∫0yn(en(y-s)+en(s-y)2n)fn(s)ds]sinnx=∑n=1∞n[(eny-e-ny2)gn+∫0y(en(y-s)+en(s-y)2n)fn(s)ds+∫0y(en(y-s)+en(s-y)2n)fn(s)ds]sinnx.
From (15), we have
(19)1nddyun(y)=(eny-e-ny2)gn+∫0y(en(y-s)+en(s-y)2n)fn(s)ds.
Combining (15) and (19), we get
(20)un(y)+1nddyun(y)=enygn+1n∫0yen(y-s)fn(s)ds.
Let y=1; we have
(21)un(1)+1nddyun(1)=engn+1n∫01en(1-s)fn(s)ds=en[gn+1n∫01e-nsfn(s)ds].
Therefore, we get
(22)gn=e-n[un(1)+1nddyun(1)]-1n∫01e-nsfn(s)ds.
From (17) and (22), we have
(23)|unϵ(y)-un(y)|=12(αα+e-ny)|eny[e-nun(1)+e-nnddyun(1)-1n∫01e-nsfn(s)ds]+eny[e-nun(1)+e-nnddyun(1)1n∫0yen(y-s)fn(s)ds|=12(αα+e-ny)|en(y-1)[un(1)+1nddyun(1)]-1n∫y1en(y-s)fn(s)ds|≤12α(e-n(1-y)α+e-n)[|un(1)|+|ddyun(1)|+∫01|enfn(s)|ds|un(1)|+|ddyun(1)|].
Moreover, one has, for τ>t>0 and α>0,
(24)e-tnα+e-τn=1(αeτn+1)t/τ(α+e-τn)1-t/τ≤αt/τ-1.
Letting τ=1, t=1-y, we get
(25)e-n(1-y)α+e-n≤α-y.
From (23), (25), we have
(26)|unϵ(y)-un(y)|≤12α1-y[|un(1)|+|ddyun(1)|+∫01|enfn(s)|ds].
Applying the inequality (a+b+c)2≤3(a2+b2+c2), we have
(27)|unϵ(y)-un(y)|2≤34α2(1-y)[|ddyun(1)|2|un(1)|2+|ddyun(1)|2+(∫01|enfn(s)|ds)2]≤34α2(1-y)[|ddyun(1)|2+01e2nfn2(s)ds|un(1)|2+|ddyun(1)|2+∫01e2nfn2(s)ds].
Thus
(28)∥uϵ(·,y)-u(·,y)∥22=π2∑n=1∞|unϵ(y)-un(y)|2≤34α2(1-y)(π2∑n=1∞|un(1)|2+π2∑n=1∞|ddyun(1)|2+π2∑n=1∞∫01e2nfn2(s)ds)=34α2(1-y)(π201∑n=1∞e2nfn2(s)ds∥u(·,1)∥22+∥uy(·,1)∥22+π2∫01∑n=1∞e2nfn2(s)ds),
or we get
(29)∥uϵ(·,y)-u(·,y)∥2≤32α1-y∥u(·,1)∥22+∥uy(·,1)∥22+π2∫01∑n=1∞e2nfn2(s)ds.
Using Theorem 1 and (29), we get
(30)∥vϵ(·,y)-u(·,y)∥2≤∥vϵ(·,y)-uϵ(·,y)∥2+∥uϵ(·,y)-u(·,y)∥2≤α-1ϵ+32α1-y∥u(·,1)∥22+∥uy(·,1)∥22+π2∫01∑n=1∞e2nfn2(s)ds=ϵ1/2+32ϵ(1-y)/2∥u(·,1)∥22+∥uy(·,1)∥22+π2∫01∑n=1∞e2nfn2(s)ds≤Mϵ(1-y)/2,
where
(31)M=1+32×∥u(·,1)∥22+∥uy(·,1)∥22+π2∫01∑n=1∞e2nfn2(s)ds.
This completes the proof of Theorem 2.
Remark 3.
From (13), as y→1, the accuracy of regularized solution becomes progressively lower. To retain the continuous dependence of the solution at y=1, we introduce the following theorem.
Theorem 4.
Assume that problem (1)–(4) has a solution u such that uy∈L2((0,1);L2(0,π)) and
(32)∫01∑n=1∞e2nfn2(s)ds<∞.
