control in heat conduction theory and in plasma physics, that is, in those problems where ... Keywords: heat equation, integral boundary conditions, non-classical ...
Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 3(6): 977-979 © Scholarlink Research Institute Journals, 2012 (ISSN: 2141-7016) jeteas.scholarlinkresearch.org Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 3(6):977-979(ISSN: 2141-7016)
On Heat Conduction Problem with Integral Boundary Condition Raid Almomani and Hasan Almefleh Department of Mathematics, Yarmouk University, Irbid – Jordan. Corresponding Author: Raid Almomani ___________________________________________________________________________ Abstract We formulate the control problem of heat conduction problem with inverse direction of time and integral boundary conditions and we show the non-well-posedness of this problem. The importance of the heat conduction problem with integral boundary conditions its non self-adjointness, which make a principle difficulties while investigating the problem. The solution of our problem play an important role in optimal control in heat conduction theory and in plasma physics, that is, in those problems where we have an integral restriction on a function. __________________________________________________________________________________________ Keywords: heat equation, integral boundary conditions, non-classical boundary conditions, optimal control for heat equation, well-posedness. __________________________________________________________________________________________ INTRODUCTION The sequence of eigenfunctions and adjoint functions We consider the control problem of heat conduction of the non selfadjoint Sturm - Liouville problem problem with inverse direction of time and integral corresponding to the problem (1)-(4) is boundary conditions, recently many works were − ( ) = ( ), 0 < < 1 devoted to this subject, see [(Benouar and Yurchuk (0) = 0, ∫ ( ) = 0 (0) = (1) , (10) 1991), (Bouziani 1996), (Bouziani 1997), (Feng, Du ( ) = 2, ( )=4 2 and Ge 2009), (Ionkin 1977), (Kamynin 1964), (Xi = 4(1 − ) 2 , = 1,2, … (11) and Jia, 2009), (Yurchuk 1986)]. Consider the equation: The sequence of eigenfunctions and adjoint function u − u = 0, 0 < < 1, 0 < < ∞ (1) of the adjoint Sturm - Liouville problem With the initial condition − ( ) = ( ), 0 < < 1 u(x, 0) = ∅(x), 0 < < 1 (2) (1) = 0, (0) = (1) (12) And the non-classical boundary conditions Which is biorthogonal with { ( )} . (0, ) = 0 (3) (4) ∫ ( , ) =0 Each eignfunction = (2 ) of the problem (10) and (12) with > 0 corresponds an eignfunction Here it is assumed that the initial function ( ) and an adjoint function ( ) of system ∅( ) satisfy (3) and (4), that is (9) and eignfunction ( ) and an adjoint function ( ) ( ) ∅ 0 = 0, ∫ ∅ = 0 . (5) ( ) of system (11). It is well known (Ionkin, 1977) that if ∅( ) is a In (Ionkin, 1977) instead of problem (1)-(4), it was continuously differentiable function that satisfy the investigated the following mixed problem: conditions − =0 (1’) ∅(0) = 0, ∫ ∅( ) = 0 , ∅ (0) = ∅ (1) (6) ( , 0) = ∅( ), 0 < < 1 (2’) Then the function (0, ) = 0, (3’) ( , )= ( )+∑ [ ( )+ ( (0, ) = (1, ), > 0 (4’) ( )−2 ( ))] (7) is the classical solution of the mixed problem (1)-(4), The boundary condition (4’) is derived from the nonWhere homogeneous condition ∫ ( , ) = (constant) taking into account equation (1). = ∅( ) ( ) , The classical solution of (1’)-(4’) with (6) on the initial function ∅( ) also has the form (7), moreover = ∫ ∅( ) ( ) , = if ≠ 0, then ≠0. ( ) ( ) ∅ , (8) ∫ It is clear that ≠0 in the solution (7) of the = 2 , problem (1)-(4). = 2 , = 1,2, … (9) 977
Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 3(6):977-979(ISSN: 2141-7016) In (Yourchuk, 1986) it was investigated a more general problem with variable coefficients − ( ( , ) )= ( , ) ( , 0) = ∅( ), 0 < < 1 (0, ) = 0, (14) ∫ ( , ) = 0, 0 < < And the existence and uniqueness of the solution were proved.
By an appropriate choice of measure units of the solution ( , ) of (1)-(4) means the temperature at point of a thin heated rod of length one at a time , under the conditions that the initial temperature is ∅( ), and at left end we have a zero temperature and the flux variation of the rod is zero (∫ ( , ) = 0). For the problem (1’)-(4’), the condition (4’), (0, ) = (1, ), means the equality of heat flux at the ends of the rod.
