# zbMATH — the first resource for mathematics

Parameter determination in a partial differential equation from the overspecified data. (English) Zbl 1080.35174
Summary: Several schemes are presented for computing the unknown coefficient $$p(t)$$ in the quasilinear equation $$u_t=u_{xx}+ p(t)u+\varphi$$, in $$R\times (0,T]$$, $$u(x,0)=f(x)$$, $$x\in R=[0,1]$$, $$u$$ is known on the boundary of $$R$$ and subject to the integral overspecification over the spatial domain $$\int^1+0k(x)u (x,t)dx=E(t)$$, $$0\leq t\leq T$$ or the overspecification at a point in the spatial domain $$u(x_0,t)=E(t)$$, $$0\leq t\leq T$$, where $$E(t)$$ is known and $$x_0$$ is a given point of $$R$$. These numerical procedures are developed for identifying the unknown control parameter which produces, at any given time, a desired energy distribution in the spatial domain, or a desired temperature distribution at a given point in the spatial domain. Several finite-difference techniques are used to determine the solution. The accuracy and stability of the methods are discussed and compared. Numerical illustrations are given.

##### MSC:
 35R30 Inverse problems for PDEs 35K55 Nonlinear parabolic equations 65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
Full Text:
##### References:
 [1] Cannon, J.R.; Yin, H.M., On a class of non-classical parabolic problems, J. differential equations, 79, 266-288, (1989) · Zbl 0702.35120 [2] Cannon, J.R.; Yin, H.M., Numerical solution of some parabolic inverse problems, Numerical methods for partial differential equations, 2, 177-191, (1990) · Zbl 0709.65105 [3] Cannon, J.R.; Lin, Y., Determination of parameter p(t) in holder classes for some semilinear parabolic equations, Inverse problems, 4, 595-606, (1988) · Zbl 0688.35104 [4] Cannon, J.R.; Lin, Y., An inverse problem of finding a parameter in a semi-linear heat equation, J. math. anal. appl., 145, 470-484, (1990) · Zbl 0727.35137 [5] Cannon, J.R.; van der Hoek, J., Diffusion subject to the specification of mass, J. math. anal. appl., 115, 527-536, (1986) [6] Cannon, J.R.; Lin, Y.; Wang, S., Determination of source parameter in parabolic equations, Meccanica, 27, 85-94, (1992) · Zbl 0767.35105 [7] 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 [8] Wang, S.; Lin, Y., A finite difference solution to an inverse problem determining a control function in a parabolic partial differential equation, Inverse problems, 5, 631-640, (1989) · Zbl 0683.65106 [9] Cannon, J.R.; Lin, Y., Determination of parameter p(t) in some quasi-linear parabolic differential equations, Inverse problems, 4, 35-45, (1988) · Zbl 0697.35162 [10] Lin, Y., Analytical and numerical solutions for a class of nonlocal nonlinear parabolic differential equations, SIAM. J. math. anal., 25, 1577-1594, (1994) · Zbl 0807.35069 [11] Deckert, K.L.; Maple, C.G., Solution for diffusion with integral type boundary conditions, (), 354-361 · Zbl 0173.12803 [12] Macbain, J.A.; Bendar, J.B., Existence and uniqueness properties for one-dimensional magnetotelluric inversion problem, Journal of mathematical physics, 27, 645-649, (1986) [13] 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, Differential equations, 23, 136-143, (1987) [14] Rundell, W., Determination of an unknown non-homogenous term in a linear partial differential equation from overspecified boundary data, Appl. anal., 10, 231-242, (1980) · Zbl 0454.35045 [15] Ionkin, N.I., Solution of a boundary value problem in heat conduction with a nonclassical boundary condition, Differential equations, 13, 204-211, (1977) · Zbl 0403.35043 [16] Macbain, J.A., Inversion theory for a parametrized diffusion problem, SIAM journal of applied mathematics, 18, 1386-1391, (1987) · Zbl 0664.35075 [17] Prilepko, A.I.; Orlovskii, D.G., Determination of the evolution parameter of an equation and inverse problems of mathematical physics, I, Differential equations, 21, 119-129, (1985) [18] Prilepko, A.I.; Orlovskii, D.G., Determination of the evolution parameter