1549301742-The_Theory_of_Difference_Schemes__Samarskii

7

Homogeneous Difference Schemes

for Time-Dependent Equations

of Mathematical Physics

with Variable Coefficients

7.1 HOMOGENEOUS DIFFERENCE SCHEMES FOR THE HEAT

CONDUCTION EQUATION WITH VARIABLE COEFFICIENTS

In the present chapter the objects of investigation are various homogeneous

difference schemes for the heat conduction equation and a second-order

equation of hyperbolic type with variable coefficients in several settings:

on nonequidistant grids, with the boundary conditions of the third kind,

etc. The results obtained in the preceding chapters find a wide range of

applications in designing homogeneous difference schemes and establishing

their stability. Especial attention here is being paid to one-dimensional

problems.

1. The original problem. vVe begin by placing the first boundary-value
   problen1 for for the heat conduction equation in which it is required to find
   a continuous in the rectangle Dr = {O < x < 1, 0 < t < T} solution to the
   equation

(1)

OU

7fi=L1.l+f(x,t),

satisfying the initial condition

(2) u(.r, 0) = u 0 (x),

459