Problem 4: Consider an infinite slab of unit width. The differential equation describing the

system 15 +

0 =

d'T

= 0

=0

dz

-0

=-1

+

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).