Then for all ϵ∈(0,1) there exists a yϵ>0 such that
(33)∥uϵ(·,yϵ)-u(·,1)∥2≤2C1(ln(1ϵ))-1/4,
where
(34)N=∫01∥uy(·,s)∥22ds,C1=max{∥u(.,1)∥22+∥uy(.,1)∥22+π201∑n=1∞e2nfn2(s)dsN,32×∥u(·,1)∥22+∥uy(·,1)∥22+π2∫01∑n=1∞e2nfn2(s)ds}.
Proof.
We have
(35)u(x,1)-u(x,y)=∫y1uy(x,s)ds.
It follows that
(36)∥u(·,y)-u(·,1)∥22≤(1-y)∫y1∥uy(·,s)∥22ds=N2(1-y).
Using (29), noticing that α=ϵ1/2, and (34), we have
(37)∥uϵ(·,y)-u(·,1)∥2≤∥uϵ(·,y)-u(·,y)∥2+∥u(·,y)-u(·,1)∥2≤C1(1-y+ϵ(1-y)/2).
For every ϵ∈(0,1), there exists uniquely a positive number yϵ such that 1-yϵ=ϵ(1-yϵ)/2; that is
(38)ln(1-yϵ)1-yϵ=lnϵ.
Using inequality ln(1-y)>-1/(1-y) for every 0<y<1, we get
(39)∥uϵ(·,yϵ)-u(·,1)∥2≤2C1(ln(1ϵ))-1/4.
This completes the proof of Theorem 4.
Theorem 5.
Let g,gϵ,u be as in Theorem 4 and (32) holds. Then one can construct a function wϵ satisfying
(40)∥wϵ(·,y)-u(·,y)∥2≤Mϵ(1-y)/2
for every y∈(0,1) and
(41)∥wϵ(·,1)-u(·,1)∥2≤C(ln(1ϵ))-1/4,
where
(42)M=2+32×∥u(·,1)∥22+∥uy(·,1)∥22+π2∫01∑n=1∞e2nfn2(s)ds,C1=max{∫01∥uy(·,s)∥22ds,32×∥u(·,1)∥22+∥uy(·,1)∥22+π2∫01∑n=1∞e2nfn2(s)ds},C=2+2C1.
Proof.
Let yϵ be the unique solution of
(43)1-yϵ=ϵ(1-yϵ)/2.
We define a function wϵ as follows:
(44)wϵ(·,y)={vϵ(·,y),0<y<1,vϵ(·,yϵ),y=1.
From Theorem 1, we have
(45)∥wϵ(·,y)-u(·,y)∥2=∥vϵ(·,y)-u(·,y)∥2≤Mϵ(1-y)/2
for every y∈(0,1). From Theorem 2, we have
(46)∥uϵ(·,yϵ)-u(·,1)∥2≤2C1(ln(1ϵ))-1/4.
Using Theorem 1, noticing that α=ϵ1/2, (43), and (46), we get
(47)∥wϵ(·,1)-u(·,1)∥2=∥vϵ(·,yϵ)-u(·,1)∥2≤∥vϵ(·,yϵ)-uϵ(·,yϵ)∥2+∥uϵ(·,yϵ)-u(·,1)∥2≤ϵ1/2+ϵ+2C1(ln(1ϵ))-1/4≤C(ln(1ϵ))-1/4,
where
(48)C=2+2C1.
This completes the proof of Theorem 5.
Remark 6.
In this theorem, we require a condition on the Fourier expansion coefficient fn in (32). This condition is very difficult to check. To improve this, in the next theorem, we only require the assumption of u, not to depend on the function f.
Theorem 7.
Let g,gϵ be as in Theorem 1. Assume that problem (1)–(4) has a solution u such that
(49)∥ux(·,y)∥22+∥uy(·,y)∥22<∞.
If we select α=ϵk for 0<k<1, then
(50)∥vϵ(·,y)-u(·,y)∥2≤ϵ1-k+P2kln(1/ϵ)
for every y∈[0,1] where
(51)P=∥ux(·,y)∥22+∥uy(·,y)∥22.
Proof.
Combining (15) and (20), we obtain
(52)unϵ(y)-un(y)=-12(αα+e-ny)[enygn+1n∫0yen(y-s)fn(s)ds]=-12(αα+e-ny)[un(y)+1nddyun(y)]=-12(ααn+ne-ny)[nun(y)+ddyun(y)].
For z>0, we consider the function φ(z)=1/(αz+e-z). By taking the derivative of φ, we get
(53)φ′(z)=-α-e-z(αz+e-z)2.
