1549301742-The_Theory_of_Difference_Schemes__Samarskii

Other problems 179

where a, d and <p satisfy the approximation conditions (18) of Section 2.

First, we consider the simplest approximation of the boundary con-

dition at the point x = 0: a 1 Yx,l = {3 1 y 0 - μ 1 and calculate the error of

approximation by inserting y = z + u:

We obtain

through the approximations

a 1 - _ k 0 + 2 l h kI 2)

0 + O(h ,

Substituting here (ku')~ = q 0 u 0 - f 0 emerged from equation (1) we find

that

V 1 =! h (q 0 u 0 - f 0 ) + O(h^2 ),

thereby justifying that the boundary condition

provides an approximation of order 2 on the solution u(x) of problem (1).

In a similar manner we derive the difference boundary condition of second-

order approximation at the point x = 1:

In this way, the third kind difference boundary-value problem (2)-( 4) of

second-order approximation on the solution of the original problem is put

in correspondence with the original problem ( 1).

2. A problem with periodicity conditions. First, we study the elementary
   problem in which it is required to find on the segment 0 < x < 1 a solution
   to the equation

(5) tt^11 (x)- q 0 u = -f(x), q 0 = const > 0, 0 < x < 1,

satisfying the condition of periodicity

(6) u(x + 1) = u(x) forall xE(0,1).

Here f(x) is a periodic function of period 1: f(x + 1) = f(x).