## Journal of Applied Mathematics

### Regularization method for parabolic equation with variable operator

Valentina Burmistrova

#### Abstract

Consider the initial boundary value problem for the equation $u_t=-L(t)u$, $u(1)=w$ on an interval $[0,1]$ for $t>0$, where $w(x)$ is a given function in $L^2(\Omega)$ and $\Omega$ is a bounded domain in $\mathbb{R}^n$ with a smooth boundary $\partial\Omega$. $L$ is the unbounded, nonnegative operator in $L^2(\Omega)$ corresponding to a selfadjoint, elliptic boundary value problem in $\Omega$ with zero Dirichlet data on $\partial\Omega$. 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.

#### Article information

Source
J. Appl. Math., Volume 2005, Number 4 (2005), 383-392.

Dates
First available in Project Euclid: 22 December 2005

https://projecteuclid.org/euclid.jam/1135272206

Digital Object Identifier
doi:10.1155/JAM.2005.383

Mathematical Reviews number (MathSciNet)
MR2204921

Zentralblatt MATH identifier
1092.35037

#### Citation

Burmistrova, Valentina. Regularization method for parabolic equation with variable operator. J. Appl. Math. 2005 (2005), no. 4, 383--392. doi:10.1155/JAM.2005.383. https://projecteuclid.org/euclid.jam/1135272206

#### References

• S. Agmon and L. Nirenberg, Properties of solutions of ordinary differential equations in Banach space, Comm. Pure Appl. Math. 16 (1963), 121--239.
• B. L. Buzbee, Application of fast Poisson solvers to $A$-stable marching procedures for parabolic problems, SIAM J. Numer. Anal. 14 (1977), no. 2, 205--217.
• B. L. Buzbee and A. Carasso, On the numerical computation of parabolic problems for preceding times, Math. Comp. 27 (1973), 237--266.
• L. Eldén, Time discretization in the backward solution of parabolic equations. I, Math. Comp. 39 (1982), no. 159, 53--68.
• --------, Time discretization in the backward solution of parabolic equations. II, Math. Comp. 39 (1982), no. 159, 69--84.
• A. Friedman, Partial Differential Equations, Holt, Rinehart, and Winston, New York, 1969.
• L. E. Payne, Improperly Posed Problems in Partial Differential Equations, Regional Conference Series in Applied Mathematics, no. 22, SIAM, Philadelphia, 1975.
• V. N. Strakhov, Solution of incorrectly-posed linear problems in Hilbert space, Differ. Equ. 6 (1970), 1136--1140 (Russian).
• A. N. Tikhonov and V. Ya. Arsenin, Methods of Decision of Ill-Posed Problems, Nauka, Moscow, 1986.