of an equation and inverse problems of mathematical physics, II, Differential equations, 21, 694-701, (1985) [19] Azari, H., Numerical procedures for the determination of an unknown coefficient in parabolic differential equations, Dynamic of continuous, discrete and impulsive systems, 9, 555-576, (2002) · Zbl 1020.35114 [20] 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 HTP, 194, 11-16, (1992) [21] Friedman, A., Partial differential equations of parabolic type, (1964), Prentice Hall · Zbl 0144.34903 [22] Ladyzenskaja, O.A.; Solonnikov, V.A.; Uralecva, N.N., Linear and quasilinear equations of parabolic type, A.M.S. tran. math. mono., 23, (1968) [23] Mitchell, A.R.; Griffiths, D.F., The finite difference methods in partial differential equations, (1980), T. Wiley Englewood Cliffs, NJ · Zbl 0417.65048 [24] Warming, R.F.; Hyett, B.J., The modified equation approach to the stability and accuracy analysis of finite-difference methods, Journal of computational physics, 14, 159-179, (1974) · Zbl 0291.65023 [25] Dehghan, M., Fully implicit finite difference methods for two-dimensional diffusion with a non-local boundary condition, Journal of computational and applied mathematics, 106, 255-269, (1999) · Zbl 0931.65091 [26] Dehghan, M., Fully explicit finite difference methods for two-dimensional diffusion with an integral condition, Nonlinear analysis, theory, methods and applications, 48, 637-650, (2002) · Zbl 1003.65094 [27] Lapidus, L.; Pinder, G.F., Numerical solution of partial differential equations in science and engineering, (1982), Wiley · Zbl 0584.65056 [28] Gerald, C.F.; Wheatley, P.O., Applied numerical analysis, (1994), Addison-Wesley New York · Zbl 0877.65003 [29] Cannon, J.R.; Prez-Esteva, S.; van der Hoek, J., A Galerkin procedure for the diffusion equation subject to the specification of mass, SIAM. J. num. anal., 24, 499-515, (1987) · Zbl 0677.65108 [30] Habetlet, G.R.; Schiffman, R.I., A finite difference method for analysing the compression of pro-viscoelastic media, Computing, 6, 342-348, (1970) · Zbl 0295.73036 [31] Miller, R.K., An integro-differential equation for rigid heat conduction equations with memory, J. math. anal. appl., 66, 318-327, (1978) [32] Raynal, M., On some nonlinear problems of diffision, (), 251-266 [33] Cannon, J.R.; van der Hoek, J., The one phase Stefan problem subject to the specification of energy, J. math. anal. appl., 86, 281-289, (1982) · Zbl 0508.35074 [34] Cannon, J.R., The solution of the heat equation subject to the specification of energy, Quart. appl. math., 21, 155-160, (1963) · Zbl 0173.38404 [35] Cannon, J.R., The one dimensional heat equation, () · Zbl 0168.36002 [36] Kamynin, L.I., A boundary value problem in the theory of heat conduction with a non-classical boundary condition, USSR comp. math. and math. physics, 4, 33-59, (1964) · Zbl 0206.39801 [37] Capasso, V.; Kunisch, K., A reaction-diffusion system arising in modeling man-environment diseases, Quart. appl. math., 46, 431-449, (1988) · Zbl 0704.35069 [38] Dehghan, M., Efficient techniques for the second-order parabolic equation subject to nonlocal specifications, Applied numerical mathematics, 52, 1, 39-62, (2005) · Zbl 1063.65079 [39] Evans, L.C., A free boundary value problem: the flow of two immiscible fluids in a one-dimensional porous medium, I, Indiana univ. math. J., 26, 915-932, (1977) · Zbl 0411.76066 [40] Evans, L.C., A free boundary value problem: the flow of two immiscible fluids in a one-dimensional porous medium, II, Indiana univ. math. J., 27, 93-111, (1978) · Zbl 0411.76067 [41] Cannon, J.R.; Zachmann, D., Parameter determination in parabolic partial differential equations from overspecified boundary data, Int. J. eng. sci., 20, 6, 779-788, (1982) · Zbl 0485.35083 [42] Dehghan, M., On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation, Numerical methods for partial differential equations, 21, 1, 24-40, (2005) · Zbl 1059.65072
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.