Abstract
We consider the problem of determining u = u(x,t) and f = f(x) which satisfy Lu = (p(x)ux)x − q(x)u − p(x)ut= p(x)f(x), 0 < x < 1, 0 < t ≤ T, u(x,0) = φ(x), u(0,t) = σ1 (t), and ux (1,t) + ßu(l,t) = σ2(t) from the additional specification of p(0)ux (0,t) = g(t). The problem is not well posed in the sense of Hadamard. Uniqueness is demonstrated. Continuous dependence of u and f upon the data is demonstrated for f twice continously differentiable in 0 ≤ x ≤ 1 with f(0) = 0 and f’(l) + ßf(l) = 0 and with first and second derivatives bounded in absolute value by the known positive constant K. The continuous dependence is shown to be asymptotically logarithmic
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© 1986 Birkhäuser Verlag Basel
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Cannon, J.R., Lin, Y. (1986). Determination of a Source Term in a Linear Parabolic Differential Equation with Mixed Boundary Conditions. In: Cannon, J.R., Hornung, U. (eds) Inverse Problems. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique, vol 77. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-7014-6_2
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DOI: https://doi.org/10.1007/978-3-0348-7014-6_2
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