## ABSTRACT

Neumann indexboundary conditions:

− d dx

( du (x)

dx

) = 0 (1.1)

u (0) = 0 (1.2) du (1)

dx = 1 (1.3)

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2 Direct Solver

One-Dimensional Problems

In the finite difference method, we selectN+1 points {xi}i=0,...,N = { iN }i=0,...,N , distributed over the domain [0, 1]. First, we take values of problem (1.3) at these pointsŁ

u(x0) = 0 (1.4)

− d dx

( du (xi)

dx

) = 0 i = 0, ..., N (1.5)

du(xN )

dx = 1 (1.6)

Second, we replace the first and second derivatives at (1.4-1.6) with their finite difference discreti-

zations:

u0 = 0 (1.7)

−ui+1 − 2ui + ui−1 h2

= 0 i = 1, ..., N − 1 (1.8) uN − uN−1

h = 1 (1.9)

Here ui = u (xi) = u ( i N

) , h = 1N . We cancel out the h

2 factors and organize the terms in rows to

obtain

u0 = 0 (1.10)

ui−1 − 2ui + ui+1 = 0 i = 1, ..., N − 1 (1.11) −uN−1 + uN = h (1.12)

Third, we construct a global system of linear equations

1 0 0 0 0 0 0

1 −2 1 0 0 0 0 0 ... ... ... 0 0 0

0 0 1 −2 1 0 0 0 0 0 ... ... ... 0

0 0 0 0 1 −2 1 0 0 0 0 0 −1 1

u0

u1

...