Kronecker Delta and Things (Term 2)

Question 1

What is the integral from 0 to L of sin(nπx/L)*sin(mπx/L) dx equal to?

Answer 1

L/2 * δnm, where δnm is the kronecker delta, which equals 1 if n=m and 0 otherwise

Question 2

What can we do with the kronecker delta form of the integral?

Answer 2

Substitute it into the integral equation from before (orthogonality thing) and get rid of the sum (change n for m

Question 3

What is the equation for Cn?

Answer 3

Cn = 2/L * integral between 0 and L of f(x)*sin(mπx/L) dx

Question 4

How do you work out the final equation for u(x,t)?

Answer 4

-Split the integral into 2 part (0 to L/2 and rest) and substitute in f(x) for these domains -Do the integrals to find the final value of Cn -Substitute Cn into general solution from before

Question 5

What does u(x,t) equal finally?

Answer 5

u(x,t) = sum of 2/L * (4*E*L*sin(nπ/2))/(π^2*n^2) * sin(nπx/L)*cos(ncπt/L) (for n odd)

Question 6

If you strike a string with hammer with head of width w, what is du(x,0)/dt

Answer 6

-0 for x < x0-w/2 -v for x0-w/2 < x < x0+w/2 -0 for x > x0 + w/2

Question 7

What do we do to the Cn constant in the hammer problem?

Answer 7

Set them to 0, otherwise the displacement at t=0 will not vanish, so only have Dn constants this time.

Question 8

What do we do with the initial velocity condition du(x,0)/dt = v0*δ(x-x0)?

Answer 8

-Take time derivative of u(x,t) with only Dn in and then substitute t=o in -Make this answer equal to the initial condition: v0*δ(x-x0)

Question 9

What do we do with the initial condition after we have subbed it in?

Answer 9

-Multiply both sides by sin(mπx/L) and integrate again, and rearrange to get the orthogonality relationship.

Question 10

What is the orthogonality relationship and what does it equal?

Answer 10

integral from 0 to L of sin(nπx/L)*sin(mπx/L) = L/2*δnm

Question 11

What do we do after using the orthogonality relationship in the hammer problem?

Answer 11

Find Dm by taking the sum out and the kronecker delta out (n=m), and rearrange for Dm.

Question 12

How do you do separation of variables for the diffusion equation?

Answer 12

u(x,t) = X(x)T(t), and sub this in. Then move around until same variables on same side, and set equal to lambda, a constant.

Question 13

What happens when lambda = 0 for the diffusion equation?

Answer 13

-d^2X/dx^2 = 0, so X(x) = A0x+ -dT/dt = 0, so T(t) = G0 -Therefore u(x,t) = (A0x+B0)*G0 = A0’x+B0’

Question 14

For the diffusion solution when lambda = 0, when are the equilibrium and steady-state solutions?

Answer 14

-Equilibrium when u(x,t) = B0 -Steady-state when u(x,t) = A0x+B0 (A0 not equal to zero)

Question 15

How can you maintain the steady state? What is the flux equal to?

Answer 15

Feed particles in at the left and pull them out on the right with exactly the right rate. Flux j=-D du(x,t)/dx

Question 16

How do you work out the equation for λ?

Answer 16

• Relabel m to n in the equation for Dm, and sub this new Dn into u(x,t)
• Set this new equation equal to v0*sin(πx/L)
• Only D1 is not equal to zero, so rearrange this and find u(x,t) = v0L/cπ * sin(πx/L)*sin(cπt/L)
• Only have one pure frequency ω = cπ/L, so λ = 2L

Question 17

What happens when lambda > 0 (lambda = k^2) for the diffusion equation?

Answer 17

-dT/dt = D*k^2*T(t) -Solution is T(t) = G*e^(D*k^2*t) -Not physical as t goes to 0, so exclude this choice

Question 18

What happens when lambda < 0 (lambda = -k^2)?

Answer 18

-Both sides equal -k^2 -T(t) = G*e^(D*k^2*t) again -X(x) = Ak*cos(kx)+Bk*sin(kx)

Question 19

What is the final solution for the diffusion equation?

Answer 19

u(x,t) = sum over k of (Ak*cos(kx)+Bk*sin(kx))*exp(-D*k^2*t) + A0+B0x

Question 20

What do we do if we specify boundary conditions of u(x=0,t)=u(x=L,t)=0 for the diffusion equation general solution? (lasers which bleach dye molecules at each end)

Answer 20

-Substitute in x=0 and x=L and find the values of Ak and k from this. -Substitute thee new values in

Question 21

What if there are no lasers? What do we do to work out the solution?

Answer 21

-No flux at ends (j=0) -Take x-derivative of solution and sub in x=0 and x=L -These must both equal zero so find Bk and k values from this -Sub back in

Question 22

What is A0 equal to in the general solution?

Answer 22

A0 = N/L (equilibrium concentration)