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2. Given is the following problem: ofu = Or (a(x)dru) , u(0, t) = 0, u(1, t) = 0, u(x, 0) = uo(x), where a(x) > 0. (5) (a) (2 points) Find the

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2. Given is the following problem: u- (a(x)u), u(0,t) , u(1, t 0, u(,u() where a(x) >0. (5) (a) 2 points) Find the finite difference scheme for (5) using the θ-method in time. You may use that central differences in space for the r (a( tem is given by (Ar)' where ajti1/2 - al) and i) b 2 points) Show that the matrix resulting from this scheme is diagonally dominant (c) (2 points) Show that a maximum principle will apply provided 2Δt(1-8)a(z)

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2. Given is the following problem:
ofu = Or (a(x)dru) ,
u(0, t) = 0, u(1, t) = 0, u(x, 0) = uo(x), where a(x) > 0. (5)
(a) (2 points) Find the finite difference scheme for (5) using the 0-method in time. You
may use that central differences in space for the ox (a(x)Oru) term is given by
Ox (a(x)0ru) "j+1/2(Uj+1 - Uj) - a;-1/2(Uj - Uj-1)
(9)
(4.r)?
where aj+1/2 = a(*j+1/2) and *j+1/2 = z(xj + Ij+1).
(b) (2 points) Show that the matrix resulting from this scheme is diagonally dominant.
(c) (2 points) Show that a maximum principle will apply provided 24t(1 -0)a(T) < (4.r)2.
(d) (5 points) Show that (6) is a consistent approximation of of (a(x)du).

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