Geodesic
Existence of a~solution of an initial-boundary value difference problem for a~linear heat equation with a~nonlinear boundary condition
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 14 (2008) no. 1, pp. 11-21

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A problem of heat propagation in the ground from a heated pipeline with a partially heat-insulating shell is considered. The possibility is proved to construct a numerical solution of a linear heat equation by using a direct finite-difference method in the case when the thermal radiation on the ground surface is taken into account. On the basis of the theorem about the solvability of a system of linear difference equations by means of the sweep method, the existence and uniqueness of a solution of a corresponding difference problem with nonlinear boundary condition are proved. 
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     author = {N. A. Vaganova},
     title = {Existence of a~solution of an initial-boundary value difference problem for a~linear heat equation with a~nonlinear boundary condition},
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}
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N. A. Vaganova. Existence of a~solution of an initial-boundary value difference problem for a~linear heat equation with a~nonlinear boundary condition. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 14 (2008) no. 1, pp. 11-21. http://geodesic.mathdoc.fr/item/TIMM_2008_14_1_a1/