Abstract
Consider the simultaneous identification of the initial field and spatial heat source for heat conduction process from extra measurements with the two additional measurement data at different times. The uniqueness and conditional stability for this inverse problem are established by using the properties of a parabolic equation and the representation of solution after reforming the equation. By combining the least squares method with the regularization technique, the inverse problem is transformed into an optimal control problem. Based on the existence and uniqueness of the minimizer of the cost functional, an alternative iteration process is built to solve this optimizing problem by the variational adjoint method. The negative gradient direction is selected as the first search direction. For further iterations, the alternative iteration algorithm is used for the initial field and heat source identification. The efficiency of the proposed scheme is tested by the numerical simulation experiments.
Highlights
Consider the following heat conduction problem: ut – u = f (x) in Ω × (0, T), (1)with initial condition u(x, 0) = φ(x) in Ω, (2)and boundary condition u(x, t) = 0 on ∂Ω, (3)where Ω ⊂ Rd (d = 1, 2) is bounded, f (x) is the space-dependent heat source and φ(x) is the initial temperature with φ|∂Ω = 0
The problem considered in this paper is to determine the initial temperature φ(x) and the space-dependent source f (x) simultaneously from two additional measurement data at times T1, T2 for 0 < T1 < T2 ≤ T : u(x, T1) = ψ1(x), x ∈ Ω, (4)
An optimization problem is presented and an alternative iteration scheme is constructed by means of the variational adjoint method in Sect
Summary
The problem considered in this paper is to determine the initial temperature φ(x) and the space-dependent source f (x) simultaneously from two additional measurement data at times T1, T2 for 0 < T1 < T2 ≤ T : u(x, T1) = ψ1(x), x ∈ Ω, (4) 2, we give uniqueness analysis and a conditional stability result for the inverse problem (1)–(5). An optimization problem is presented and an alternative iteration scheme is constructed by means of the variational adjoint method in Sect.
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