On inverse problems of determining a coefficient in a parabolic equation. II

Prilepko, A. I.; Костин, А. Б.

doi:10.1007/bf00971406

Cited by 26 publications

(22 citation statements)

References 2 publications

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“…Hence, the agreement condition is satisfied for problem (7), (8), (9 ). This condition, the above membership, and the conditions on the data of the problem in the statement of the assertion guarantee the existence of a solution to problem (7), (8), (9 ) in W 2,1 2 (Q) ∩ L ∞ (0, T ; W 1 2 (D)); see [18].…”

mentioning

confidence: 88%

“…is a solution to the boundary value problem (7), (8), (9 ). Moreover, passing to the limit as k → ∞ in the inequalities of (10)-(12) with indices m k yields the estimates…”

mentioning

confidence: 98%

“…Integrating the first equality by parts and using the conditions of the assertion, estimate (10), and Young's inequality, we can easily obtain the estimate…”

mentioning

confidence: 98%

“…Thus, for every function v(x, t) in H, there is a function u(x, t) in H solving the boundary value problem (7), (8), (9 ).…”

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confidence: 99%

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On solvability of an inverse problem with an unknown coefficient and right-hand side for a parabolic equation

Кожанов

2002

Journal of Inverse and Ill-posed Problems

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We study the inverse problem for a parabolic equation in which the unknowns are the solution, coefficient of the function, and right-hand side (exterior load). We prove existence and uniqueness theorems.Let D be a bounded domain in R n with boundary Γ and let Q be the cylinder D × (0, T ), 0 < T < +∞. Consider the following equation in Q:where λ is a given positive constant, f (x, t) and a(x, t) are given functions subject to the conditions to be specified below, and q(x) and q 0 (x) are functions to be determined together with a solution u(x, t).The problems of this kind are called inverse, and there are many articles devoted to their study. Among the latter the most closed to us are [1-17]. The problem under consideration relates to the class of problems in which the unknowns are not only a solution but also the coefficient of the equation and right-hand side. The fact that the coefficient is unknown makes this problem nonlinear.We turn to the exact statement of the problem. Considering inverse problems, we must in general provide extra information or take extra measurements alongside the initial information corresponding to the information about boundary values of some direct boundary value problem (i.e., a problem with available coefficients and right-hand side). In our case we suggest using as complementary the information on the state and velocity of the medium at a fixed point of time.

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confidence: 88%

“…is a solution to the boundary value problem (7), (8), (9 ). Moreover, passing to the limit as k → ∞ in the inequalities of (10)-(12) with indices m k yields the estimates…”

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confidence: 98%

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On solvability of an inverse problem with an unknown coefficient and right-hand side for a parabolic equation

Кожанов

2002

Journal of Inverse and Ill-posed Problems

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“…Note that the problem we consider below is not reduced to this case. In this paper, the results of [6,7] are extended to the more general overdetermination condition (1.2) and an iterative method for finding the coefficient r(x) in the class of bounded functions is proposed and justified. The case of integral observation for a smooth weight function ω(t), which has not been addressed earlier for such a problem, is considered separately.…”

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confidence: 99%

Recovery of the coefficient of u t in the heat equation from a condition of nonlocal observation in time

Костин

2015

Comput. Math. and Math. Phys.

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The inverse problem of finding the coefficient ρ(x) = ρ 0 + r(x) multiplying u t in the heat equation is studied. The unknown function r(x) 0 is sought in the class of bounded functions, and ρ 0 is a given positive constant. In addition to the initial and boundary conditions (data of the direct problem), a nonlocal observation condition is specified in the form u(x, t)dμ(t) = χ(x) with a given measure dμ(t) and a function χ(x). The case of integral observation (i.e., dμ(t) = ω(t)dt) is considered separately. Sufficient conditions for the existence and uniqueness of a solution to the inverse problem are obtained in the form of easy to check inequalities. Examples of inverse problems are given for which the assumptions of the theorems proved in this work are satisfied.Keywords: coefficient inverse problems, inverse problem for the heat equation, nonlocal observation (or overdetermination) condition, sufficient conditions for the existence and uniqueness of a solution. ≥ 0 T ∫ COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS Vol. 55 No. 1 2015 KOSTINwhere μ(t) and χ(x) are given functions. Observation condition (1.2) is nonlocal in time t, and it was pro posed for a linear inverse problem in [19]. More specifically, it includes the final observation and the integral observation .If, for example, ω(t) = 1/T, then we have the condition of the integral mean of the function u(x, t):.Another typical example of a condition of form (1.2) is , where α k > 0 and t k are given numbers. This condition can be interpreted as setting a weighted mean value of u(x, t).The problem of determining the coefficient of u t in a parabolic equation from a final observation con dition was studied in [3,8]. The solution was considered in Hölder classes, which ruled out the possibility of discontinuous coefficients. In [6, 7] a similar coefficient inverse problem with final and integral obser vations was investigated in Sobolev classes. In [11] the coefficient of Δu was recovered in the case of inte gral observation. Note that the problem we consider below is not reduced to this case. In this paper, the results of [6,7] are extended to the more general overdetermination condition (1.2) and an iterative method for finding the coefficient r(x) in the class of bounded functions is proposed and justified. The case of integral observation for a smooth weight function ω(t), which has not been addressed earlier for such a problem, is considered separately. As a result, a unique solvability theorem for the inverse problem is proved under weaker constraints on the given functions. Note also that the inverse problem of recovering r(x) with a general observation of form (1.2), where μ(t) has a bounded variation on [0, T], has not been studied previously. However, even in the case of integral and final observations, Theorem 1 and 2 provide new (in comparison with [6-8]) unique solvability conditions for the inverse problem. Specifically, in contrast to [6][7][8], the initial data are no longer required to be zero, and an explicit estimate for the desir...

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Iterative Process for Numerical Recovering the Lowest Order Space-Wise Coefficient in Parabolic Equations

Ivanov

Vabishchevich

2019

Finite Difference Methods. Theory and Applications

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On inverse problems of determining a coefficient in a parabolic equation. II

Cited by 26 publications

References 2 publications

On solvability of an inverse problem with an unknown coefficient and right-hand side for a parabolic equation

On solvability of an inverse problem with an unknown coefficient and right-hand side for a parabolic equation

Recovery of the coefficient of u t in the heat equation from a condition of nonlocal observation in time

Iterative Process for Numerical Recovering the Lowest Order Space-Wise Coefficient in Parabolic Equations

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