4. Boundary Value Problems Flashcards
General Linear 2nd-Order ODEs
A(x;ε)y’’ + B(x;ε)y’ + C(x;ε)y = D(x;ε), x1
Regular Perturbation Problems
Steps
-write:
y = y0 + εy1 + ε²y2 + …
-sub in to the differential equation and boundary conditions
-gather together terms O(ε^0), use boundary conditions to find constants in expression for y0
-repeat for terms O(ε) to find y1
-repeat for terms in successively higher powers of ε as required
-sub the y0,y1,… into the original equation for y
Singular Perturbation Problems
Approach
-write:
y = y0 + εy1 + ε²y2 + …
-sub in to the differential equation
-consider terms O(ε^0), it is clear that only one of the two boundary conditions can be satisfied at a time
Singular Perturbation Problems
Outer Solution
-the solution which satisfies one boundary condition and covers most of the range over which y is defined is called the outer solution
Singular Perturbation Problems
Inner Solution
- we suppose that there is a boundary layer that is thin, of width δ(ε)«1 as ε->0
- introduce a stretched coordinate s=x/δ(ε) so s goes from 0 to Os(1) over the boundary layer
Prandtl’s Matching Theorem
-the simplest but least powerful technique for matching
-for ε=0 :
lim yout = lim y in
-first limit taken as x->0, second limit taken as s->∞
-this formula allows you to calculate the unknown constants in yin
-then a composite solution is constructed, ycomp = yin + yout + value of the limit above
Van Dyke’s Matching Rule
Definition
- the M-term inner expansion of the N-term outer expansion matches with the N-term outer expansion of the M-term inner expansion
- a composite solution is then constructed, ycomp = yin + yout + expansion above
Van Dyke’s Matching Rule
M-Term Inner Expansion of N-Term Outer
1) calculate the N-term expansion of the outer solution in terms of outer variable x
2) rewrite in terms of the inner variable s
3) expand in gauge functions of the inner expansion
4) truncate to M terms
5) re-express in terms of the outer variable
Van Dyke’s Matching Rule
N-Term Outer Expansion of M-Term Inner
1) calculate the M-term expansion of the inner solution in terms of stretched coordinate s
2) rewrite in terms of outer variable x
3) expand in gauge functions
4) truncate to N-terms
Boundary Layer Location
Prandtl Matching
-consider:
εy’’ + a(x)y’ + b(x)y = c(x)
-with y(xl)=yl and y(xr)=yr over interval (xl,xr)
-to match an inner (boundary layer) solution at some location x=xc (either xl or xr) to the outer solution we use Prandtl matching:
lim x->xc yout = lim s->∞ yin
-since lim x->xc yout is typically finite we need the limit of yin yo also be finite for matching
Boundary Layer Location
xl
- if a(xl)>0 the inner solution can match on to an outer solution via Prandtl matching since its limit as s->∞ is finite
- however, if a(xl)<0, this inner solution blows up as s->∞ and it cannot be matched onto an outer solution so a boundary layer cannot exist at x=xl
Boundary Layer Location
xr
- if a(xR)<0 the inner solution can match onto an outer solution as s->∞ (exiting the boundary layer)
- if a(xr)>0, the inner solution blows up as s->∞ and a boundary layer at x=xr cannot exist
Solution by Inspection
a(x) one sign throughout [xl,xr]
- if a(x) remains of one sign throughout [xl,xr], things are straightforward; there is a BL at xl if a(x)>0 and a BL at xr if a(x)<0
1) identify if BL is at xl or xr
2) find the outer solution by setting ε=0
3) find constants in outer solution by satisfying the BC at the other end of the range
4) sketch outer solution with boundary layer to force meeting of the other BC
Solution by Inspection
a(x) changes sign at xi∈(xl,xr)
- maybe boundary layers at both ends, neither ends but an internal BL at xi or neither ends but a near-corner at xi
1) check where boundary layers could be
2) calculate yout
3) sketch
Boundary Layer Thickness
-make sure you don’t assume boundary layer thickness to early
-calculate stretched coordinate as:
s = x-xl / δ(ε)
OR
s = xr-x / δ(ε)
-find δ(ε) such that the order of all terms in the differential equation is the same
