Methods of Mathematical Modelling 3

what are the 3 types of second order PDEs

elliptical, hyperbolic, parabolic

what is the general form of a second order PDE

let u=u(x,t)

A δ_xxu + B δ_xtu + C δ_ttu + G (u, δ_xu, δ_tu) = 0

what is the discriminant of a second order PDE

D= B² - 4AC

how to determine the type of a second order PDE

• elliptical if D>0

• hyperbolic if D<0

• parabolic if D=0

what are two types of domains we can study PDEs on

ℝ or on bounded domains

what do we call problems studied on ℝ

initial value problems or Cauchy problems

what do we call problems studied on bounded domains

boundary value problems, or BVPs

what are the four key types of boundary conditions

• Dirichlet

• Neumann

• Robin or mixed

• Periodic 

what is a Dirichlet boundary condition

when the value of the solution to the PDE is fixed at the boundary (or at parts of it)

what is a Neumann boundary condition

when the function does not change across the boundary

what is a Robin/ mixed boundary condition

when the change of the function in the normal direction is proportional to the value of the function

why is a mixed boundary condition called “mixed”

mixed boundary condition is when
δₙu(x,t) = α(u(x,t) −u_R(x)) ∀x ∈ δΩ or parts of the boundary (α>0 and u_R given)

as α→∞, this tends towards Dirichlet, and as α→0, this tends towards Neumann. hence, it is a mix of Dirichlet and Neumann

what is a periodic boundary condition

used for example on a torus, when the edges of the domain are “glued together”. if Ω = [0,1], this would imply

u(0,t) = u(1,t) for all t ≥ 0

three conditions for well-posedness

existence, uniqueness, stability

how to define stability

the solutions depends continuously on the given data

general form of the transport equation

u_t(x,t) + v(x,t)u_x(x,t) = 0

what extra information do you need to get a unique solution

• a function u(x₀,0)= Φ(x₀) ie initial conditions

how to solve TE using method of characteristics, given u(x₀,0)= Φ(x₀)

• set ξ’(t) = v(x,t), ξ(0)=x₀

• solve for ξ(t)

• set x=ξ(t) and solve for x₀ in terms of x

• as u doesn’t change along the characteristic curve,  u(x,t)=u(ξ(t),t)=u(ξ(0),0)= u(x₀,0)=Φ(x₀)

• sub x₀ for the expression in terms of x

• done!

general form of wave equation

u_tt(x,t) = c²u_xx(x,t)

what extra information do you need to get a unique solution (on the infinite line)

• a function u(x,0)= Φ(x)

• a function u_t(x,0)= V(x)

how to find general solution of wave equation (without any other conditions)

• set the characteristic coordinates ξ= x+ct, η= x-ct, and have v(ξ, η) = u(x,t)

• sub these new coordinates into the PDE, giving v_ξη(ξ, η)=0

• integrate with respect to η then ξ to obtain u(x,t)= v(ξ, η) = f(ξ) + g(η) = f(x+ct) + g(x-ct)

how to get unique solution for wave equation as given above

• note that u(x,0)= Φ(x) = f(x) + g(x)
  and u_t(x,0)=V(x)= cf’(x) - cg’(x)

• solve (like sim eqs) for f’(x) and g’(x) then integrate for f(x) and g(x)

• note that f(x) + g(x) = Φ(x) gives that the sum of the integration constants for f and g is 0

• then u(x,t) = f(x+ct) + g(x-ct)

• this is d’Alembert’s formula

state d’Alembert’s formula

u(x,t)= ½ (Φ(x+ct) + Φ(x-ct)) 1/2c _x-ct∫^x+ct V(r) dr

what extra information do you need to have a well-posed question on an interval

what is Leibniz’ rule

• if f(x) = _a(x)∫^b(x) g(x,y) dy

• a,b must be ctsly diffable, and g and g_y must be cts

• then f’(x)= g(x,b(x)) b′(x) − g(x,a(x)) a′(x) + _a(x)∫^b(x)g_x(x,y) dy