QUASI-BOUNDARY VALUE METHOD FOR AN ILL POSED DIFFUSION SYSTEM

In this paper we investigate an ill posed diffusion system with a non diagonal diffusion matrix. Based on the quasi-boundary value method, we regularize the problem, we prove that the approximate solutions depend continuously on the data and we establish some convergence results. Finally numerical results are presented to illustrate the accuracy and efficiency of the proposed method. AMS Subject Classification: 35F35, 65F22


Introduction
In this paper we consider the inverse problem of determining the source term u(0, x) in a system of partial differential equations of the form where Ω is a sufficiently regular bounded domain in R N and D is an n × n Received: July 5, 2015 c 2015 Academic Publications, Ltd.
url: www.acadpubl.eu§ Correspondence author real matrix with semi-simple and positive eigenvalues.The diffusion equation together with their modified forms and systems of diffusion equations have many important applications in mathematical physics, mathematical finance, chemistry, biology and environmental science [2], [3] , [5], [6], [7].In this study as previously mentioned, our purpose is to identify u(0, x) from the final data u(T, x), this problem is ill posed, even a unique solution exists it does not depend continuously on the data.Hence, a regularization is in order.In the mathematical literature various methods have been proposed for solving illposed problems, we can notably mention the quasi-boundary value method (QBVM).It has been used by many authors, such as L.S. Abdulkerimov [1], P.N.Vabishchevich et al [15], I.V. Melnikova et al [8], [11], [12], R.E.Showalter [14], G.W. Clark and S.F.Oppenheimer [4] and it has been successfully applied to various classes of elliptic and parabolic ill-posed problems.
In the present work, we extend the QBVM to systems of partial differential equations of the form (1.1) The study is based on the semigroups theory precisely the characterization of the C 0 -semigroup generated by the system (1.1) which is given and analyzed by Leiva [9] and Oliveira [10].
We note that though the diffusion systems forward in time have been extensively studied in literature, the case of diffusion systems backward in time does not seem have been widely investigated in spite of their physical and practical importance.
The paper is organized as follows, in Section 2 we introduce some preliminary results, Section 3 gives an abstract formulation of the problem and shows the ill posed-ness of the problem.In Section 4, we introduce the regularized solution and we give some convergence results.Finally numerical implementation is described in Section 5.

Preliminaries
Let H = L 2 (Ω) with the inner product (., .)and consider the following classical boundary-eigenvalue problem for the laplacien: where Ω is sufficiently regular bounded domain in R N (N ≥ 1), and Each eigenvalue λ j with finite multiplicity γ j equal to the dimension of the corresponding eigenspace S j .
1. Therefore, there exists a complete orthonormal set {ϕ jk } k=γ j k=1 of eigenvectors of −∆ such that, • for all w ∈ D(−∆), we have where So, {E j } is a family of complete orthogonal projections in H and for all w ∈ H, we have w = ∞ j=1 E j w. 2. The operator ∆ generates an analytic semigroup {T (t)} t≥0 on H, defined by e −λ j t E j w.Now, we denote by Z the Hilbert space (L 2 (Ω)) n of the square integrable functions u : Ω → R N with the usual inner product We define the following unbounded operator is a family of complete orthogonal projections in Z.
where A j = −λ j D.

Ill-Posedness of the Problem
In the Hilbert space Z, the system (1.1) can be written as an abstract functional differential equation Let us consider the following direct problem corresponding to the backward Cauchy problem (3.1) Since A is the generator of an analytic semigroup, then for all g ∈ Z, the problem (3.2) has a unique solution v ∈ C([0, T ], Z) given by where g = ∞ j=1 P j g, see [13], Chap.4, theorem 1.4, p. 104).
Remark The matrix D does not necessarily have distinct eigenvalues.Let 0 < d 1 < .... < d s , s ≤ n be the distinct eigenvalues, then D admits the following spectral decomposition where {Q i } 1≤i≤s is a complete family of projections.
Then the solution of the problem (3.2) can be written as follows Proof.If the problem (3.1) admits a solution u then u(t) = S(t)u(0).
and consider the problem (3.5) Since (3.5) is the direct well-posed, so it has a unique solution given by Let t = T in (3.6), we obtain Hence, u is the unique solution to (3.1).
Since t < T, we can see from (3.6) that the terms e λ j d i (T −t) are the instability causes.In the following section, it is our aim to restore the stability of the problem (3.1) by using a regularization method.

