Inverse problems in mathematical physics

Glasko, V. B.

doi:10.1515/9783112313930-045

Cited by 50 publications

(55 citation statements)

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“…Proof. Following [4] and [1], we reduce the inverse problem to the operator equation for the coefficient k. On the other hand, the difference a − a σ satisfies the equation…”

Section: Existence and Uniqueness Theoremmentioning

confidence: 99%

“…The main goal of the paper is to establish the existence, uniqueness, and stability (in the sense of the continuous dependence on the data) of a strong solution to the problem (1.1)- (1.3). We use the method developed in [1]- [3] and based on reducing the inverse problem to an operator equation of the second kind for the unknown coefficient [4]. We note that the inverse problem for the linear equation Mu + ku = f with unknown constant coefficient k was studied in [5] and [6] in the case of boundary conditions of the first and third kind respectively.…”

Section: Introductionmentioning

confidence: 99%

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An Inverse Problem for a Quasilinear Elliptic Equation

Lyubanova

Velisevich

2023

J Math Sci

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We consider the inverse problem for a quasilinear second order elliptic equation with the unknown coefficient at the lower-order term and the first kind boundary condition and the integral overedetermination condition on the boundary. We prove a theorem on the local existence and uniqueness of a strong solution to the problem. The result is illustrated by an example of a nonlinear equation satisfying all the assumptions of the theorem. Bibliography: 6 titles.

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“…Proof. Following [4] and [1], we reduce the inverse problem to the operator equation for the coefficient k. On the other hand, the difference a − a σ satisfies the equation…”

Section: Existence and Uniqueness Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An Inverse Problem for a Quasilinear Elliptic Equation

Lyubanova

Velisevich

2023

J Math Sci

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“…On the other hand, for the hyperbolic inverse problem (2)-( 5), or (2), (3), ( 5)-( 7), or (2)-( 4), (6), or (2)-( 4), ( 7), uniqueness results are not known yet, but they could be investigated in the future using, e.g. the techniques from (Lesnic et al, 2016) (Carleman estimates) and Chapter 8 of (Prilepko et al, 2000) (semi-group theory).…”

Section: Mathematical Formulationmentioning

confidence: 99%

“…acting on the space X 5 H 1 (D) 3 L 2 (D) equipped with the family of equivalent norms, see p.590 of (Prilepko et al, 2000),…”

Section: Mathematical Formulationmentioning

confidence: 99%

Determination of the space-dependent blood perfusion coefficient in the thermal-wave model of bio-heat transfer

Alosaimi

Lesnic

2023

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PurposeWhen modeling heat propagation in biological bodies, a non-negligible relaxation time (typically between 15-30 s) is required for the thermal waves to accumulate and transfer, i.e. thermal waves propagate at a finite velocity. To accommodate for this feature that is characteristic to heat transfer in biological bodies, the classical Fourier's law has to be modified resulting in the thermal-wave model of bio-heat transfer. The purpose of the paper is to retrieve the space-dependent blood perfusion coefficient in such a thermal-wave model of bio-heat transfer from final time temperature measurements.Design/methodology/approachThe non-linear and ill-posed blood perfusion coefficient identification problem is reformulated as a non-linear minimization problem of a Tikhonov regularization functional subject to lower and upper simple bounds on the unknown coefficient. For the numerical discretization, an unconditionally stable direct solver based on the Crank–Nicolson finite difference scheme is developed. The Tikhonov regularization functional is minimized iteratively by the built-in routine lsqnonlin from the MATLAB optimization toolbox. Both exact and numerically simulated noisy input data are inverted.FindingsThe reconstruction of the unknown blood perfusion coefficient for three benchmark numerical examples is illustrated and discussed to verify the proposed numerical procedure. Moreover, the proposed algorithm is tested on a physical example which consists of identifying the blood perfusion rate of a biological tissue subjected to an external source of laser irradiation. The numerical results demonstrate that accurate and stable solutions are obtained.Originality/valueAlthough previous studies estimated the important thermo-physical blood perfusion coefficient, they neglected the wave-like nature of heat conduction present in biological tissues that are captured by the more accurate thermal-wave model of bio-heat transfer. The originalities of the present paper are to account for such a more accurate thermal-wave bio-heat model and to investigate the possibility of determining its space-dependent blood perfusion coefficient from temperature measurements at the final time.

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“…An inverse problems for this Boussinesq system have not been studied a lot, for instance, in previous studies [1,8,10], the results of existence and uniqueness of solutions of such inverse problems have been established in two dimensional case, by different methods. But, there are many works on inverse source problems of hydrodynamics, in particulary for Navier-Stokes system, we refer to the literature [28,7,9,11], and the references therein. To our best knowledge, an inverse source problem for heat convection for Kelvin-Voigt system has not been studied; however, there are several inverse problems for Kelvin-Voigt equations, which one can find in the literature [5,12,15,16,21].…”

Section: Introductionmentioning

confidence: 99%

Inverse problems for a Boussinesq system for incompressible viscoelastic fluids

Antont︠s︡ev

Khompysh

2023

Math Methods in App Sciences

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In this paper, we study two inverse problems for the nonlinear Boussinesq system for incompressible viscoelastic nonisothermal Kelvin–Voigt fluids. The studying inverse problems consist of determining an intensities of density of external forces and heat source under given integral overdetermination conditions. Two types of boundary conditions for the velocity boldv$$ \mathbf{v} $$ are considered: sticking and sliding conditions on boundary. In both cases of these boundary conditions, the local and global in time existence and uniqueness of weak and strong solutions are established under suitable assumptions on the data. The large time behavior of weak solutions is also studied.

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Inverse problems in mathematical physics

Cited by 50 publications

References 0 publications

An Inverse Problem for a Quasilinear Elliptic Equation

An Inverse Problem for a Quasilinear Elliptic Equation

Determination of the space-dependent blood perfusion coefficient in the thermal-wave model of bio-heat transfer

Inverse problems for a Boussinesq system for incompressible viscoelastic fluids

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