The function φ(z) attains maximum value at the z0 such that φ′(z0)=0. It follows that ez0=1/α or z0=ln(1/α). Hence
(54)1αz+e-z≤1αz0+e-z0=1αln(1/α)+α≤1αln(1/α).
Using this inequality, we obtain
(55)∥uϵ(·,y)-u(·,y)∥22=π2∑n=1∞|unϵ(y)-un(y)|2≤π2∑n=1∞14(ααn+e-n)2[nun(y)+ddyun(y)]2≤12ln2(1/α)π2∑n=1∞[n2|un(y)|2+|ddyun(y)|2]≤12ln2(1/α)(∥ux(·,y)∥22+∥uy(·,y)∥22).
By Theorem 1 and α=ϵk(0<k<1) and using the triangle inequality, we get
(56)∥vϵ(·,y)-u(·,y)∥2≤∥vϵ(·,y)-uϵ(·,y)∥2+∥uϵ(·,y)-u(·,y)∥2≤ϵ1-k+P2kln(1/ϵ).
Remark 8.
Condition (49) is natural and reasonable.
3. A Numerical Experiment
To illustrate the theoretical results obtained before, we will discuss the corresponding numerical aspects in this section. We consider a simple problem as follows:
(57)Δu=38(e2y+e-2y)sinx,(x,y)∈(0,π)×(0,1),u(0,y)=u(π,y)=0,uy(x,0)=0,u(x,0)=g(x).
Consider the exact data g(x)=sinx/4; then the exact solution to this problem is
(58)u(x,y)=e2y+e-2y8sinx.
Considering the measured data gϵ(x)=(32/πϵ+1)g(x), we have
(59)∥gϵ-g∥2=(∫0π32πϵ2(g(x))2dx)1/2=(2πϵ2∫0πsin2xdx)=ϵ.
Let ϵ be ϵ1=10-1, ϵ2=10-5, ϵ3=10-10, respectively. If we put
(60)y={0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9}
we get Tables 1, 2, and 3 for the case 0<y<1 and we have the graphic that is displayed in Figures 1, 2, 3, and 4 on the interval [0,π]×[0,0.9].
ϵ1=10-1
y
∥vϵ-u∥2
0
0.0053
0.1
0.0285
0.2
0.0019
0.3
0.0309
0.4
0.0722
0.5
0.1245
0.6
0.1914
0.7
0.2769
0.8
0.3865
0.9
0.5268
ϵ2=10-5
y
∥vϵ-u∥2
0
4.8387×10-4
0.1
7.6623×10-4
0.2
0.0011
0.3
0.0016
0.4
0.0023
0.5
0.0032
0.6
0.0043
0.7
0.0059
0.8
0.0080
0.9
0.0109
ϵ3=10-10
y
∥vϵ-u∥2
0
1.5665×10-6
0.1
2.4632×10-6
0.2
3.6404×10-6
0.3
5.1995×10-6
0.4
7.2769×10-6
0.5
1.0056×10-5
0.6
1.3786×10-5
0.7
1.8801×10-5
0.8
2.5551×10-5
0.9
3.4647×10-5
The exact solution in the case 0<y<1.
The regularized solution with ϵ1=10-1.
The regularized solution with ϵ2=10-5.
The regularized solution with ϵ3=10-10.
For each figure, we can find that the smaller the ϵ is, the better the computed approximation is. And the bigger the y is, the worse the computed approximation is. Figure 5 shows the comparisons of the exact solution u(x,y) and the approximation vϵ(x,y) at the point y=1. In the case y=1, from (43) and using inequality ln(1-y)>-1/(1-y) for every 0<y<1, we get
(61)yϵ>1-12ln(1/ϵ).
Therefore, we will choose yϵ1=0.4, yϵ2=0.8, and yϵ3=0.99, with ϵ1=10-1, ϵ2=10-5, and ϵ3=10-10, respectively. Numerical results are given in Table 4.
