Analytic approximation for nonlinear problems

The Case of Blasius Similarity Solution

Saleh Tanveer

(Ohio State University)

Collaborator: O. Costin

Background

Analytic solution for nonlinear problems, including ones a rising

in vortex dynamics are very desirable.

Unfortunately, this can be only achieved for only a special c lass

of problems

When small asymptotic parameters are available; we can make

some headway if the limiting problem is solvable, even rigor ously.

N [u; ǫ] = 0 , suppose N [u0; 0] = 0

Define u = u0 + E, then L[E] = −δ − N1[E], where

δ = N [u0; ǫ], L := ∂N

∂u[u0; ǫ], N1[E] := N [u0 + E] − L[E]

Inversion of L and contraction leads to E = O(ǫ) estimates

New non-perturbative methods

Recently, in some problems with no perturbation parameter,

Costin, Huang, Schlag [2012] and Costin, Huang & T [2012]

determined u0 for which N [u0] is small. Proof of error estimates

E = u − u0 found by inversion of L in L[E] = −δ − N1[E]

The idea is simple and natural. Use empirical data to constru ct

orthogonal polynomial approximation in a finite part of doma in;

for the rest use far-field asymptotics; matching gives a comp osite

u0. The method can in principle be applied to any nonlinear

problem that allows convenient theoretical estimates of L−1

We illustrate this for the classical Blasius Similarity sol ution:

f ′′′ + ff ′′ = 0 , with f(0) = 0 = f ′(0) = 0 , limx→∞

f ′(x) = 1 ,

Blasius equation: Background

Blasius (1912) introduced this as similarity solution for P randtl

Boundary layer equation for flow past a flat plate

Much work since then. Topfer (1912) introduced transformat ion

f(x) = a−1/2F (xa−1/2) that makes this equivalent to

F ′′′ + FF ′′ = 0 F (0) = 0 = F ′(0) , F ′′(0) = 1 ,

provided limx→∞ F ′(x) = a > 0 exists. Weyl (1942) proved

existence and uniqueness. Series representation using

hodograph transformations (Calleghari-Friedman, 1968) h ave

been applied, but representation not very convenient for fin ding

f . Liao (’99) obtained an emprically accurate solution throu gh a

formal process. No rigorous control on the error available f or

accurate and efficient approximation.

Definitions

Define P (y) =12∑

j=0

2

5(j + 2)(j + 3)(j + 4)pjy

j (1)

where [p0, ..., p12] are given by

[

−

510

10445149,−

18523

5934,−

42998

441819,

113448

81151,−

65173

22093,

390101

6016,−

2326169

9858,

4134879

7249,−

1928001

1960,

20880183

19117,−

1572554

2161,

1546782

5833,−

1315241

32239

]

(2)

Define t(x) =a

2(x + b/a)2, I0(t) = 1 −

√πteterfc(

√t)

J0(t) = 1 −√2πte2terfc(

√2t),

q0(t) = 2c√te−tI0 + c2e−2t

(

2J0 − I0 − I20

)

, (3)

Main Theorem

Theorem: Let F0 be defined by

F0(x) =

x2

2+ x4P

(

25x)

for x ∈ [0, 52]

ax + b +√

a2t(x)

q0(t(x)) for x > 52

(4)

Then, there is a unique triple (a, b, c) in S close to

(a0, b0, c0) =(

32211946

,−27631765

, 3771613

)

S =

{

(a, b, c) :

√

(a − a0)2 +1

4(b − b0)2 +

1

4(c − c0)2 ≤ 5 × 10−5

}

(5)

with the property that F ≈ F0 in the sense

F (x) = F0(x) + E(x) , (6)

Rigorous Error bounds for approximation

‖E′′‖∞ ≤ 3.5×10−6 , ‖E′‖∞ ≤ 4.5×10−6 , ‖E‖∞ ≤ 4×10−6 on [0,5

2]

(7)

and for x > 52

,

∣

∣

∣E∣

∣

∣≤ 1.69 × 10−5t−2e−3t ,

∣

∣

∣

d

dxE∣

∣

∣≤ 9.20 × 10−5t−3/2e−3t

∣

∣

∣

d2

dx2E∣

∣

∣≤ 5.02 × 10−4t−1e−3t (8)

Remark: These rigorous bounds are likely overestimates since

comparison with numerical solutions give at least ten times

smaller errors.

FN − F0, F ′N − F ′

0, F ′′N − F ′′

0 for x > 52

O O

x

3 4 5 6 7 8 9 10

0

2.#10 - 7

4.#10 - 7

6.#10 - 7

8.#10 - 7

FN − F0, F ′N − F ′

0, F ′′N − F ′′

0 for x ∈ [0, 52]

O O

x

0.5 1 1.5 2 2.5

K5.#10- 7

K4.#10- 7

K3.#10- 7

K2.#10- 7

K1.#10- 7

0

1.#10- 7

Guess F0

Guess F0 in[

0, 52

]

found by using empirical data for F ′′′ = −FF ′′

on [0, 52] and projecting it to a low order Chebyshev polynomial.

Guess for F0 in(

52,∞)

found from asymptotics for large x,

rigorously controlled, that includes upto O[

exp{

−a(

x + ba

)2}]

contribution exactly, for each a > 0, b and c.

Parameters (a, b, c) determined from continuity of solution

representation of F at x = 52

, casting it as a contraction map in R3

Once guess F0 is found so that residual R = F ′′′0 + F0F

′′0 is

uniformly small, rest is a mathematical analysis of E, where

F = F0 + E, helped by the smallness of R, which allows a

contraction mapping argument.

Conclusion

We have shown how an accurate and efficient analytical

approximation of Blasius solution with rigorous error boun ds is

possible.

The method is general and simply relies on finding a candidate F0

so that N [F0] is small.

Candidate F0 on finite domain I = [0, 52] obtained through

orthogonal projection of empirical solution using a small n umber

of Chebyshev polynomials. More accuracy requires more mode s.

Outside I rigorous exponential asymptotics ideas was used to

come up with highly accurate F ≈ F0 approximation

The ideas equally applicable to problems with parameters or to

class of nonlinear PDEs

Full analysis available in

http://www.math.ohio-state.edu/ ∼tanveer