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Determination of a control parameter in the two-dimensional diffusion equation. (English) Zbl 0982.65103
Summary: This paper considers the problem of finding $w= w(x,y,t)$ and $p= p(t)$ which satisfy $$w_t= w_{xx}+ w_{yy}+ p(t)w+ \phi,$$ in $R\times (0,T]$, $w(x,y,0)= f(x,y)$, $(x,y)\in R= [0,1]\times [0,1]$, $w$ is known on the boundary of $R$ and also $\int^1_0 \int^1_0 w(x,y,t) dx dy= E(t)$, $0< t\le T$, where $E(t)$ is known. Three different finite-difference schemes are presented for identifying the control parameter $p(t)$, which produces, at any given time, a desired energy distribution in a portion of the spatial domain. The finite difference schemes developed for this purpose are based on the (1,5) fully explicit scheme, and the (5,5) Noye-Hayman (N-H) fully implicit technique, and the Peaceman and Rachford (P-R) alternating direction implicit (ADI) formula. These schemes are second-order accurate. The ADI scheme and the 5-point fully explicit method use less central processor (CPU) time than the (5,5) N-H fully implicit scheme. The P-R ADI scheme and the (5,5) N-H fully implicit method have a larger range of stability than the (1,5) fully explicit technique. The results of numerical experiments are presented, and CPU times needed for this problem are reported.

MSC:
65M32Inverse problems (IVP of PDE, numerical methods)
65M06Finite difference methods (IVP of PDE)
35K15Second order parabolic equations, initial value problems
35R30Inverse problems for PDE
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Full Text: DOI
References:
[1] Cannon, J. R.; Lin, Y.: Determination of parameter $p(t)$ in some quasi-linear parabolic differential equations. J. inverse probl. 4, 35-45 (1988) · Zbl 0697.35162
[2] Cannon, J. R.; Lin, Y.: Determination of parameter $p(t)$ in holder classes for some semilinear parabolic equations. J. inverse probl. 4, 595-606 (1988) · Zbl 0688.35104
[3] Cannon, J. R.; Lin, Y.: An inverse problem of finding a parameter in a semi-linear heat equation. J. math. Anal. appl. 145, No. 2, 470-484 (1990) · Zbl 0727.35137
[4] J.R. Cannon, Y. Lin, S. Wang, Determination of a control function in a parabolic partial differential equations, Research report 89-10, Department of Mathematics and Statistics, McGill University, 1989
[5] Cannon, J. R.; Lin, Y.; Wang, S.: Determination of source parameter in parabolic equations. Meccanica 27, 85-94 (1992) · Zbl 0767.35105
[6] Cannon, J. R.; Yin, H. M.: On a class of non-classical parabolic problems. J. differential equations 79, 266-288 (1989) · Zbl 0702.35120
[7] Cannon, J. R.; Yin, H. M.: Numerical solution of some parabolic inverse problems. Numer. methods partial differential equations 2, 177-191 (1990) · Zbl 0709.65105
[8] Day, W. A.: Extension of a property of the heat equation to linear thermoelasticity and other theories. Quart. appl. Math. 40, 319-330 (1982) · Zbl 0502.73007
[9] Dehghan, M.: Implicit locally one-dimensional methods for two-dimensional diffusion with a non-local boundary condition. Math. comput. Simul. 49, 331-349 (1999) · Zbl 0949.65085
[10] Dehghan, M.: A finite difference method for a non-local boundary value problem for 2-dimensional heat equation. Appl. math. Comput. 112, No. 1, 133-142 (2000) · Zbl 1023.65087
[11] Gerald, C. F.; Wheatley, P. O.: Applied numerical analysis. (1994) · Zbl 0877.65003
[12] Lapidus, L.; Pinder, G. F.: Numerical solution of partial differential equations in science and engineering. (1982) · Zbl 0584.65056
[13] Mitchell, A. R.; Griffiths, D. F.: The finite difference methods in partial differential equations. (1980) · Zbl 0417.65048
[14] Noye, B. J.; Hayman, K. J.: Implicit two-level finite-differences methods for the two-dimensional diffusion equation. Internat. J. Comput. math. 48, 219-228 (1993) · Zbl 0792.65065
[15] Prilepko, A. I.; Orlovskii, D. G.: Determination of the evolution parameter of an equation and inverse problems of mathematical physics, part I. J. differential equations 21, 119-129 (1985)
[16] Prilepko, A. I.; Soloev, V. V.: Solvability of the inverse boundary value problem of finding a coefficient of a lower order term in a parabolic equation. J. differential equations 23, No. 1, 136-143 (1987)
[17] Rundell, W.: Determination of an unknown non-homogeneous term in a linear partial differential equation from overspecified boundary data. Appl. anal. 10, 231-242 (1980) · Zbl 0454.35045
[18] Wang, S.: Numerical solutions of two inverse problems for identifying control parameters in 2-dimensional parabolic partial differential equations. Modern developments in numerical simulation of flow and heat transfer 194, 11-16 (1992)
[19] Wang, S.; Lin, Y.: A finite difference solution to an inverse problem determining a control function in a parabolic partial differential equations. J. inverse probl. 5, 631-640 (1989) · Zbl 0683.65106
[20] Warming, R. F.; Hyett, B. J.: The modified equation approach to the stability and accuracy analysis of finite-difference methods. J. comput. Phys. 14, No. 2, 159-179 (1974) · Zbl 0291.65023