PDE's

Question 1

Represent the PDE d_tu - (d_xxu + d_yyu) + u³ - u = 0 as a function F

Answer 1

F: R⁴ to R, F(u, d_tu, d_xxu, d_yyu)(x,y,t) = 0

Question 2

Define linear and the order of a PDE

Answer 2

A PDE is linear if the associated function F is linear. The oreder of the PDE is the order of the highest derivative appearing in the PDE

Question 3

When is a PDE well-posed

Answer 3

When it has a unique solution which continuously depends on data:

well-posedness = existence + uniqueness + stability

Question 4

State the transport equation

Answer 4

d_tu(x,t) + v(x,t)d_x(u,t) = 0 for a function u where v(x,t) is a given function and we seek a solution in a time interval t in [0,T] with T in (0, inf) and x in R

Question 5

State the characteristic method for solving the transport equation

Answer 5

Question 6

With initial conditions u_0: R to R s.t u(x₀,0) = u₀(x₀), how do we solve the transport equation

Answer 6

Question 7

How do we solve the transport equation now equal to a source term s(x,t, u(x,t))

Answer 7

Question 8

State the wave equation

Answer 8

d_ttu - c²d_xxu = 0

Question 9

Derive the general solution to the wave equation

Answer 9

Question 10

State the initial value problem for the wave equation in 1D

Answer 10

Question 11

state D’Alembert’s formula

Answer 11

Question 12

State Leibniz’s rule

Answer 12

Question 13

Answer 13

Question 14

Theorem: Show that the solution to the initial value problem wave equation is unique and given by D’Alembert’s formula

Answer 14

Question 15

If we have the wave equation on a finite domain ie x in (0,L), how do we rescale the problem

Answer 15

Question 16

State the homogeneous Dirichlet boundary conditions and the general solution

Answer 16

Question 17

State the homogenous Neumann boundary conditions for the wave equation and the general solution

Answer 17

Question 18

What are trigonometric polynomials of the degree 2n, give the complex and real versions

Answer 18

Question 19

Answer 19

Question 20

Given a fourier series exists, what are it’s coefficients?

Answer 20

Question 21

State Bessel’s inequality

Answer 21

Question 22

State Parseval’s equality and the conditions for it to hold

Answer 22

Question 23

State the Riemann-Lebesgue lemma

Answer 23

Question 24

When is a function odd and even

Answer 24

Question 25

Let phi: R to R be 2pi periodic. If phi is even what is it's fourier series? Likewise for odd

Answer 25

Question 26

Let f, f_n map [-pi, pi] to C. Define f_n converging to f i) pointwise ii) uniformly iii) mean square sense

Answer 26

Question 27

Define the Dirichlet kernel and state its maximum

Answer 27

Question 28

What is S_n(phi)(x) in terms of the Dirichlet kernel

Answer 28

Question 29

Answer 29

Question 30

Answer 30

Question 31

State the heat equation

Answer 31

d_tu(x,t) = kd_xxu(x,t)

Question 32

State the initial boundary value problem for the heat equation in 1D with homogeneous dirichlet boundary conditions

Answer 32

Question 33

Define the space-time rectangle and the parabolic boundary

Answer 33

Question 34

State the maximum principle for the heat equation

Answer 34

Question 35

Prove the maximum principle for the heat equation

Answer 35

Question 36

State the initial boundary value problem for the heat equation in 1D with neumann boundary conditions

Answer 36

Question 37

Show uniqueness for the heat equation using energy methods

Answer 37

Question 38

How do we solve this problem?

Answer 38

Question 39

State and prove Duhamel's principle

Answer 39

Question 40

Answer 40

Question 41

What is the Cauchy problem for the heat equation

Answer 41

Question 42

Whats the fundamental solution for the Cauchy problem

Answer 42

Question 43

When is a PDE i) elliptic ii) hyperbolic iii) parabolic

Answer 43

Question 44

Define the Laplacian

Answer 44

Question 45

State poissons equation

Answer 45

Question 46

What is the laplacian in polar co-ordinates

Answer 46

Question 47

State the wedge problem and its solution

Answer 47

Question 48

When is u harmonic

Answer 48

When -(laplacian)u = 0

Question 49

What is the principle of causality

Answer 49

the wave equation has finite propogation speed

Question 50

Which equation smooths information

Answer 50

the heat equation, singularities persist in the wave equation

Question 51

If f is in C²(Rⁿ) and has a compact support, what equation u(x) solves -(laplacian)u = f on Rⁿ

Answer 51