8. WKBJ Theory Flashcards Preview

MATH5366M Advanced Mathematical Methods > 8. WKBJ Theory > Flashcards

Flashcards in 8. WKBJ Theory Deck (13)
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1
Q

What is WKBJ Theory?

A

-a tool for solving ODEs of the form:

ε² y’’ - q(x) y = 0, 0

2
Q

The Leading Order Solution

Form of y(x)

A
y(x) = A exp[Σ δn(ε) Sn(x)]
= A exp[δ0(ε)S0(x) + δ1(ε)S1(x) + ...]
= AE(x)
-sum from n=0 to n=∞
-with {δn(ε)} and {Sn(x)} to be found and δ0>>δ1>>δ2>>... as ε->∞
3
Q

The Leading Order Solution

Steps

A
  • taking the WKBJ form of y, find y’ and y’’ then sub in to the original equation
  • find dominant balance including q(x) to find δ0 and S0
  • find the next balance to get δ1 and S1
4
Q

The Leading Order Solution

Eikonal Equation

A

-the equation for So’

So’ = ± √[q(x)]

5
Q

The Leading Order Solution

Transport Equation

A

-equation for S1’

S1’ = - So’‘/2So’

6
Q

The Leading Order Solution

Solution

A

y(x) = A E(x)
-where:
E(x) = exp[1/ε (±∫√[q(x)]dx) + 1 (log|q|^(-1/4) + c) + O(ε)]
-integral from x0 to x where x0 is chosen based on the region of interest for the problem

7
Q

The Leading Order Solution

General Solution

A

y(x) ~ A+ / |q(x)|^(1/4) exp[1/ε ∫√[q(x)]dx] + A- / |q(x)|^(1/4) exp[-1/ε ∫√[q(x)]dx]

8
Q

The Leading Order Solution

-when is the two term approximation the exact solution?

A

-when q is constant

9
Q

The Leading Order Solution

q(x)<0

A

-the exp terms correspond to oscillations with length scale ε which are modulated on the length scale of q(x)

10
Q

WKBJ Theory

Boundary Value Problems

A

-write in the form of a WKBJ problem:
ε² y’’ - q(x) y = 0,
-find general solution as normal
-use boundary conditions to find constants

11
Q

WKBJ Theory

Initial Value Problem: y’’ + fy

A
  • follow the same steps as for the BVP
  • when using the initial conditions, for the y’ condition there will be terms that can be ignored due to their relative size, so check order of terms
12
Q

WKBJ Theory

Initial Value Problem: y’‘+fy’+gy

A
  • make the substitution T=εt
  • seek a WKBJ solution x(T)=AE(T)
  • sub in to initial equation
  • consider O(1) terms to find So
  • consider O(ε) terms to find S1
  • sub in to y form for WKBJ to get general solution
  • find unknown constants
  • convert back to t from T
13
Q

WKBJ Theory

Eigenvalue Problems

A
  • an eigenvalue problem:
  • y’’ = λ w(x) y, y(a)=0, y(b)=0
  • with w(x)>0 on [a,b] so λ>0
  • in WKBJ form, ε=1/λ & q(x)=-w(x)<0 and suppose that λ»1
  • sub in to general solution with x0=a
  • solution will contain a constant n with some constraints corresponding to the fact that λ in the original equation can only take on certain values