Homogeneous difference schemes for the heat conduction equation with variable coefficients

1 HOMOGENEOUS DIFFERENCE SCHEMES FOR THE HEAT CONDUCTION EQUATION WITH VARIABLE COEFFICIENTS 

The origiiral problem. We begin by placing the first boundary-value problem for for the heat conduction equation in which it is required to find a continuous in the rectangle Dt = 0 x 1, 0 t T solution to the equation 

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we take for granted that problem (l)-(3) possesses a unique solution with all necessary derivatives. 

Homogeneous difference schemes with weights. In a common setting it seems natural to expect that a difference scheme capable of describing this or that nonstationary process would be suitable for the relevant stationary process, that is, for du/dt = 0 we should have at our disposal a difference scheme from a family of homogeneous conservative schemes, whose use permits us to solve the equation Lu + / = 0. 

One way of covering this for the heat conduction equation is to construct a homogeneous conservative scheme by means of the integro-interpo-lation method. To make our exposition more transparent, we may assume that the coefficient of heat conductivity k = k x) is independent of t. The general case k = k(x,t) will appear on this basis in Section 8 without any difficulties. 

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Homogeneous difference schemes for the heat conduction equation with variable coefficients... 

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