5. Initial Value Problems Flashcards
First Order IVPs
General Form
x’ = f(x(t),t)
-with x(0)=xo
Second Order IVPs
General Form
x’’ = f(x’(t),x(t),t)
-with x(0)=xo and x’(0)=xo’
First Order IVPs - Standard Expansion
Steps
1) suppose that: x ~ xo(t) + ε*x1(t) + ε²x2(t)+... as ε->0 2) sub into equation 3) consider O(1) terms, then O(ε), then O(ε²) etc.
Second Order IVPs
Most Fundamental Form
x’’ + ωo²x = 0
-with initial conditions
Second Order IVPs
Fundamental Form - General Solution
x = a cos(ωot) + b sin(ωot) OR x = Ae^(iωot) + Be^(-iωot) = Ae^(iωot) + A* e^(-iωot) -since x(t) needs to be real -to fit the initial conditions, use a and b i the first equation or Re(A) and Im(A) in the second
Second Order IVPs
Fundamental Form - Period and Frequency
- solution is oscillatory and periodic in time, once cycle corresponds to ωot increasing by 2π
- so the period is, T=2π/ωo
- and ωo is the frequency
Second Order IVPs
Forced Systems
x’’ + ωo²x = α cosωt + β sinωt
-where ωo is the natural frequency and ω is the forcing frequency
Second Order IVPs
Forced Systems - Complimentary Function
xcf = a cosωot + b sinωot
-determine a and b with initial conditions
Second Order IVPs
Forced Systems - Particular Integral
xpi = C cosωot + D sinωot -where: C = α / (ωo²-ω²) -and D = β / (ωo² - ω²)
Second Order IVPs
Forced Systems - Resonant Response
-there is a resonant response when ω->ωo
Second Order IVPs
Forced Systems - Forcing at Resonant Frequency
-governing equation:
x’’ + ωo²x = α cosωot + β sinωot
=>
x(t) = a cosωot + b sinωot - β/2ωo t cos(ωot) + α/2ωo t sin(ωot)
Motion of a Non-Linear Pendulum
Equation
d²θ/dq² + g/L sinθ = 0 -with: θ(0)=θo dθ(0)/dq = 0 -where θo>0
Motion of a Non-Linear Pendulum
Non-Dimensionalising
-introduce a non-dimensional time, t, with t∝q where q has dimensions of seconds:
t = √[g/L] q
Motion of a Non-Linear Pendulum
Conservation of Energy
- the non-linear pendulum system conserves energy
- prove this by multiplying by dθ/dt
Motion of a Non-Linear Pendulum
Solution
-make the substitution x(t) = θ/θo
-apply a Taylor expansion to sine
=>
x’’ + x = εx³/6 - ε²x^5/120 + O(ε³)
-gather terms of like order and apply the boundary conditions
Motion of a Non-Linear Pendulum
Time Period at Leading Order
-at leading order:
x(t) ~ cost
=>
T = 2π
Motion of a Non-Linear Pendulum
Time Period at O(ε)
-we anticipate:
T = 2π + 4α1ε + O(ε²)
-we know that x=0 when t=T/4=π/2 + α1ε + O(ε²)
-sub in to the O(ε) expression for x(t)
Motion of a Non-Linear Pendulum
Secular Term
- the order ε expansion for x(t) contains a secular term ∝ tsint which grown in time
- the expansion is valid when t=O(1) but the secular term makes it invalid when εt=Os(1)
- to improve this, alternative techniques are needed
Method of Multiple Scales
Aim
-to develop an asymptotic expansion that works for both t=O(1) and t=Os(1/ε), as ε->0
Method of Multiple Scales
Time
-define fast and slow timescales:
τ = t and T = εt
-so if t=O(1) then τ=O(1) but T is small
-if t=Os(1/ε) then τ is large but T is O(1)
-as the name suggests, multiple of these timescales can be defined
Method of Multiple Scales
Steps
1) define time scales e.g. τ=t and T=εt
2) we then seek solutions x(τ ,T) rather than x(t), so rewrite the governing equation in terms of τ and T
3) assume that x ~ xo + εx1 + ε²x2 +…
4) consider O(1) terms from the governing equation, then O(ε) terms etc.
5) make choices about the unknown constants which will eliminate the secular terms
6) convert back to original timescale, t
Linstedt-Poincare Technique
Aim
-we use LPT when the initial value problem with ε≠0 (the perturbed problem) has a frequency shift from the unperturbed problem
Linstedt-Poincare Technique
Strained Coordinate
-replace t by a strained coordinate:
τ = (1 + εω1 + ε²ω2 + …)t
-where we choose ω1,ω2,… to remove secularities
Linstedt-Poincare Technique
Steps
1) introduce strained coordinate τ = (1 + εω1 + ε²ω2 + …)t
2) rewrite governing equation and boundary conditions in terms of τ instead of t
3) let x ~ xo + ε x1 + ε² x2 + ….
4) consider O(1) terms, choose ω’s to remove forcing at resonant frequency (i.e. to avoid secularities)
5) then do the same for O(ε) terms and so on
Renormalisation
Aim
-similar technique to LPT
-starts from the results of a completely standard asymptotic expansion using:
x(t) ~ xo(t) + εx1(t) + ε²x2(t) + …
-the results of which is known to be secular
Renormalisation
Straining Functions
-introduce:
t = τ + ε f1(τ) + ε² f2(τ) + …
-where the straining functions fj(τ) will be chosen to eliminate secularities
Renormalisation
Steps
1) introduce strained functions, t = τ + ε f1(τ) + ε² f2(τ) + …
2) take the standard expansion with secularities that need removing and rewrite in terms of τ
3) choose fj(τ) to eliminate the secularities
4) truncate series at desired order
