REGULARIZATION METHOD FOR PARABOLIC EQUATION WITH VARIABLE OPERATOR

Consider the initial boundary value problem for the equation ut =−L(t)u, u(1)= w on an interval [0,1] for t > 0, where w(x) is a given function in L2(Ω) and Ω is a bounded domain in Rn with a smooth boundary ∂Ω. L is the unbounded, nonnegative operator in L2(Ω) corresponding to a selfadjoint, elliptic boundary value problem in Ω with zero Dirichlet data on ∂Ω. The coefficients of L are assumed to be smooth and dependent of time. It is well known that this problem is ill-posed in the sense that the solution does not depend continuously on the data. We impose a bound on the solution at t = 0 and at the same time allow for some imprecision in the data. Thus we are led to the constrained problem. There is built an approximation solution, found error estimate for the applied method, given preliminary error estimates for the approximate method.


Introduction
Consider the problem of solving a parabolic partial differential equation with variable operator backwards in time.For convenience we write the equation in the following abstract form u t = −L(t)u, 0≤ t ≤ 1, u(1) = w. (1.1) Here w(x) is a given function in L 2 (Ω), and Ω is a bounded domain in R n with a smooth boundary ∂Ω.L is the unbounded, nonnegative operator in L 2 (Ω) corresponding to a selfadjoint, elliptic boundary value problem in Ω with zero Dirichlet data on ∂Ω.The coefficients of L are assumed to be smooth and dependent of time.
The system (1.1) is ill-posed because the solution does not depend continuously on the data.We impose a bound on the solution at t = 0 and at the same time allow for some imprecision in the data.Now we are led to the constrained problem.

Get any solution of
where the norm is the L 2 (Ω)-norm, and δ and M are given positive constants, δ M. Using logarithmic convexity (see [1], [7, page 11]), we have that any two solutions of (1.2), u 1 and u 2 , satisfy (1.3) Write down system (1.4) Thus for 0 < t ≤ 1 we have continuous dependence on the data.
It is difficult to solve (1.2), because solutions are not unique.There are some methods for approximating solutions of (1.2), which are optimal in the sense that Hölder type error estimates (1.3) can be obtained for them.
We consider a method related to the regularization method of Tikhonov [5] and Phillips.This method for parabolic equation with operator L independent of time is learned in [4].Now we consider more generalized case: parabolic equation with variable coefficients.

Valentina Burmistrova 385
We now raise the following question.Can we discretize (1.5) in such a way that for the discrete approximation v a we get an error estimate of type (1.6) for some constant C?
The answer to this question will have significance for the possibilities of solving numerically problems in two (or more) space dimensions, with nonrectangular geometry or nonconstant coefficients, since for such problems we must discretize in time and space.
In this paper, we give a partial answer to the above question.We consider approximating the exponential function in (1.5) in a way which corresponds to a time discretization.In Section 3, we show that if exp(−λ) is approximated well enough for 0 ≤ λ ≤ log(M/δ), we can get error estimates of the form (1.7) with C = 2.

The regularization method for parabolic equation with variable coefficients
We show that the estimate (1.6) holds for the regularization method (1.5).The proof is quite simple and we use the same method in connection with discretization of (1.5).We also show that the same error estimate is valid if we use (1.5) in a step-by-step manner.
We assume that δ and M have been chosen so that there exist solutions of (1.2).
The assumption about the existence of solutions of (1.2) is equivalent to there being functions u 0 and ψ such that where the operator norm is defined such way A = sup{ Au : u = 1}.We now use (2.2) and the fact that L is selfadjoint and nonnegative to get where . (2.5) We have We use the fact from [4].We have for 0 ≤ p, t ≤ 1 the inequality is valid, we obtain (look at the definition (1.5) of µ(t)).Therefore we can estimate (2.4) The numerical of a forward parabolic problem is usually computed by a marching procedure, that is, a procedure which is recursive in time.We show that the method (1.5) for the backward problem can be generalized to a recursive formula in such a way that the procedure remains optimal in the above sense.Make a (possibly nonuniform) partitioning of the interval [0,1] and let the recursion be Valentina Burmistrova 387 where v(t s ) is given by (1.5) and Corollary 2.2.Let u(t) denote an arbitrary solution of (1.2), and let (v i ) s i=1 be defined by (2.10).Then (2.12) Proof.The result is obviously true for i = s.Then assume that it is true for i = k, and consider The recursion formula (2.10) is a straightforward generalization of (1.5) to the interval [0,t k ], and, putting τ k = t k−1 /t k , we obtain (2.14)

Preliminary error estimation
We get now error estimates for which is (1.5) with the exponential function replaced by an approximation f , such that f (λ) ≈ e −λ .In the next section f (λ) will depend on N, where k = 1/N is a step length parameter, but here this dependence is suppressed.There we will be dealing explicitly with the class of approximations defined by exp−λ ≈ (Q(λ/N)/P(λ/N)) N (see [1, p. 54]), but in this section it will be sufficient to distinguish between two subclasses characterized by the following inequalities: First we give an error estimate for approximations satisfying (3.3).
Theorem 3.1.Let u(t) denote an arbitrary solution of (1.2), let v a (t) be defined by (3.2), and assume that f satisfies (3.3).If Proof.As in Theorem 2.1 we have Look at it in details ψ . (3.8) Here (3.9)By (2.6) we have

.10)
We have then We have then (3.12) We have then (3.13) Here if −λ(ξ) − lng < 0, then A < 0 and A 1 ≥ A 2 , that is, we have earlier observed case.

and so we have
(3.20) As we have Thus we obtain the same error estimate as for case with operator independent of time [4].It means that the method can be applied for more wide field of problems.