Partial Differential Equations

Define a PDE (and the independent variables)

A PDE for u(x,t) is an equation of the form F(x, t, u, u_x, u_t, u_xx, …) = 0.

The variables x and t are called the independent variables.

Define the order of a PDE

A PDE is nth order if it only involves the function and its derivatives up to and including derivatives of order n.

When is an operator, L, linear?

If L(λu+µv) = λL(u) +µL(v) for any two functions u and v and arbitrary constant λ and µ.

A PDE is linear if…

… it can be written as L(u) = f where L is a linear differential operator and f is known.

(Otherwise we say it is nonlinear.)

A linear PDE L(u) = f is homogeneous if…

… L(u) = 0,

i.e. there is no term that is independent of u and its derivatives

What is always a solution for a linear homogeneous equation?

0

What is the Principle of Superposition?

If u and v are two solutions to a homogeneous linear PDE, then λu+µv is also a solution for constants λ and µ.

What three ingredients make up a boundary value problem (BVP) for a PDE?

1. A PDE

2. A set in which it is posed called the domain

3. Some conditions that the solution must satisfy on the boundary of the domain

If a time variable, t, is involved than what may a BVP also be called?

An initial value problem (IVP)

How many boundary conditions are needed for nth order ODE?

n

For space independent variables across an interval, where do we normally impose the conditions?

On the endpoints, using the value of u or u' usually.

What are the three required conditions to call a BVP ‘well-posed’?

1. There is at least one solution satisfying the PDE and boundary conditions

2. There is at most one solution

3. The unique solution depends continuously on the data given

Considering au_xx + bu_xt + cu_tt = 0 for a general second-order PDE, if b² > 4ac, the PDE is called…

… hyperbolic

e.g. the wave equation

Considering au_xx + bu_xt + cu_tt = 0 for a general second-order PDE, if b² < 4ac, the PDE is called…

… elliptic

e.g. Laplace’s equation

Considering au_xx + bu_xt + cu_tt = 0 for a general second-order PDE, if b² = 4ac, the PDE is called…

… parabolic

e.g. the heat equation

If the coefficients a, b and c are not constant…

… the characterisation of the PDE can change in different regions of the PDE

By a linear transformation of the independent variables, if b² > 4ac, the equation can be reduced to…

… u_ζζ - u_ηη + … = 0,

where … denotes the terms of order 1 or 0 and ζ and η are now our independent variables.

By a linear transformation of the independent variables, if b² < 4ac, the equation can be reduced to…

… u_ζζ + u_ηη + … = 0,

where … denotes the terms of order 1 or 0 and ζ and η are now our independent variables.

By a linear transformation of the independent variables, if b² < 4ac, the equation can be reduced to…

… u_ζζ + … = 0,

where … denotes the terms of order 1 or 0 and ζ and η are now our independent variables.

Define a periodic function

A function f(t) is periodic with period T, if f(t) = f(t + T) for all t.

What is the fundamental period?

The smallest positive T such that f(t) = f(t+T) for all t holds.

The sum of any two T-periodic functions is…

… T-periodic.

The product of any two T-periodic functions is…

… T-periodic.

If f is T-periodic then, for any a∈R, ∫_a^a+T f(t) dt = ?

∫₀^T f(t) dt

We say that S(t) is a trigonometric polynomial or Fourier series with coefficients a_k, k = 0,1,2…, and b_k, k = 1,2,…, if S(t) = ?

a₀/2 + ∑^∞_k=1 (a_kcos(2πkt/T) + b_ksin(2πkt/T))

S(t) is…

… periodic with period T

The series can also be written as S(t) = ?

∑_k=0^∞ c_ke^i2πkt/T with c_k the complex coefficients

sin(nx)cos(mx) = ?

1/2[sin((n-m)x) + sin((n+m)x)]

cos(nx)cos(mx) = ?

1/2[cos((n-m)x) + cos((n+m)x)]

sin(nx)sin(mx) = ?

1/2[cos((n-m)x) - cos((n+m)x)]

If n,m∈Z, with n,m≥0, ∫₀^2π sin(nx)cos(mx) dx = ?

0

If n,m∈Z, with n,m≥0, ∫₀^2π cos(nx)cos(mx) dx = ?