Quasi-Boundary Value Method
We shall regularize problem (3.1) using the quasi-boundary value method.Let us consider the approximate problem for f ∈ Z, α > 0 and t ∈ [0, T ].
Theorem 4.1.The function u α (t) is the unique solution of (4.1) and it depends continuously on f.Proof.Consider the problem: where This problem is well-posed and its solution is given by Observe that Thanks to (4.5) and uniqueness of solution to direct problem (4.3), we deduce that the problem (4.1) admits the unique solution u α given by (4.2).
To show the continuous dependence of u α on f, we compute Theorem 4.2.For all f ∈ Z, α > 0 and t ∈ [0, T ], we have that Now, let α be such that α 2 < ε(2M 2 ℵ j=1 e 2λ j dsT P j f 2 ) −1 .Hence we are done.Proof.Let k ∈ (0, 2).Fix natural number j, put γ j = λ j d s , and Differentiating with respect to α, once obtain Since h j (α) > 0, h j (0) = 0 and lim α→0 h j (α) = 0, we have that α ′ = k 2−k e −γ j T is the critical point at which h j achieves its maximum.Thus we have We calculate If we choose k = 2 − θ, we obtain e θT λ j ds P j f 2 .
This completes the proof.
Lemma 4.1.For all f ∈ Z, the problem (3.1) has a solution u(t) if u α (0) converges in Z. Furthermore, we then have that u α (t) converges to u(t) as the parameter α tends to zero uniformly in t.
Proof.Working as in the proof of theorem 4.5 we obtain as above, letting k = 2 − θ we obtain the result.
From the inequality (4.6) and theorem 4.6, we arrive at the following Corollary 4.1.Assume that the problem (3.1) admits a solution u(t).If there exists 0 < θ < 2 so that f ∈ C θ (A), then u α (t) converges to u(t) as the parameter α tends to zero with order α θ uniformly in t.
Let us now construct a family of regularizing operators for the problem (3.1).Definition 4.3.A family of bounded linear operators R α (t) : Z → Z, α > 0, t ∈ [0, T ] is called a family of regularizing operators for the problem (3.1) if for each u(t) (0 ≤ t ≤ T ) solution of the problem (3.1)with the final element f, and for any δ > 0, there exists α(δ) > 0, such that Theorem 4.7.Assuming that f satisfies (3.4).Then the family {R α (t)} defined above is a family of regularizing operators for (3.1).
Remark All the previous results obtained in this note remain true if 1. the matrix D has semi-simple eigenvalues in the half plane Re σ(D) > 0.
2. the −△ in (1.1) is replaced by a positive self-adjoint linear operator with compact resolvent.This consideration should be useful to study abstract versions of the problem.
The data error at t = 1, Thus, arbitrarily small data errors can lead to arbitrarily large errors in the result.By applying the quasi-boundary value method, the regularized solution is given by ) sin mx ) sin mx ) sin mx    .
If we choose n = 300, the error of the quasi-boundary value method is given in the table 1.
Table 1 shows that the approximate solutions converge to the exact solution as epsilon tends to zero.

Conclusion
In this paper, we considered a quasi boundary value method to solve an ill posed diffusion system.In the theoretical results, it was shown that under certain conditions stability estimate was obtained and convergence results were established.Meanwhile, the numerical results verified the efficiency of this method.

Theorem 3 . 1 .
The problem (3.1) admits a solution if and only