∥vϵ(·,yϵ)-u(·,1)∥2
ϵ1=10-1
yϵ1=0.4
0.8319
ϵ2=10-5
yϵ2=0.8
0.3792
ϵ3=10-10
yϵ3=0.99
0.0225
The exact solution and the regularized solution in the case y=1.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
LavrentevM. M.RomanovV. G.ShishatskiiS. P.198664Providence, RI, USAAmerican Mathematical SocietyTranslations of Mathematical MonographsGorenfloR.Funktionentheoretische Bestimmung des Aussenfeldes zu einer zweidimensionalen magnetohydrostatischen Konfiguration19651622792902-s2.0-004225558510.1007/BF01587651ZBL0127.16903JohnsonC. R.Computational and numerical methods for bioelectric field problems199725181QianZ.FuC.LiZ.Two regularization methods for a Cauchy problem for the Laplace equation200833814794892-s2.0-3454883017610.1016/j.jmaa.2007.05.040ZBL1132.35493BeskosD. E.Boundary element methods in dynamic analysis: part II (1986–1996)19975031491972-s2.0-003109581110.1115/1.3101695HarariI.BarboneP. E.SlavutinM.ShalomR.Boundary infinite elements for the Helmholtz equation in exterior domains1998416110511312-s2.0-0032024388ZBL0911.76035MarinL.ElliottL.HeggsP. J.InghamD. B.LesnicD.WenX.BEM solution for the Cauchy problem associated with Helmholtz-type equations by the Landweber method2004289102510342-s2.0-314268621510.1016/j.enganabound.2004.03.001ZBL1066.80009KlibanovM. V.SantosaF.Computational quasi-reversibility method for Cauchy problems for Laplace's equation1991516165316752-s2.0-002637057110.1137/0151085ZBL0769.35005QianA.XiongX.WuY.On a quasi-reversibility regularization method for a Cauchy problem of the Helmholtz equation20102338196919792-s2.0-7045027887010.1016/j.cam.2009.09.031ZBL1185.65171TuanN. H.TrongD. D.QuanP. H.A new regularization method for a class of ill-posed Cauchy problems20106(19)2189201ZBL1236.35200QianZ.FuC.XiongX.Fourth-order modified method for the Cauchy problem for the Laplace equation200619222052182-s2.0-3364608298010.1016/j.cam.2005.04.031ZBL1093.65107ShiR.WeiT.QinH. H.A fourth-order modified method for the Cauchy problem of the modified Helmholtz equation20092326340ZBL1212.65355FuC.FengX.QianZ.The Fourier regularization for solving the Cauchy problem for the Helmholtz equation20095910262526402-s2.0-6764983132310.1016/j.apnum.2009.05.014ZBL1169.65333QianA.MaoJ.LiuL.A spectral regularization method for a cauchy problem of the modified helmholtz equation20102010132-s2.0-7795531485410.1155/2010/212056212056ZBL1198.65114TuanN. H.TrongD. D.QuanP. H.A note on a Cauchy problem for the Laplace equation: regularization and error estimates20102177291329222-s2.0-7804927931810.1016/j.amc.2010.09.019ZBL1206.65225HonY. C.WeiT.Backus-Gilbert algorithm for the Cauchy problem of the Laplace equation20011722612712-s2.0-003530617610.1088/0266-5611/17/2/306ZBL0980.35167EldénL.SimonciniV.A numerical solution of a Cauchy problem for an elliptic equation by Krylov subspaces2009256222-s2.0-7035030948610.1088/0266-5611/25/6/065002065002ZBL1169.65089EldénL.BerntssonF.A stability estimate for a Cauchy problem for an elliptic partial differential equation2005215164316532-s2.0-2544451603810.1088/0266-5611/21/5/008ZBL1086.35115MarinL.LesnicD.The method of fundamental solutions for the Cauchy problem associated with two-dimensional Helmholtz-type equations2007834-52672782-s2.0-1134427705210.1016/j.compstruc.2004.10.005QinH. H.WeiT.Modified regularization method for the Cauchy problem of the Helmholtz equation2009335233423482-s2.0-5824913210710.1016/j.apm.2008.07.005ZBL1185.65203RegińskaT.TautenhahnU.Conditional stability estimates and regularization with applications to cauchy problems for the helmholtz equation2009309-10106510972-s2.0-7344911609610.1080/01630560903393170ZBL1181.47009TautenhahnU.Optimal stable solution of cauchy problems for elliptic equations19961549619842-s2.0-2144444691110.4171/ZAA/740ZBL0865.65076HàoD. N.Van DucN.SahliD. A.A non-local boundary value problem method for the Cauchy problem for elliptic equations20092552-s2.0-70350308017055002ZBL1170.35555FengX.EldénL.FuC.A quasi-boundary-value method for the Cauchy problem for elliptic equations with nonhomogeneous Neumann data20101866176452-s2.0-7865073575310.1515/JIIP.2010.028ChengH.FengX.FuC.A mollification regularization method for the cauchy problem of an elliptic equation in a multi-dimensional case20101879719822-s2.0-7795649572710.1080/17415977.2010.492519ZBL1206.65224