Determination of a time-dependent heat source under nonlocal boundary and integral overdetermination conditions

Ismailov, Mansur I.; Kanca, Fatma; Lesnic, D.

doi:10.1016/j.amc.2011.09.044

Cited by 51 publications

(34 citation statements)

References 10 publications

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“…Inverse time-dependent source problems for the heat equation with local, nonlocal, integral or nonclassical (boundary) conditions have become the point of interest in many recent papers, [10,12,13,14,17,31,33], to name only a few. In the present paper, we consider yet another reconstruction of a time-dependent heat source from an integral over-determination measurement of the thermal energy of the system and a new dynamic-type boundary condition.…”

Section: Introductionmentioning

confidence: 99%

An inverse time-dependent source problem for the heat equation with a non-classical boundary condition

Hazanee

Lesnic

Ismailov

et al. 2015

Applied Mathematical Modelling

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This paper investigates the inverse problem of determining the time-dependent heat source and the temperature for the heat equation with a non-classical boundary and an integral over-determination conditions. The existence, uniqueness and continuous dependence upon the data of the classical solution of the inverse problem is shown by using the generalised Fourier method. Furthermore in the numerical process, the boundary element method (BEM) together with the second-order Tikhonov regularization is employed with the choice of regularization parameter based on the generalised cross-validation (GCV) criterion. Numerical results are presented and discussed.

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Section: Introductionmentioning

confidence: 99%

An inverse time-dependent source problem for the heat equation with a non-classical boundary condition

Hazanee

Lesnic

Ismailov

et al. 2015

Applied Mathematical Modelling

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“…[6][7][8] and the references therein. The integral is usually taken over the whole domain (or over a subdomain).…”

Section: G(u(t X))dx = M(t) In (0 T )mentioning

confidence: 99%

Full discretization of a nonlinear parabolic problem containing volterra operators and an unknown dirichlet boundary condition

Grimmonprez

Slodička

2015

Numer. Methods Partial Differential Eq.

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The reconstruction of an unknown solely time-dependent Dirichlet boundary condition in a nonlinear parabolic problem containing a linear and a nonlinear Volterra operator is considered. The inverse problem is converted into a variational problem in which the unknown Dirichlet condition is eliminated using a given integral overdetermination. A time-discrete recurrent approximation scheme is designed, using Backward Euler's method. The convergence of the approximations towards a solution of the variational problem is proved under appropriate assumptions on the data and on the Volterra operators. The uniqueness of this solution is shown in the case that the nonlinear Volterra operator satisfies a particular inequality. Moreover, the Finite Element Method is used to discretize the time-discrete approximation scheme in space. Finally, full-discrete error estimates are derived for a particular choice of the finite elements. The corresponding convergence rates are supported by a numerical experiment.

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“…Crucial in the analysis is that the term depending on the solution u in the right-hand side of (1) is in integral form. Note that other parameter (time-dependent) identification problems can be found in [16,17,18,19,20,21,22,23,24,25,26,27,28].…”

Section: Introductionmentioning

confidence: 99%

Identification of a memory kernel in a nonlinear integrodifferential parabolic problem

Bockstal¹,

Slodička²,

Gistelinck

2017

Applied Numerical Mathematics

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In this contribution, the reconstruction of a solely time-dependent convolution kernel in an nonlinear parabolic equation is studied. The missing kernel is recovered from a global integral measurement. The existence, uniqueness and regularity of a weak solution is addressed. More specific, a numerical algorithm based on Rothe's method is designed. Numerical experiments support the obtained results.

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Determination of a time-dependent heat source under nonlocal boundary and integral overdetermination conditions

Cited by 51 publications

References 10 publications

An inverse time-dependent source problem for the heat equation with a non-classical boundary condition

An inverse time-dependent source problem for the heat equation with a non-classical boundary condition

Full discretization of a nonlinear parabolic problem containing volterra operators and an unknown dirichlet boundary condition

Identification of a memory kernel in a nonlinear integrodifferential parabolic problem

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