# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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:
 65M32 Inverse problems (IVP of PDE, numerical methods) 65M06 Finite difference methods (IVP of PDE) 35K15 Second order parabolic equations, initial value problems 35R30 Inverse problems for PDE
Full Text:
##### 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