Determination of Compactly Supported Sources for the One-Dimensional Heat Equation
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ination of Compactly Supported Sources for the One-Dimensional Heat Equation V. V. Solov’eva,* a National
Research Nuclear University “MEPhI”, Moscow, 115409 Russia *e-mail: [email protected]
Received May 10, 2019; revised February 3, 2020; accepted April 9, 2020
Abstract—The inverse problem of determining a source in the one-dimensional heat equation in the case of a Dirichlet boundary value problem is investigated. The trace of the solution of the direct problem on straight-line segments inside the domain at the final time is specified as overdetermination (i.e., additional information on the solution of the direct problem). A Fredholm alternative theorem for this problem is proved, and sufficient conditions for its unique solvability are obtained. The inverse problem is considered in classes of smooth functions with derivatives satisfying the Hölder condition. Keywords: heat equation, unknown source, inverse problem, uniqueness of solution, existence of solution DOI: 10.1134/S0965542520090146
1. FORMULATION OF THE INVERSE PROBLEM Let T > 0 , 0 < α < 1, and l > 0 be fixed numbers, and let Ω = (0, l ) and Ω = [0, l ] denote an open and a closed interval on the x axis. In the plane of points ( x, t ) with Cartesian coordinates, define a rectangle ΩT = Ω × (0, T ] with an upper cap and the parabolic boundary ∂ΩT = ({0} × [0,T ]) ∪ ({l} × [0,T ]) ∪ ([0, l ] × {0}) . In the closed rectangle ΩT = Ω × [0, T ], consider the Dirichlet initial-boundary value problem for the heat equation with a source of special kind, namely, consider the problem of finding a function u : ΩT → ℜ determined by the conditions
(Lu)( x, t ) = ut ( x, t ) − uxx ( x, t ) = fk ( x)hk ( x, t ) + g( x, t ) = q( x, t ) + g( x, t ), u(0, t ) = μ1(t ),
u(l, t ) = μ2(t ),
t ∈ [0,T ],
u( x, 0) = ϕ( x ),
( x, t ) ∈ ΩT , x ∈ Ω.
(1) (2)
In Eq. (1), g, hk : ΩT → ℜ, μ1, μ2 : [0, T ] → ℜ , and fk : ℜ → ℜ are given functions; k = 1,..., N , where N is a fixed positive integer; and summation from 1 to N is implied over diagonally written indices k . The functions fk are assumed to have a compact support on the interval Ω ; more precisely, there exist numbers 0 < a1 < b1 < .... < aN < bN < l such that each of the functions fk is nonzero only on the respective interval [ak , bk ], k = 1,..., N . To simplify the subsequent constructions, in the rectangle ΩT , we define the sets
P (k ) = (ak , bk ),
P
(k )
= [ak , bk ],
N
k = 1,..., N ,
P =
∪ k =1
(k )
P ,
ωk = P
T = P
N
∪
(k )
× {T },
T(k ) = [a , b ] × (0.T ], P k k
PT(k ) = P (k ) × (0, T ],
N
T(k ), P
PT =
k =1
∪P
(k ) T ,
A = {a1,..., aN },
B = {b1,..., bN },
k =1
T , QT = ΩT \ P
= Ω \P . Q T T T
Throughout this paper, the norm sign without additional indices for a function without indicated arguments is understood as the usual sup norm; moreover, the supremum of its absolute value is taken over the entire domain of this function. Additionally, for functions of several variables, we use the notation g(., t ) = sup x g( x, t ) . By the norm of the dire
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