Direct and Inverse Problems for the Heat Equation with a Dynamic type Boundary Condition (original) (raw)
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3 Direct and Inverse Problems for the Heat Equation with a Dynamic Type Boundary Condition
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This paper considers the initial-boundary value problem for the heat equation with a dynamic type boundary condition. Under some regularity, consistency and orthogonality conditions, the existence, uniqueness and continuous dependence upon the data of the classical solution are shown by using the generalized Fourier method. This paper also investigates the inverse problem of finding a time-dependent coefficient of the heat equation from the data of integral overdetermination condition. au t (1, t) + du x (1, t) + (ap(t) − b)u(1, t) = af (1, t).
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Let u t = u xx − q x u 0 ≤ x ≤ 1 t > 0 u 0 t = 0 u 1 t = a t u x 0 = 0, where a t is a given function vanishing for t > T a t ≡ 0 T 0 a t dt < ∞. Suppose one measures the flux u x 0 t = b 0 t for all t > 0. Does this information determine q x uniquely? Do the measurements of the flux u x 1 t = b t give more information about q x than b 0 t does? These questions are answered in this note.
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