Determination of a time-dependent term in the right hand side of linear parabolic equations

Tf* r*.

Bùi Việt Hương Tap chi KHOA HOC & CONG NGHE 135(05): 139 - 143

D E T E R M IN A T IO N O F A T IM E -D E P E N D E N T T E R M IN T H E R IG H T H A N D

S ID E O F L IN E A R P A R A B O L IC E Q U A T IO N S

B ù i V iệ t H ương*

College of Science, Thainguyen University, Vietnam

A BSTRA CT

We propose a variational method for determining a time-dependent term in the

right hand side of parabolic equations from integral observations. It is proved that

the functional to be minimized is Frechet differentiable and a formula for its gradient

is derived via an adjoint problem. The problem is discretized by the finite difference

methods and then he conjugate gradient method in coupling with Tikhonov regular￾ization is applied for numerically solving it. Numerical results are presented showing

that our technique is efficient.

K ey w o rd s: Inverse problems, ill-posed problems, integral observations, finite differ￾ence method, conjugate gradient method.

1 Introduction

Let O be an open bounded domain in R” . De￾note by <90 the boundary of 0 , Q := 0 x (0, T],

and S := <90 x (0, T]. Consider the following

problem

— g(x, t). (X, t) G S.

ut - A u = f{t)<p(x, t) + ip(x, t), (x, t) 6 Q,

zi(rr, 0) = u o ( x ) .x £ f l,

du

. dv

(1.1)

Here, v is the outer normal to <90. And G

L°°(Q), ip{x,t) G L 2(Q), e, I 2(0),

g{x,t) G L 2{S). Furthermore, it is assumed

that ip > ip > 0, with \p being a given con￾stant.

To introduce the concept of weak solution,

we use the ’ standard Sobolev spaces

//¿(O ), H 1,0(Q) and t f u (Q) [3, 5, 6]. Fur￾ther, for a Banach space B , we define

L2(0 ,T ;B ) = {u : u (t) G B a.e. t G

(0, T) and ||'u||i2(0,r;B) < °°}! with the norm

f T IIuIIl2(0,T;S) = JI ll«(0llfl*-

In the sequel, we shall use the space

W {0 ,T ) defined as W (0 ,T ) ■= {u : u G

L 2(0,T; H 1 (Ci)),ut G L2(0, T; ( i i 1(f2))/)},

equipped with the norm

ll'u !lx,2 ( 0 , T ; i i 1 ( n ) ) + ll ^ t llz-2 ( 0 ,'T ; ( ir 1 ( « ) ) /) '

D e fin itio n 1. The function u G W (0 ,T ) is

said to be a weak solution of (1.1) if for all

ry G L 2(0, T; ii^ O ) ) satisfying ry(-,T) = 0, the

following identity holds

/ (ut,v){H1{n)y,H1(tt)dt+ ■

Jo

J J \7 u \7 rjdxdt + J J ag(x.t)i](x.t)dsdt

Q ‘ S

= J J {f(t)ip(x.t) + i>(x,t))r](x,t)dxdt,

(1.2)

and u|t=o = uo- ^ Proved in [5] th at there

exists a unique solution in W (0, T ) of the prob￾lem (1.1). Furthermore, there is a positive con￾stant Cd > 0 independent of f,ip ,ip ,g and uq

such th at .

II u \\w(o,T) ^ cd(lI f 'P IIl2(Q) + HV-; IIi2(q ) +

; .)l S j l i ^ S ) + || Uo 11Z/2 (i 2)) •

In this paper, we Will consider the inverse prob￾lem of determ ining th e tim e-dependent term