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Problem 4: Consider an infinite slab of unit width. The differential equation describing the
system 15 +
0 =
= 0
g(=) =10-20=
1) Using 4 eigenfunctions o(=) = H, cos(@;=), where @; = (i-1)x, H1=1, Hi = 12, i=
2...4, approximate the differential equation with a set of algebraic equations using the
Galerkin method. Report your answer in the form of Ax = b by identifying x, A and b
and indicating how an approximation of (z) can be constructed from x. You should
find the following identity useful+
=cos(@ =)d= = co
cos(@ =) = sin( ()=)
2) Show that the solution to the system of equations from part 1 is +
T ( = ) = a + ( b, / 1 ) ( = ) + (b; / 4. )$ (=)+(b, 195 ) (=)+
where by are element of the vector b and of is any arbitrary number. Using MATLAB,
plot 7(z) with a = 303.+
3) Repeat part 1 with g(=) =10 and show that the resulting set of algebraic equations has
no solution. Discuss this conclusion in the context of Example 2.13 from the book
(Graham and Rawlings).

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