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Determination of a source term in a linear parabolic partial differential equation

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Abstract

The determination of a spatially dependent source term in a linear parabolic differential equation whose coefficients depend only on the space variable from the specification of the heat flux at one of the boundaries in addition to the usual boundary-initial conditions is a not wellposed problem in the sense of Hadamard. Continuous dependence upon the data is studied and it is demonstrated that the asymptotic dependence as the norm of the data tends to zero is no worse than logarithmic.

Zusammenfassung

Wir betrachten eine lineare parabolische Differentialgleichung, deren Koeffizienten zeitunabhängig sind. Die Bestimmung der unbekannten zeitunabhängigen Wärmequelle aus dem seitlichen Wärmefluss, zusätzlich zu den üblichen Randbedingungen ist kein wohlbestimmtes Problem im Sinne von Hadamard. Wir betrachten die stetige Abhängigkeit der Lösungen von diesen Daten und zeigen, dass die asymptotische Abhängigkeit höchstens logarithmisch ist, wenn die Norm der Daten gegen Null strebt.

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Authors and Affiliations

  1. Dept. of Mathematics, The University of Texas at Austin, Austin, USA

    J. R. Cannon

  2. Dept. of Mathematics, Oakland University, Rochester, Michigan, USA

    Richard E. Ewing

Authors
  1. J. R. Cannon
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  2. Richard E. Ewing
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Cannon, J.R., Ewing, R.E. Determination of a source term in a linear parabolic partial differential equation. Journal of Applied Mathematics and Physics (ZAMP) 27, 393–401 (1976). https://doi.org/10.1007/BF01590512

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  • DOI: https://doi.org/10.1007/BF01590512

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