• 0 if n≠m

• π if n=m≠0

• 2π if n=m=0

If n,m∈Z, with n,m≥0, ∫₀^2π sin(nx)sin(mx) dx = ?

• 0 if n≠m

• π if n=m≠0

• 0 if n=m=0

What is the relationship between orthonormal basis and Fourier?

The functions (1/2π, sin(t)/ √π, cos(t)/√π, sin(2t)/√π) is an orthonormal basis for the infinite-dimensional vector space consisting of 2π-periodic functions who Fourier series converge with the inner product of functions f and g given by ∫₀^2π fg dt

If f(t) = a₀/2 + ∑^∞_k=1 (a_kcos(kt) + b_ksin(kt)) then what are the Fourier coefficients?

a₀ = 1/π ∫₀^2π f(t) dt, a_k = 1/π ∫₀^2π f(t)cos(kt) dt, b_k = 1/π ∫₀^2π f(t)sin(kt) dt

Let f be a periodic function with period T. The Fourier coefficients of f are defined by…

… a_k = 2/T ∫₀^T f(t)cos(2πkt/T) dt, b_k = 2/T ∫₀^T f(t)sin(2πkt/T) dt

The partial Fourier sum is…

… S_N(t) = a₀/2 + ∑^N_k=1 (a_kcos(2πkt/T) + b_ksin(2πkt/T))

Graphs of the Fourier series reveal…

… Gibbs phenomenon

What is the Riemann-Lebesgue Lemma?

If f is continuous, then a_k → 0 and b_k → 0 as k → infty.

The Fourier coefficients of a 2π-periodic function f can also be written as…

… a_k = 1/π ∫_-π^π f(t)cos(kt) dt, b_k = 1/π ∫_-π^π f(t)sin(kt) dt

f(t) is even if…

… f(t) = f(-t)

f(t) is odd if…

… if f(t) = -f(-t)

The product of two even functions is…

… even

The product of two odd functions is…

… even

The product of an even and an odd function is…

… odd.

If f is odd, then ∫_-L^L f(t) dt = ?

0 for all L > 0

If f is even, then ∫_-L^L f(t) dt = ?

2∫₀^L f(t) dt for all L > 0

If f(t) is even, then…

… b_k = 0 for all k.

If f(t) is odd, then…

… a_k = 0 for all k.

Given a function f(t) defined by 0≤t≤2π, its periodic extension is defined by…

… f(t+2kπ) = f(t) for k∈Z\{0}, t∈[0,2π)

Given f(t) defined for t∈[0,π], its even periodic extension is defined by…

… f(t) = f(-t) for t∈(-π, 0) and f((t) + 2kπ) = f(t) for k∈Z\{0}, t∈(-π,π]

Given f(t) defined for t∈[0,π], its odd periodic extension is defined by…

… f(t) = -f(-t) for t∈(-π, 0) and f((t) + 2kπ) = f(t) for k∈Z\{0}, t∈(-π,π]

Given an odd function f(t) on (-π, π), the Fourier sine series of f is…

… S(t) = ∑_k=1^∞ b_ksin(kt) where b_k = 2/π ∫₀^π f(t)sin(kt) dt

Given an even function f(t) on (-π, π), the Fourier sine series of f is…

… S(t) = a₀/2 + ∑_k=1^∞ a_kcos(kt) where a_k = 2/π ∫₀^π f(t)cos(kt) dt

The smoother a function is…

… the faster its Fourier coefficients decay and the faster the Fourier series converges

A periodic function f(t) is piecewise differentiable if there exist a finite number of points t_i, i = 1,…,n such that…

• f(t) is differentiable on (0,t), (t_i, t_i+1) for i = 1,…,n-1 and (t_n, T)

• at t_i, the left and right limits of f(t) exist, at 0 the right limit of f(t) exists and at T the left limit of f(t) exists

If f(t) is piecewise differentiable then:

If f is differentiable at t, S_N(t) → ? as N → ∞

f(t)

(pointwise convergence)

If f(t) is piecewise differentiable then:

At a point t_i, then S_N(t_i) → ? as N → ∞

1/2[f(t_i^-) + f(t_i⁺)]

(converges to average value)

If f(t) is piecewise differentiable then:

At the ‘edge’ of the periodic window, S_N(0) → ? as N → ∞

1/2[f(0⁺) + f(T^-)]

We say S_N(t) converges in L² to f(t) if…

… ∫₀^T (S_N(t) - f(t))² dt → 0 as N → ∞

(also called mean square convergence)

If f is piecewise differentiable then S_N → ? in L²

f

The Fourier series converges in L² for…

… any f that is square integrable

Parseval’s Theorem:

If f(t) is a piecewise differentiable function on (0,T), with Fourier coefficients a_k, b_k and Fourier sum S(t), then…

… ∫₀^T S²(t) dt = ∫₀^T f²(t) dt and T/2(a₀²/2 + ∑_k=1^∞ (a_k² + b_k²)) = ∫₀^T f²(t) dt

Abstract Parseval’s Theorem

Suppose that {e_n}_n=0^∞ is an othonormal basis for a real inner-product space.
Given a vector f, let f_m := f _° e_m and S_N(f) := ∑^N_m=1 f_me_m.
Suppose that f is such that S_N(f) → f as N → ∞.

Then…

… ||S_N(f)||² → ||f||² as N → ∞ and ||S_N(f)||² = ∑_m=1^N f_m²

What is the Wave Equation?

u_tt - c²u_xx = 0, where c is the wave speed

What is the physical derivation of wave equation for motion of a string?

Let u(x,t) be the displacement of a string at time t and point x where the displacement is measured from where the string at rest.

We derive from here using Newton’s second law.

Define the Dirichlet problem for the wave equation

Given c > 0 and u₀(x), v₀(x) for 0<x<L find u(x,t) satisfying:

• the wave equation, u_tt - c²u_xx = 0 in 0<x<L, 0<t<∞

• the initial conditions, u(x,0) = u₀(x) and u_t(x,0) = v₀(x) for 0<x<L

• the Dirichlet boundary conditions, u(0,t) = 0 and u(L,t) = 0 for 0<t<∞

(Dirichlet corresponds to the ends being clamped)

What’s the difference for the Neumann problem?

The Neumann boundary conditions: u_x(0,t) = 0 and u_x(L,t) = 0 for 0<t<∞

Neumann corresponds to the string being allowed to slide freely on a movable loop

What is the frequency of the wave?

The number of cycles per 2π time intervals, ω

What is the wave number?

The density of waves, k

What must the separation constant λ be?

Less than zero

Given an odd function f(x) on (-L, L), the Fourier sine series of f is given by…

… S(x) = ∑_n=1^∞ b_nsin(nπx/L) where b_n = 2/L ∫₀^L f(x)sin(nπx/L) dx

If u₀(x) has a jump discontinuity then…

… the Fourier series expression for u(x,0) will converge to the average of the jump

What is the solution of the Dirichlet problem?

u(x,t) = ∑_n=1^∞ sin(nπx/L)(A_ncos(nπct/L) + B_nsin(nπct/L)) if A_n = 2/L ∫₀^L u₀(x)sin(nπx/L) dx and B_n = 2/L L/nπc ∫₀^L v₀(x)sin(nπx/L) dx

What is the separation of variables algorithm?

0. Reduce the BVP to one with zero boundary conditions (heat only)

1. Assume that u(x,t) = X(x)T(t), substitute into the PDE and obtain separated ODEs for X(x) and T(t)

2. Solve the ODEs for X(x) and T(t)

3. Impose the boundary conditions to obtain a family of solutions depending on n∈Z

4. Use the principle of superposition to find a solution oas a sum over n∈Z

5. FIx the unknown coefficients in the sum by applying the initial conditions

What is d’Alamberts formula for the solution?

Given u₀(x) for 0<x<L, let u^~₀ be the odd periodic extension of u₀(x) from (0,L) to R.

The solution of the Dirichlet problem for the wave equation with v₀ = 0 is then given by u(x,t) = 1/2(u^~₀(x-ct) + u^∼₀(x+ct))

What is the Heat Equation?

u_t - κu_xx = 0, where κ>0 is the thermal diffusivity or diffusion coefficient

What is the physical derivation of the heat equation?

Consider a one-dimension bar. Let u(x,t) be the temperature at a point x and time t.

The second law of thermal dynamics says that heat flows from hot areas to cold temperature.