If the function ∅( ) ∈ satisfy (5), ∅(0) = 0, ( ) ( ) = 0, and , = 1, ( , ) = 0, then ∫ ∅ the series (7) ( = 0) is the generalized solution of (14) in the space , where and are the space of solutions and the space of the right hand side of this problem respectively equipped with the following norms ‖ ‖ = 0≤ ≤ ∫ ( ‖∅‖ = ∫
(1 − ) [| | + | (
)
|
| +| |
| ]
STATEMENT OF THE PROBLEM We set the solution of the control problem (1)-(4) ( , ) by means of the initial condition ∅( ), for a given number > 0 and some function ( ) defined on (0,1), we have to find an initial function ∅( ) such that the solution of (1)-(4) at = differs a little from ( ).
+
Let ( , , ∅)be the solution of mixed problem (1)-(4) in the rectangle = {( , ): 0 < < 1, 0 < < }. For a given > 0 and the functions ( ) ∈ (0,1) satisfying the conditions (0) = 0, ∫ ( ) = 0 (21) We minimize the functional (∅) = ∫ | ( , , ∅) − ( )| (22)
(15)
)
|∅ | + |∅( )| (16) = {( , ): 0 < < 1, 0 < < }
The proof of the last statements implies from a priori estimate ‖ ‖ ≤ ‖∅‖ (17) Where is the known constant from (Yourchuk, 1986).
(0,1) is first order Sobolev space. Note Where that the proximity (closeness) of the function ( , , ∅) and ( ) in (22) is defined by the metric (0,1)and of the space (0,1). In the case of (0,1) spaces it is necessary to minimize the functional (∅) = ∫ ( , , ∅) − ( ) (23) and
This a priori estimation established in (Yourchuk, 1986) with the help of integration by parts and some estimation forms which derived by multiplying in ( ) the equation of problem (14) by a chosen according to the operator domain generated by this problem- integrating factor = (1 − ) + 2(1 − ) Where = ∫ (∅, ) ∅ (18) The importance of the considered problem 1-4 is its non self-adjointness. This importance is an implication of the non-classical boundary conditions. The integral boundary condition also gives more importance.
(∅) = ∫ ( , , ∅) − ( ) (24) An “ideal” solution of the given problem could be if we set ∅( ) = ( , 0) (25) Where ( , ) is the solution of the problem with inverse direction of time, that is the solution of the heat equation (1) with boundary conditions (3), (4) and the condition ( , ) = ( ) instead of the initial condition (2), − = 0, 0 < < 1, < ( , ) = ( ), 0 < < 1 (0, ) = 0, (26) ∫ ( , ) = 0, 0 <
0, ∈ (0, ), ( , 0) = ( ), (0, ) = 0, (27) ∫ ( , ) = 0, Which has the following solution ( , )=
[
( )+
S. Xi, M. Jia, H. Ji (2009); Positive solutions of boundary value problems for systems of second order differential equations with integral boundary condition on the half-line, Electron. J. Qual. Theory Differ. Equ. 1, 1-12. N. I. Yurchuk (1986), Mixed problem with an integral condition for certain parabolic equations, Diff. Eqs., Vol. 22, No. 12, pp. 2117-2126.
(
( )+2 ( ))] (28) We change anyhow the small initial function ( ). Let ( ) = ( ) + ( ). ( ) sin 2 where , − , = − sufficiently large number. Then ( , ) = ( , ) + ≫ ( , ). This means that the solution of (27) can be changed anyhow by changing anyhow the small initial function ( ), ∀ > 0 from which implies the instability of this problem and this means the instability of (26) to small variations of ( ), that is, it’s non-well-posdness consequently, we cannot apply the considered “ideal” approach to the solution of the given problem. REFERENCES N. E. Benouar and N. I. Yurchuk (1991), Mixed problem with an integral condition for parabolic equations with the Bessel operator, Di erentsial'nye Uravneniya, 27, 2094-2098. A. Bouziani (1996), Mixed problem with boundary integral conditions for a certain parabolic equation, J. Appl. Math. Stochastic Anal., 9, 323-330. A. Bouziani (1997), Strong solution for a mixed problem with nonlocal condition for a certain pluriparabolic equations, Hiroshima Math. J., 27, 373-390. M. Feng, B. Du, W. Ge (2009); Impulsive boundary value problems with integral boundary conditions and one-dimensional p-Laplacian, Nonlinear Anal. 70, 3119-3126. N. I. Ionkin (1977), Solution of a boundary-value problem in heat conduction with a nonclassical boundary condition, Diff. Eqs., Vol. 13, No. 2, pp. 294-304.
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