Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 4(2):216-218 (ISSN: 2141-7016)

216

Reducing the Solution of the Control Problem of Inhomogeneous

Heat Equation to the Homogeneous Case

Hasan Almefleh

Dept. Of Mathematics, Yarmouk University, Irbid – Jordan.

__________________________________________________________________________________________ Abstract In the earlier study of Raid Almomani and Hasan Almefleh, the control problem of heat conduction problem with inverse direction of time and integral boundary conditions was formulated, showing the non-well-posedness of the problem. In this paper we reduce the solution of the control problem of the inhomogeneous heat equation to the homogeneous case. The solution of our problem plays 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: integral boundary conditions, inhomogeneous heat equation, control problem of heat conduction,

inverse direction of time. INTRODUCTION The importance of the heat conduction problem with integral boundary conditions its non self-adjointness, which make principle difficulties while investigating the problem. The solution of our problem plays 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. Many works were devoted to this problem lately, see Benouar and Yurchuk (1991), Bouziani (1996), Bouziani (1997), Feng, B. Du, W. Ge (2009), S. Xi, M. Jia, H. Ji (2009), Yurchuk (1986). Nonhomogeneous Equation and Nonhomogeneous Boundary Conditions. Consider in the domain

the inhomogeneous heat equation

(1)

With initial condition

(2)

And nonhomogeneous boundary conditions

(3)

(4)

Here the following consistency conditions hold

(5)

The function where is the Sobolev space of first order.

We set the control problem as the solution of problem (1)-(4). For a given and function that satisfy conditions

(6)

We minimize the functional

(7)

Where is the solution of (1)-(4) at

. The problem (1)-(4) turns to a problem with homogeneous boundary conditions by the following substation

(8)

Where

(9)

Then if we set

(10)

Where is

a known function, , then the minimization problem of the functional (7) turns to the minimization problem of the following functional

(11)

Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 4(2): 216-218 © Scholarlink Research Institute Journals, 2013 (ISSN: 2141-7016) jeteas.scholarlinkresearch.org

Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 4(2):216-218 (ISSN: 2141-7016)

217

Where is the solution of the problem with homogeneous boundary conditions

(12)

,

(13)

So, if the problem for the functional (11) is solved, i.e. the function such that at

. Then the function for which

can be found by the formula (14)

Nonhomogeneous Equation and Homogeneous Boundary Conditions Consider in the domain the nonhomogeneous equation

, (15)

With initial condition

(16) And homogeneous boundary conditions

(17)

(18)

We consider the minimization problem of the functional

(19)

Let be the solution of the problem,

(20)

And let be the solution of

(21)

Then

And

(22)

Therefore, the functional minimization problem (19) turns to the already considered functional minimization problem for the homogeneous equation by substituting in place of

. If the function is found such that

as at for the

homogeneous equation, then as

for a given function for the considered nonhomogeneous equation. CONCLUSION We reduced the solution of the control problem of the inhomogeneous heat equation to the homogeneous case, and this makes the problem much easier to deal with. The solution of our problem plays an important role in optimal control in heat conduction theory and in plasma physics. REFERENCES 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.

Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 4(2):216-218 (ISSN: 2141-7016)

218

N. E. Benouar and N. I. Yurchuk (1991), Mixed problem with an integral condition for parabolic equations with the Bessel operator, Differentsial'nye Uravneniya, 27, 2094-2098. N. I. Yurchuk (1986), Mixed problem with an integral condition for certain parabolic equations, Diff. Eqs., Vol. 22, No. 12, pp. 2117-2126 Raid Almomani, Hasan Almfleh (2012), On Heat Conduction Problem with Integral Boundary Condition, (JETEAS) 3(6): 977-979. 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.