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Determination of a control parameter in a parabolic partial differential equation

  • J. R. Cannon (a1), Yanping Lin (a2) and Shingmin Wang (a3)
Abstract
Abstract

The authors consider in this paper the inverse problem of finding a pair of functions (u, p) such that

where F, f, E, s, αi, βi, γi, gi, i = 1, 2, are given functions.

The existence and uniqueness of a smooth global solution pair (u, p) which depends continuously upon the data are demonstrated under certain assumptions on the data.

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[3] J. R. Cannon and J. van der Hoek , “The classical solution of the one-dimensional two-phase Stefan problem with energy specification”, Annali di Mat. Pura ed appl, (4) 130 (1982) 385398.

[4] J. R. Cannon and J. van der Hoek , “The one phase Stefan problem subject to the specification of energy”, J. Math. Anal. and Appl. 86 (1982) 281291.

[5] J. R. Cannon and J. van der Hoek , “Diffusion subject to the specification of mass”, J. Math. Anal. Appl. 115 (1986) 517529.

[6] J. R. Cannon , S. P. Esteva and J. van der Hoek , “A Galerkin procedure for the diffusion equation subject to the specification of mass”, SIAM J. Numer. Anal. 24 (1987) 499515.

[7] J. R. Cannon and Yanping Lin , “Determination of parameter p(t) in some quasi-linear parabolic differential equations”, Inverse Problems 4 (1988) 3545.

[8] J. R. Cannon and Yanping Lin , “Determination of parameter p(t) in Hölder classes for some semilinear parabolic equations”, Inverse Problems 4 (1988) 595606.

[9] J. R. Cannon and D. Zachmann , “Parameter determination in parabolic partial differential equations from overspecified boundary data”, Int. J. Eng. Sci. 20 No. 6 (1982) 779788.

[16] L. A. Kamynin , “A boundary value problem in the theory of heat conduction with a non-classical boundary condition”, (English translation) USSR Comp. and Math. Phys. 4 (1964) 3359.

[18] J. L. Lions and E. Magenes , Non-homogeneous boundary valued problems and applications, Vols. 1–III (Springer-Verlag, Berlin, 1972).

[22] W. Rundell , “Determination of an unknown nonhomogeneous term in a linear partial differential equation from overspecified boundary data”, Applicable Analysis 10 (1980) 231242.

[23] K. L. Teo , Computational methods for optimizing distributed systems (Academic Press, Orlando, 1984).

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The ANZIAM Journal
  • ISSN: 1446-1811
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