II. FINITE DIFFERENCE METHOD 1 Difference formulae

Inserting such approximation in the equation (1) for i = 1, 2, . . . , n ¡ 1 and
replacing y(xi) by ¯yi, we obtain the following linear equations
or
¡ ¯yi+1 ¡ 2¯yi + ¯yi−1
h 2 + q(xi)¯yi = f(xi), i = 1, 2, . . . , n ¡ 1
¡¯yi+1 + ¯yi(2 + h 2 qi) ¡ ¯yi−1 = h 2 fi, i = 1, 2, . . . , n ¡ 1 (3)
where qi = q(xi), fi = f(xi). In particular, for i = 1 ad i = n ¡ 1, taking
into account the boundary conditions, we have the equations
for i = 1, (2 + h 2 q1)¯y1 ¡ ¯y2 = α + h 2 f1 (4)
for i = n ¡ 1, (2 + h 2 qn−1)¯yn−1 ¡ ¯yn−2 = β + h 2 fn−1 (5)
Thm. II.2.1 (Existence and uniqueness of the numerical solution)The difference
method applied to the problem (1) leads to a discrete problem which
is the linear system of n ¡ 1 equations
where
and
A =
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A¯y = b (6)
¯y = (¯y1, . . . , ¯yn−2, ¯yn−1) T ,
(2 + h 2 q1) ¡1 0
¡1 (2 + h 2 q2) ¡1
.
.
.
¡1 (2 + h 2 qn−2) ¡1
0 ¡1 (2 + h 2 qn−1)
b = (h 2 f1 + α, h 2 f2, ..., h 2 fn−2, h 2 fn−1 + β) T .
Under the hypothesis q(x) ¸ 0 such a system admits a unique solution.
PROOF
Equations (3), (4), (5) can be written in vector form as stated. Now from
linear algebra
A is non-singular () z T Az 6= 0, 8z 6= (0, ..., 0).
6
.
.
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⎟
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