PARTIAL DIFFERENTIAL EQUATIONS
EXAM QUESTIONS WITH 100%
CORRECT ANSWERS (VERIFIED
ANSWERS)
What is the derivative of a Fourier series f(x) ? When does this hold?(2) - Answer-f'(x) =
∑[(-n.π.a_n/L)*sin(nπx/L) + (n.π.b_n/L)*cos(nπx/L)]
This only holds when:
(1) f(x) is continuous everywhere
(2) f'(x) satisfies the Dirichlet's conditions
Can we always integrate the Fourier seires? Does it give a Fourier series? - Answer-
The Fourier series may ALWAYS be integrated term by term.
If it contains an 'x' term it is not a Fourier series, but we only need to replace x with the
Fourier series for x
What kind of PDE do we get if:
(1) b^2 > 4ac;
(2) b^2 = 4ac;
(3) b^2 < 4ac ?
Give an example for each one. - Answer-(1) Hyperbolic. eg Wave equation
(2) Parabolic. e.g diffusion equation
(3) Elliptic. e.g Laplace's equation
What is the general form of a second order linear PDE? - Answer-a(x,y)∂^2u/∂x^2 +
b(x,y)∂^2u/∂xy + c(x,y)∂^2u/∂y^2 + d(x,y)∂u/∂x + e(x,y)∂u/∂y + f(x,y).u = 0
What is the wave equation? - Answer-∂^2y/∂t^2 = c^2.∂^2y/∂x^2
where c is a constant
What is the change of variables we make in D'Alembert's solution?
When we substitute it into the wave equation what does it become? - Answer-ξ = x + c.t
η = x -c.t
When we partially differentiate our change of variables twice and sub into the wave
equation it becomes:
∂^2y/∂ξ∂η = 0
, When we have T**/T = c^2.X''/X , what can we conclude about X and T and what does
this give us? - Answer-We can only have a function of t equal to a function of x, if both
are the same constant function λ.
This gives us,
X'' - λX = 0
T** - λc^2.T = 0
What are the normal mode solutions? What do they mean? - Answer-y_n(x,t) =
[C_n.cos(nπct/L) + D_n.sin(nπct/L)].sin(nπx/L)
They're the set of solutions which only satisfy the wave equation and boundary
conditions but not initial conditions.
What is the general solution of the wave equation using the superimposed normal mode
solutions? - Answer-Y_n(x,t) = ∑[C_n.cos(nπct/L) + D_n.sin(nπct/L)].sin(nπx/L)
How do we find the coefficients C_n and D_n in the general solution of the wave
equation? - Answer-Evaluating our general solution at t=0 gives:
p(x) = ∑C_n.sin(nπx/L) (Fourier sine series for C_n)
Evaluating the time derivative of the general solution at t=0 gives:
q(x) = ∑D_n.(nπc/L)sin(nπx/L) (Fourier sine series for D_n)
Using Euler formulae we get:
C_n = (2/L)∫p(x).sin(nπx/L)dx
D_n = (2/nπc)∫q(x).sin(nπx/L)dx
(Where the integration interval is 0 to L)
What seperation of variable in 6 steps, where y(x,t) = X(x).T(t) ? - Answer-(1) Determine
equations for X,T
(2) Use boundary conditions of y in order to obtain boundary conditions of X
(3) Solve eigenvalue problem for X to get λ_n and X_n
(4) Inset λ_n into T and solve to get T_n
(5) The normal modes are y_n = X_n.T_n and the general solution obtained by
superposition y(x,t) = ∑X_n(x).T_n(t)
(6) Use the initial conditions y(x,0), ∂y(x,0)/∂t to determine all undertimed coefficients.
