A numerical method for an inverse source problem for parabolic equations and its application to a coefficient inverse problem

Abstract

<jats:title>Abstract</jats:title><jats:p>Two main aims of this paper are to develop a numerical method to solve an inverse source problem for parabolic equations and apply it to solve a nonlinear coefficient inverse problem. The inverse source problem in this paper is the problem to reconstruct a source term from external observations. Our method to solve this inverse source problem consists of two stages. We first establish an equation of the derivative of the solution to the parabolic equation with respect to the time variable. Then, in the second stage, we solve this equation by the quasi-reversibility method. The inverse source problem considered in this paper is the linearization of a nonlinear coefficient inverse problem. Hence, iteratively solving the inverse source problem provides the numerical solution to that coefficient inverse problem. Numerical results for the inverse source problem under consideration and the corresponding nonlinear coefficient inverse problem are presented.</jats:p>

Stable Identification of Sources Located on Interface of Nonhomogeneous Media

“…, for all θ ∈ [0, 2π], θ 0 = π, and β 2 = 0.1. In this case, g belongs to L 2 (S 1 ) and satisfies condition (6). For this exact source g, we approximated function g ∈ L 2 (S 1 ) by its truncated Fourier series g N using the first N terms.…”

“…Theorem 3. For a measurement V defined on S 2 , if a source g defined on S 1 exists and satisfies condition (6), which is a solution to the inverse problem, then it is unique.…”

“…In this example, we considered the circular region Ω defined above, with σ 1 = 3, σ 2 = 1, R 1 = 1, and R 2 = 1.2. Let g(x, y) = x 2 − y 2 for all (x, y) ∈ S 1 be the exact source; therefore, it belongs to L 2 (S 1 ) and satisfies condition (6). In polar coordinates, g is expressed as…”

“…, m denotes the Lebesgue measure on R, β 2 = 0.1, and (a 1 , a 2 ) = (−1.0067, 0.0093) ∈ S 1 . In this case, g belongs to L 2 (S 1 ) and satisfies condition (6).…”

“…Source identification problems are widely investigated in many research fields [1][2][3][4][5][6][7][8][9], and they are modeled by boundary value problems for which the analysis of the associated forward problem and its corresponding inverse problem must be considered, see, e.g., [10,11]. The latter problem involves determining the source that yields the measurement on the boundary of the region [12][13][14].…”