1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference equations 25

A;

.0..i,i+I = - B; .0..i,i-I.

Due to this fact the condition .0..; 1 , ; 1 + 1 -::/= 0 for some i = i 1 yields .0..;, ;+ 1 -::/= 0

for all possibilities of i.

Taking into account the equation£ [y;] = 0 we see that, for any m > 1,

.0..;,i+m can be expressed through .0..i,i+l and, hence, .0..;,i+m-::/= 0 is ensured

by the condition .0..;,i+I -::/= 0.

If y?) and yp) are linearly independent solutions of the homogeneous

equation ( 44), then its general solution can be designed as a linear combi-

nation of y?) and y~^2 ) with arbitrary constants C 1 and C 2 :

(!) (2)

Yi = C1 Y; + C2 Yi ·

The constants are free to be chosen from the initial or boundary conditions,

since .0..i,i+m -::/= 0 for all admissible subscripts i and m.

The general solution of any nonhomogeneous difference equation

£ [y;] = -F; is representable by

(!) (2) ~

Yi = C1 Y; + C2 Y; + Yi ,

where fj; is a particular solution to the equation£ [fJ;] =-Fi.

In the case of the Cauchy problem with assigned values y 0 and y 1 , we

have at our disposal the system of algebraic equations for constants cl and

C2:

{

cl y~I) + C2 Y~^2 ) = Yo '

C 1 Y1 (1) + c 2 Y1 (2) = Y1 ·

As far as y~^1 ) and y~^2 ) are linearly independent, the condition .0.. 0 , 1 -::/= 0
provides a possibility of finding C 1 and C 2. In the case of the boundary-
val ue problem with Yo = μl and YN = μ2 incorporated, the constants cl
and C 2 Can Uniquely be determined Under the agreement .0._0, N -::/= 0.
If the coefficients involved in equation ( 44) are constant, that is, A; =
a, C; = c and B; = b, then particular solutions can be found in explicit
form. This can be done by attempting particular solutions to equation ( 45)
in the form Yk = qk, while the number q -::/= 0 remains as yet unknown.
Substituting Yk into ( 45) we obtain the quadratic equation for q:

( 46) b q^2 - c q +a= 0