Approximation of mild solutions of the linear and nonlinear elliptic equations
In this paper, we investigate the Cauchy problem for both linear and semi-linear elliptic equations. In general, the equations have the form
where is a positive-definite, self-adjoint operator with compact inverse. As we know, these problems are well-known to be ill-posed. On account of the orthonormal eigenbasis and the corresponding eigenvalues related to the operator, the method of separation of variables is used to show the solution in series representation. Thereby, we propose a modified method and show error estimations in many accepted cases. For illustration, two numerical examples, a modified Helmholtz equation and an elliptic sine-Gordon equation, are constructed to demonstrate the feasibility and efficiency of the proposed method.
Keywords and phrases: Elliptic equation; Cauchy problem; Ill-posed problem; Regularization method; Contraction principle.
Mathematics subject Classification 2000: 35K05, 35K99, 47J06, 47H10x.
The Cauchy problem of elliptic equation plays an important role in inverse problems. For example, in optoelectronics, the determination of a radiation field surrounding a source of radiation (e.g., a light emitting diode) is a frequently occurring problem. As a rule, experimental determination of the whole radiation field is not possible. Practically, we are able to measure the electromagnetic field only on some subset of physical space (e.g., on some surfaces). So, the problem arises how to reconstruct the radiation field from such experimental data (see, for instance, ). In the paper of Reginska , the authors considered a physical problem which is connected with the notion of light beams. Some applications of this model can be established in more detail in . Another application in inverse obstacle problems (cf. ), which are investigated in connection with inclusion detection by electrical impedance tomography when only one pair of boundary current and voltage is used for probing the examined body .
Let be a real Hilbert space, and let be a positive-definite, self-adjoint operator with compact inverse on . In this paper, we consider the problem of finding a function satisfying
associated with the initial conditions
where is a mapping from , and are the exact data in . Physically, the exact data can only be measured, there will be measurement errors, and we thus would have as data some function and in for which
where the constant represents a bound on the measurement
error, denotes the norm.
Since Hadamard, it is well known that the Cauchy problem of elliptic equation, for example, Problem (1)-(2), is severely ill-posed: although it has at most one solution, it may have none, and if a solution exists, it does not depend continuously on the data in any reasonable topology. Therefore, regularization is needed to stabilize the problem. In recent years, many special regularization methods for the homogeneous and nonhomogeneous Cauchy problem of elliptic equation have been proposed, such as Backus-Gilbert algorithm , the method of wavelet , quasi-reversibility method , truncation method , non-local boundary value method  and the references therein.
Although we have many works on the linear homogeneous case of Cauchy problem for elliptic equation, however, regularization theory and numerical simulation for nonlinear elliptic equations are still limited. Especially, the nonlinear cases for elliptic equation appear in many real applications. For example, let us see a simple one infered by giving and in the problem (1)-(2). In particular, it is given by
If in (4), then it is called Helmholtz equation which has many applications related to wave propagation and vibration phenomena. This equation is often used to describe the vibration of a structure, the acoustic cavity problem, the radiated wave and the scattering of a wave. With in (4), we obtain the elliptic sine-Gordon equation. From the point of view of the modelling of physical phenomena, the motivation for the study of this equation comes from its applications in several areas of mathematical physics including the theory of Josephson effects, superconductors and spin waves in ferromagnets, see e.g. . With , we have the Allen-Cahn equation originally formulated in the description of bi-phase separation in fluids.
Switch back to the considered problem, it is more complicated than the ones above. Hence, the purpose of this paper is to introduce a new method of integral equation that is based on a modification of the exact solution formulation. As the regularization parameter tends to zero, the solution of our regularized problem converges monotonically to the solution of the Cauchy problem with the exact data.
Prior to the approach of main results, we would like to introduce the representation of solution in problem (1)-(2) for linear and semi-linear cases. We can see that the operator , as a consequence, admits an orthonormal eigenbasis in , associated with the eigenvalues such that
Let be the Fourier series of in the Hilbert space . For homogeneous problem, i.e, in (1), by a seperable method, we get the homogeneous second order differential equation as follows
and its solution leads to
where denotes the inner product in . From F. Browder terminology, as in [Dan Henry, Geometric Theory of Semi-linear Parabolic Equations, Springer-Verlag, Berlin Heildellberg, Berlin, 1982], in (6) is called the mild solution of (1)-(2) with .
For the nonlinear problem , we say that is a mild solution if satisfies the integral equation
where . The transformation from problem (1)-(2) into (7) is easily proved by a separation method which is similar above process. From now on, to regularize Problem (1)-(2), we only consider the integral equation (7) and find a regularization method for it. The main idea of integral equation method can be found in a paper  on nonlinear backward heat equation.
The paper is organized as follows. In Section 2, we present our regularization method for the linear problem implied by letting in (1). The theoretical results in the Section 2 are inspirable for us to suggest a new regularization method for semi-linear case in Section 3. New convergence estimates are given under some different priori assumptions for the exact solution. Proofs of the results in these sections will be showed in the appendix in the bottom of paper. In Section 4, simple numerical examples aimed to illustrate the main results in Section 3 are analyzed.
2 The linear homogeneous problem
In , C.L. Fu and his group applied the quasi-reversibiity (QR ) method to approximate problem (4) in case and . The main idea of the original QR method  is to approach the ill-posed second order Cauchy problem by a family of well-posed fourth order problems depending on a (small) regularization parameter. In particular, they considered approximate problem
The solution of (8) is defined by
and the authors proved that converges to the solution of homogeneous problem as .
