Identification of the parameters of a convection diffusion system
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SYSTEMS ANALYSIS IDENTIFICATION OF THE PARAMETERS OF A CONVECTION DIFFUSION SYSTEM I. V. Sergienkoa† and V. S. Deinekaa‡
UDC 519.6
Computational algorithms are proposed for the realization of gradient methods based on the solution of direct and conjugate problems in weak formulations for some complex inverse problems of the recovery of parameters of multicomponent parabolic distributed systems. The approach proposed makes it unnecessary to set up Lagrange functionals in explicit form and to use the Green function. Keywords: convection diffusion, identification of parameters, gradient methods.
The papers [1, 2] consider the derivation of explicit expressions for gradients of quadratic residual functionals for the realization of gradient identification methods for various parameters of multicomponent distributed systems. Similar problems are considered in [3, 4, 5] for pseudoparabolic, hyperbolic, and pseudohyperbolic multicomponent distributed systems, respectively. In the present paper, we will obtain explicit expressions for gradients of quadratic residual functionals in terms of the solutions of direct and conjugate problems for the identification of parameters of boundary conditions, diffusion coefficients, convective-transport rate, and the parameter of a weakly permeable thin constituent medium of a distributed convection diffusion system. We will consider the application of the parametric identification of parameters. 1. IDENTIFICATION OF THE PARAMETERS OF BOUNDARY CONDITIONS AT THE OPPOSITE ENDS OF AN INTERVAL Let a parabolic equation be defined on a domain W T conditions
¶y ¶ æ ¶y ö ~ = çk - uy ÷ + f (1) ¶t ¶x è ¶x ø = W ´ ( 0, T ) (W = W 1 È W 2 , W 1 = ( 0, x ) , W 2 = ( x , l ) , 0 < x < l < ¥) and the boundary æ ¶y ö -çk - uy ÷ = u1 , x = 0, t Î ( 0, T ) , è ¶x ø k
¶y - uy = - u 2 y + u 3 , x = l, t Î ( 0, T ) , ¶x
( 2)
( 3)
be specified at the ends of the interval [ 0, l] . Let us specify the following imperfect contact conditions at the point x = x: ù é ¶y êëk ¶x - uyúû = 0, t Î ( 0, T ) , a
(4 )
V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine, [email protected]; ‡[email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 42–63, January–February 2009. Original article submitted September 18, 2008. †
38
1060-0396/09/4501-0038
©
2009 Springer Science+Business Media, Inc.
ì ¶y ü - u yý ík î ¶x þ
±
= r [ y], t Î ( 0, T ) ,
( 5)
where [ j] = j + - j - , j ± = {j}± = j ( x ± 0, t ), and r = const ³ 0. The initial condition is
| t= 0 = y0 ,
y
x Î W1 È W 2 .
(6)
Solutions of the initial–boundary-value problem (1)–(6) are assumed known at N points d i ÎW : y ( d i , t ) = f i ( t ), t Î ( 0, T ), i = 1, N .
(7)
The problem (1)–(7) is as follows: find a vector u = {u i }i3 = 1 Î U such that the solution y = y ( u ) = y ( u; x, t ) of the initial–boundary-value problem (1)–(6) satisfies Eqs. (7), where U = C ( 0, T ) ´ C + ( 0, T ) ´ C ( 0, T ) , C + ( 0, T ) = {u ( t ) Î C ( 0, T ): u > 0, t Î [ 0, T ]}
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