1549301742-The_Theory_of_Difference_Schemes__Samarskii

Heat conduction equation with several spatial variables 343

Comparison with the general difference equation

(6) A(P) y(P) = B(P, Q) y(Q)+F(P),

QEPatt'(P)

which has been considered in Section 2 of Chapter 4, shows that in the case

of interest

p

A(P) = 1, B(P, Q) = 1 - 2 """"' ~ h2

T

°'

The boundary r of the grid w for equation (5) consists of the nodes (x, 0),

x E wh, and (x,tj'), x E fh> tj' < tj+i· It is straightforward to check that

D(P) = A(P) - B(P, Q) = 0, B(P, Q) > 0

QEPatt'(P)

if

p

(7) 1-2 2=

Common practice involves the decomposition for the solution of the problem
concerned as a sum y = y + y, where y is a solution to the homogeneous
equation

Yt = Ay, :YI "ih = μ, y(x, 0) = u 0 (x),

and f; is a solution 'to the nonhornogeneous equation

Yt =A f; + 'P, y -1 "ih --^0 ) f;(x,0)=0.

As a matter of fact, we obtain for y equation (6) with F = 0. On the
strength of the maximum principle (see corollaries to Theorem 2, Chapter
4, Section 2.3) we have