Very recently, homogeneous problem has been considered by Hao, Duc and Lesnic . They applied the method of non-local boundary value problems (also called quasi-boundary value method) to regularized the above problem as follows
with being given and is the regularization parameter. They proved that the solution to (10) is
and as with some assumptions on the exact solution .
Following the work , in  Tuan, Trong and Quan used a Fourier truncated method to treat the following Cauchy problem of an elliptic equation with nonhomogeneous Dirichlet and Neumann data. From the simple analysis about the exact solution (6), we know that the data error can be arbitrarily amplified by the “kernel” . That is the reason why the Cauchy problem of elliptic equation is ill-posed. Since the general regularization theory  and paper , we now give a more general principle of regularization methods for the Cauchy problem of (6). Our idea on regularization method is of constructing a new kernel and replacing by where the new kernel should satisfy
(A) If is fixed, is bounded.
(B) If is fixed, then .
Following properties (A) and (B), one can construct other kernels. Furthermore, the idea of properties (A) and
(B) can be applied to other ill-posed problems when the solution has the similar form of (6), e.g., the inverse heat
conduction problem . In this sense, we say that the properties (A) and (B) are useful and interesting. Now, from above discussion, it is easy to check that the kernels in  and in  satisfy (A) and (B).
We now have a look at the solution in (6). To find a regularization solution for , the unstability terms and in (6) should be replaced by two kernels and respectively. Here the kernel satisfies (A), (B) and kernel satisfies the following conditions
(C) If is fixed, is bounded.
(D) If is fixed, then .
In , we choose
In this section, we consider the homogeneous problem of (1) (also given in  ) by other choices for kernels. From the formula of and , we realize that the term is unstability cause while the term is stable under the boundedness of the unity. Hence, by a simple and natural way, we replace and by two new kernels
to obtain a regularization solution
Here is called parameter reguarization and satisfies . It is easy to check that and satisfy and respectively. Moreover, (13) leads to
Under the inexact data and , the regularized solution becomes
The main results of this section are in the following theorem.
Let for .
If there is a positive constant such that
then we have
If there is a positive constant such that
then we have
If there is a positive constant such that
then we have
In order to prove this theorem, we have to obtain some auxiliary results given by the lemmas below.
Let and let as introduced in Remark 1. Then, we have the following estimate
At , the error in case (i) is useless while it is useful in case (ii). Moreover, in case (iii), under the strong assumptions of , we get the error of Holder-logarithmic type. In fact, if is fixed then the right-hand side of (21) get its maximum value at . Thus, we obtain the error of order .
On the other hand, the condition in (18) is accepted and natural. Thus, we prove that
Then the condition
is easy to check.
3 The semi-linear problem
As we introduced, many previous papers only regularized problems related to (1) in which . This condition makes the applicability of the method very narrow. Until now, the results in nonlinear case are very rare. In this section, we consider the problem (1) where is a Lipschitz continuous function, i.e., there exists independent of such that
Since , we know from (7) that, when becomes large, the terms
increases rather quickly. Thus, these terms are the unstability causes. Hence, to find a regularization solution, we have to replace these terms by new kernels (called stability terms). These kernels have some common properties . In fact, we define a following regularization solution
Here, are bounded by for any . Moreover, if fixed then
By direct computation, we see that the kernels in Theorem 1 is not applied to nonlinear problem. For solving this problem, we find some suitable kernels as follows
Then, we show error estimates between the solution and the regularized solution in norm under some supplementary error estimates and assumptions. Simultaneously, the uniqueness of solution is proved by contraction principle.
Generally speaking, we obtain the following theorem.
Let be the solution as denoted in (7). Suppose there is a positive constant such that
Then by letting the problem
has a unique solution satisfying
where for each , such that for
The following lemmas will lead to proof of the main theorem.
Let and be defined in (34), then it follows that
The following integral equation
has a unique solution .
has a unique solution and the error estimate holds
Let be a function defined in (39), then the following estimate holds
4 Numerical examples
In this section, we aim to show two numerical examples to validate the accuracy and efficiency of our proposed regularization method for 1-D semi-linear elliptic problems including both linear and nonlinear cases. The examples are involved with the operator and taken by Hilbert space . Particularly, we give examples of a modified Helmholtz equation and an elliptic sine-Gordon equation to demonstrate how the method works.
The aim of numerical experiments is to observe for . The couple of plays as measured data with a random noise. More precisely, we take perturbation in couple of exact data to define by the following way.
where rand is a random number determined in .
Then, the regularized solution (with choosing ) is expected to be closed to the exact solution under a proper discretization. For convergence tests, we would like to introduce two errors: the absolute error at the midpoint and the relative root mean square (RRMS) error. Also, the 2-D and 3-D graphs are applied and analysed.
To be more coherent, we are going to divide this section into two subsections. The first one is to consider the modified Helmholtz equation and the second one is for the elliptic sine-Gordon equation. As we introduced, they are simply outstanding for many applied problems.
Generally, the whole process is summarized in the following steps.
Step 1. Given and to have
Step 2. Choose , put and set . We find
Step 3. For and put and , we find the matrices in containing all discrete values of the exact solution and the regularized solution , denoted by and , respectively.
Step 4. Calculate the errors and present 2-D and 3-D graphs.
4.1 Example 1
We will consider the following equation.
Based on , we get an orthonormal eigenbasis associated with the eigenvalue in . In order to ensure the problem (44) has solution with a given Cauchy data , we will construct the exact solution from a function as follows
where is a truncation term and will be chosen later. Then, this problem has a unique solution by applying method of separation of variables.
Thus, we have
Simultaneously, the regularized solution defined in (31) becomes
Now when we divide the time , it turns out that a simple iterative scheme in time is applied to (LABEL:eq:41). Particularly, we will compute from as follows.