Define the inhomogeneous Dirichlet problem for the heat equation

Given κ>0, u₀(x) for 0<x<L and T₀,T₁>0, find u(x,t) satisfying:

• the heat equation, u_t - κu_xx = 0 for 0<x<L, 0<t<∞

• the intial condition, u(x,0) = u₀(x) for 0<x<L

• the Dirichlet boundary conditions, u(0,t) = T₀ and u(L,t) = T₁

(corresponds fixing the temperature at the ends of the bar)

What’s the difference for the inhomogeneous Neumann problem?

Neumann boundary conditions: u_x(0,t) = F₀ and u_x(L,t) = F₁ for 0<t<∞

(corresponding to specificying the flux of heat through the two ends of the bar is F₀ and F₁)

Define the homogeneous Dirichlet problem for the heat equation

Given κ>0 and u^₀(x) for 0<x<L, find u^(x,t) satisfying:

• the heat equation: u^_t - κu^_xx = 0 for 0<x<L, 0<t<∞

• the initial condition: u^(x,0) = u^₀(x) for 0<x<L

• the Dirichlet boundary conditions: u^(0,t) = 0 and u^(L,t) = 0 for 0<t<∞

If u(x,t) is the solution of the inhomogeneous Dirchlet problem and u^(x,t) is the solution to the homogeneous Dirichlet problem, then what is the relationship between u(x,t) and u^(x,t)?

u(x,t) = U(x) + u^(x,t) where U(x) := T₀ + (T₁ - T₀)x/L and u^₀(x) = u₀(x) - U(x)

What is the solution of the Dirichlet problem by separation of variables?

u^(x,t) = ∑_n=1^∞ A_nexp(-κn²π²t/L)sin(nπx/L) if A_n := 2/L ∫₀^L u^₀(x)sin(nπx/L) dx

What is Laplace’s Equation?

u_xx + u_yy = 0 (2D since 1D is trivial)

What is the physical relevance of Laplace’s equation?

• Maxwell’s equations of electromagnetism reduce to Laplace’s equation

• If a fluid is incompressible and irrotational then the fluid velocity equals

Define the Dirichlet problem for Laplace’s equation

Given a domain Ω⊂R² with piecewise smooth boundary ∂Ω and a function f(x,y) defined for (x,y) ∈ ∂Ω, find u(x,y) in Ω such that:

• ∇²u(x,y) = 0 for (x,y) ∈ Ω

• u(x,y) = f(x,y) for (x,y) ∈ ∂Ω

What is the Neumann problem?

u(x,y) = f(x,y) is replaced by u_n(x,y) = g(x,y) for (x,y) ∈ ∂Ω where u_n = n_°∇u (n is the OPUNV)

Define the Dirichlet problem in a square

• u_xx + u_yy = 0 for (x,y) ∈(0,L)x(0,L)

• u(0,y) = u(L,y) = u(x,L) = 0 and u(x,0) = f(x)

What is the solution of the Dirichlet problem for the square?

u(x,y) = ∑_n=1^∞ B_nsin(nπx/L)sinh(nπ(L-y)/L) if B_n = 2/Lsinh(nπ) ∫₀^L f(x)sin(nπx/L) dx

How do you obtain the solution to Laplace’s equation in a square with non-zero Dirichlet conditions on each of the four sides?

Superimposing four solutions of the type found above with u(x,0) = f₁(x), u(0,y) = f₂(y), u(x,L) = f₃(x), u(L,y) = f₄(y)

Define the Dirichlet problem in a disc

• Disc: Ω := {(x,y): x²+y² < R²}

• Use polar coordinates to make applying the boundary conditions easy

What is the solution of the Dirichlet problem in a disc?

u(r, θ) = a₀/2 + ∑_n=1^∞ (r/R)ⁿ(a_ncos(nθ) + b_nsin(nθ)), where a_n = 1/π ∫₀^2π f(θ)cos(nθ) dθ and b_n = 1/π ∫₀^2π f(θ)sin(nθ) dθ

What changes when finding the solution outside a circular disc?

The consideration of the bounds of R (it will kill A not B)

What is the Poisson formula for a disc?

u(r,θ) 1/2π ∫₀^2π (R² - r²) / (R² + r² - 2rRcos(θ-θ')) f(θ') dθ'