Integro-differential equation modelling heat transfer in conducting, radiating and semitransparent materials

Laitinen, M.; Tiihonen, Timo

doi:10.1002/(sici)1099-1476(19980325)21:5<375::aid-mma953>3.0.co;2-u

Cited by 38 publications

(18 citation statements)

References 9 publications

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“…Note that for d ≤ 2 it is in fact possible to use X itself as the state space, but for d = 3 we cannot guarantee that e 2 is well defined due to the fourth-order nonlinearity in T (compare [7]). …”

Section: The Optimal Control Problemmentioning

confidence: 99%

Analysis of optimal boundary control for radiative heat transfer modeled by the $SP_{n}$-system

Pinnau¹

2007

Communications in Mathematical Sciences

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Abstract. We present an analytic study of an optimal boundary control problem for the diffusive SP 1 -system modeling radiative heat transfer. The cost functional is of tracking-type and the control problem is considered as a constrained optimization problem, where the constraint is given by the nonlinear parabolic/elliptic SP 1 -system. We prove the existence, uniqueness and regularity of bounded states, which allows for the introduction of the reduced cost functional. Further, we show the existence of an optimal control, derive the first-order optimality system and analyze the adjoint system, for which we prove existence, uniqueness and regularity of adjoint states.

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Section: The Optimal Control Problemmentioning

confidence: 99%

Analysis of optimal boundary control for radiative heat transfer modeled by the $SP_{n}$-system

Pinnau¹

2007

Communications in Mathematical Sciences

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“…The transfer equation together with energy conservation is considered in [8,15]. The issue of heat conduction is addressed in [11,12,10]. Convection, conduction and radiation are treated in [18,14].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Existence and bounds for the half-moment entropy approximation to radiative transfer

Frank¹,

Pinnau²

2007

Applied Mathematics Letters

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“…Earlier, in [9] (see also [10]), the existence of the solution to the stationary problem of radiative-conductive heat transfer in a system of semitransparent bodies in a simplified formulation without regard for reflection and refraction at the boundaries of bodies was proven. It is also worth mentioning references [11][12][13], in which similar problems were considered.…”

Section: Introductionmentioning

confidence: 99%

Stationary problem of complex heat transfer in a system of semitransparent bodies with boundary conditions of diffuse reflection and refraction of radiation

Амосов

2017

Comput. Math. and Math. Phys.

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A boundary value problem describing complex (radiation-conductive) heat transfer in a system of semitransparent bodies is considered. Complex heat transfer is described by a system consisting of a stationary heat equation and an equation of radiative transfer with the boundary conditions of diffuse reflection and diffuse refraction of radiation. The dependence of the radiation intensity and optical properties of bodies on the frequency of radiation is taken into account. The unique existence of the weak solution to this problem is established. The comparison theorem is proven. Estimates of the weak solution are derived, and its regularity is established.

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Integro-differential equation modelling heat transfer in conducting, radiating and semitransparent materials

Cited by 38 publications

References 9 publications

Analysis of optimal boundary control for radiative heat transfer modeled by the $SP_{n}$-system

Analysis of optimal boundary control for radiative heat transfer modeled by the $SP_{n}$-system

Existence and bounds for the half-moment entropy approximation to radiative transfer

Stationary problem of complex heat transfer in a system of semitransparent bodies with boundary conditions of diffuse reflection and refraction of radiation

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