8 PreliminariesFor the second-order difference equations capable of describing the
basic mathematical-physics problems, boundary-value problems with addi-
tional conditions given at different points are rnore typical. For example, if
we know the value Yo for i = 0 and the value yN for i = N, the correspond-
ing boundary-value problem can be formulated as follows: it is necessary
to find the solution Yi, 0 < i < N, of problem ( 6) satisfying the boundary
conditions(8)with known numbers μ 1 and μ 2.
Common boundary conditions may be specified by( 8')that is, at the boundary nodes i = 0 and i = N not only the function values,
but also the first difference values or linear combinations of the function
and difference values are yet to be known. Substituting y 1 = Yo+ b. Yo into
the first condition (8') yields(8")
and needs certain clarification. The case x 1 = 0 corresponds to the first
kind boundary condition: Yo is given at the boundary node i = 0.
When x 1 = 1 we deal with the second kind boundary condition: b. Yo
is given at the same node. All the cases with x 1 -::f. O; 1 reflect the third
kind boundary condition as a linear cornbination of the function and
the first difference at the node i = 0.
Due to serious achievements of the Russian and foreign mathemati-
cians in applied mathematics the majority of rnathematical-physics prob-
lems may be reduced to computational algorithms, at every step of which
3-point equations like (6) with conditions (8') must be solved.
Moreover, a lot of rather complicated problems in numerical analysis
gives rise to the canonical problem, where a square (N + 1) x (N + 1)-matrix
of the corresponding system acquires a tridiagonal form
(^1) -X1 0 0 0 0 0 0 0
A1 -C1 B1^0 0 0 0 0 0
0 0 0 A; -C; B;^0 0 0
0 ()^0 0 0 0 A iV-1 -C N-l B 1V-l
0 0 0 0 0 0 0 -X2^1