1. Use the Galerkin method to solve the following boundary value problem using

(a) one-tei approximation and

(b) two-term approximation. Compare your results with the exact soluti by plotting them on the same graph.

d^u o

^r+X"=0, 0

•C": dx2 I^i it(0) = 1 1 f—> Boundary conditions tt(l)=0j cr;

S;

Hint: Use the following one- and two-term approximations:

One-term approximation:

u{x) = (1 — x) + Ci(pi(x)

= (1 — x) + Cix{\ — x)

Two-term approximation:

u{x) = (1 - X) +C\^\ (x) + C202 W

= (1 — x) + C\x{\ — x) + C2X2{1 — X)

The exact solution is u(x) = 1 — x(x3 + 11)/12 The approximate solution is split into two parts. The first term satisfies the given essential boi dary conditions exactly; i.e., «(0) = 1 and u(l) = 0. The rest of the solution containing (

unknown coefficients vanishes at the boundaries.