This step involves Fourier series
In Cartesian coordinates, what is the heat equation in:
(1) One dimension;
(2) Two dimensions;
(3) Three dimensions? - Answer-(1) ∂u/∂t = κ^2.(∂^2u/∂x^2)
(2) ∂u/∂t = κ^2.(∂^2u/∂x^2 + ∂^2u/∂y^2)
(3) ∂u/∂t = κ^2.(∂^2u/∂x^2 + ∂^2u/∂y^2 + ∂^2u/∂z^2)
EXAM QUESTIONS WITH 100%
CORRECT ANSWERS (VERIFIED
ANSWERS)
What is the derivative of a Fourier series f(x) ? When does this hold?(2) - Answer-f'(x) =
∑[(-n.π.a_n/L)*sin(nπx/L) + (n.π.b_n/L)*cos(nπx/L)]
This only holds when:
(1) f(x) is continuous everywhere
(2) f'(x) satisfies the Dirichlet's conditions
Can we always integrate the Fourier seires? Does it give a Fourier series? - Answer-
The Fourier series may ALWAYS be integrated term by term.
If it contains an 'x' term it is not a Fourier series, but we only need to replace x with the
Fourier series for x
What kind of PDE do we get if:
(1) b^2 > 4ac;
(2) b^2 = 4ac;
(3) b^2 < 4ac ?
Give an example for each one. - Answer-(1) Hyperbolic. eg Wave equation
(2) Parabolic. e.g diffusion equation
(3) Elliptic. e.g Laplace's equation
What is the general form of a second order linear PDE? - Answer-a(x,y)∂^2u/∂x^2 +
b(x,y)∂^2u/∂xy + c(x,y)∂^2u/∂y^2 + d(x,y)∂u/∂x + e(x,y)∂u/∂y + f(x,y).u = 0
What is the wave equation? - Answer-∂^2y/∂t^2 = c^2.∂^2y/∂x^2
where c is a constant
What is the change of variables we make in D'Alembert's solution?
When we substitute it into the wave equation what does it become? - Answer-ξ = x + c.t
η = x -c.t
When we partially differentiate our change of variables twice and sub into the wave
equation it becomes:
∂^2y/∂ξ∂η = 0
, When we have T**/T = c^2.X''/X , what can we conclude about X and T and what does
this give us? - Answer-We can only have a function of t equal to a function of x, if both
are the same constant function λ.
This gives us,
X'' - λX = 0
T** - λc^2.T = 0
What are the normal mode solutions? What do they mean? - Answer-y_n(x,t) =
[C_n.cos(nπct/L) + D_n.sin(nπct/L)].sin(nπx/L)
They're the set of solutions which only satisfy the wave equation and boundary
conditions but not initial conditions.
What is the general solution of the wave equation using the superimposed normal mode
solutions? - Answer-Y_n(x,t) = ∑[C_n.cos(nπct/L) + D_n.sin(nπct/L)].sin(nπx/L)
How do we find the coefficients C_n and D_n in the general solution of the wave
equation? - Answer-Evaluating our general solution at t=0 gives:
p(x) = ∑C_n.sin(nπx/L) (Fourier sine series for C_n)
Evaluating the time derivative of the general solution at t=0 gives:
q(x) = ∑D_n.(nπc/L)sin(nπx/L) (Fourier sine series for D_n)
Using Euler formulae we get:
C_n = (2/L)∫p(x).sin(nπx/L)dx
D_n = (2/nπc)∫q(x).sin(nπx/L)dx
(Where the integration interval is 0 to L)
What seperation of variable in 6 steps, where y(x,t) = X(x).T(t) ? - Answer-(1) Determine
equations for X,T
(2) Use boundary conditions of y in order to obtain boundary conditions of X
(3) Solve eigenvalue problem for X to get λ_n and X_n
(4) Inset λ_n into T and solve to get T_n
(5) The normal modes are y_n = X_n.T_n and the general solution obtained by
superposition y(x,t) = ∑X_n(x).T_n(t)
(6) Use the initial conditions y(x,0), ∂y(x,0)/∂t to determine all undertimed coefficients.
This step involves Fourier series
In Cartesian coordinates, what is the heat equation in:
(1) One dimension;
(2) Two dimensions;
(3) Three dimensions? - Answer-(1) ∂u/∂t = κ^2.(∂^2u/∂x^2)
(2) ∂u/∂t = κ^2.(∂^2u/∂x^2 + ∂^2u/∂y^2)
(3) ∂u/∂t = κ^2.(∂^2u/∂x^2 + ∂^2u/∂y^2 + ∂^2u/∂z^2)
