Topic 3 - linear partial differential equations

Question 1

linear PDE

Answer 1

solutions obey principle of superposition ie y1,y2 solutions implies y = Ay1 + By2 solution therefore solution space is vector space ( ∞ dim)

Question 2

second order linear PDE

Answer 2

if u = u(x,t)
Auxx + 2Buxt + Cutt + Dut + Eux + Fu = 0
A, B, C, D, E, F can be any functions of x we shall take generally as constants
PDEs are classified into 3 types - sign of det(M)
M = (A B
B C)
detM = AC - B^2

Question 3

3 types of PDE

Answer 3

• det M < 0 hyperbolic
  eg ytt = c^2 uxx
• det M = 0 parabolic
  eg ut = k^2 uxx
• det M > 0 elliptic
  eg uxx + uyy = 0

Question 4

boundary conditions for ODEs

Answer 4

suppose we know all at t = 0
want to know solution to nth order linear ODE at t>0
y = A1y1 + A2y2 + … + Anyn
specify n pieces of info to fix Ai’s and get particular solution
eg y(x), dy/dy (0)… d^(n-1)y/dt^(n-1) (0)

Question 5

boundary conditions for PDEs

Answer 5

as solution space is ∞ dim need ∞pieces of info to fix Ai’s - generally in form of functions specified at t = 0 eg u(x,0) = R(x), du/dt (x,0) = S(x)
can have boundary at t= 0 but also in space need to specify boundary here as well

Question 6

boundary condition examples PDE

Answer 6

• wave eq on whole line t ≥0
  need to specify position u(x,0) = R(x) and velocity ut(x,0) = S(x)
• heat eq on whole line t ≥ 0
  specify u(x,0) = R(x) initial temp
• finite string from a to b
  specify position u(x,0) = R(x) and velocity ut(x,0) = S(x) - need boundary at a and b
• finite metal bar
  specify u(x,0) = R(x) initial temp - need boundary at a and b
• laplaces x from a to b and y from c to d
  need 1 boundary on each edge (Dirichlet or Neumann)

Question 7

boundary conditions in space

Answer 7

can have a fixed end u(a,t) = 0 - dirichlet
or vibrating vertically ux(a,t) = 0 - neumann

Question 8

method of separation of variables

Answer 8

u(x,t) = ∑ Ai ui(x,t) look for ui in nice form
ui(x,t) = X(x)T(t)
- sub into differential eq
- sub in position boundary conditions and solve
- sub in time boundary conditions

Question 9

Bn and An in Ancos(t) +Bn sin(t) term

Answer 9

Bn control initial velocity
An control initial position

Question 10

finding An method 1 - fourier series

Answer 10

R(x) = ∑An sin(nx)
sum is odd/ even periodic extension of the R(x)
An = 1/pi ∫R(x)sin(nx) dx from pi to -pi

Question 11

finding An method 2 - orthogonality

Answer 11

R(x) = ∑An sin(nx)
sin(nx) is a solution to x’’ = λx so sin(x) is a eigenfunction of d^2/dx^2 with eigenalue λ
if d^2/dx^2 is self adjoint then sin(nx) orthogonal basis so easy to find components
chose and inner product (f,g) = ∫fg dx from pi to 0 - show is self adjoint - then sin(nx) orthogonal so (sin(nx),sin(mx)) = 0 for n≠m
so Am = (R(x),sin(mx))/(sin(mx), sin(mx))