Department of Mathematics
MA3AM/MA4AM Asymptotic Methods
Problems 8
1. Consider the problem
8x.. − x. + x = 0 , 0 < 8 << 1
x(0; 8) = α, x(1; 8) = β ; β ≠αe .
Show that it is not possible to match one term outer and inner expansions using Prandtl’s matching condition by assuming that the boundary layer is at t = 0, and using a stretching
, for any a > 0. By assuming that the boundary layer is at t = 1, and stretching , a > 0, show that it is possible to match the one term outer and inner solutions
provided a = 1. Hence show that a uniformly valid expansion throughout [0, 1] is
xcomp (t ; 8) = αet + (β− αe)e−(1−t )8 + O(8) . (see §8.3.1)
2. Show that a uniformly valid expansion in [0, 1] for the solution of
8x.. + x. = 1 , 0 < 8 << 1 ,
x(0; 8) = α, x(1; 8) = β ; α− β+ 1≠ 0 is
xcomp (t ; 8) = t + β− 1+ (α− β+ 1)e−ti8 + O(8)
using (i) Prandtl’smatching condition
(ii) Van Dyke’s matching condition with m = n = 1.
3. Show, with the aid of Van Dyke’s matching condition with m = n = 3, that a uniformly valid expansion in [0, π] for the solution of
8 x x+ = sint , 0 < 8 << 1 , x(0; 8) = α, x(π; 8) = β ; α − β + 2 ≠ 0
is
xcomp (t ; 8) = β− 1− cost − 8sint + 82 (cost + 1) + e−ti8 (α− β+ 2 − 282 ) + O(83 ) .
4. Show that a uniformly valid expansion in [0, 1] for the solution of the problem
εx.. − x. − x = t , 0 < ε << 1 ,
x(0; ε) = α , x(1; ε) = β , β ≠ (α− 1)e−1 , is
xcomp (t ; ε) = (1− t) + (α− 1)e−t (1+ εt)
+e−(1−t ) ε ((β− (α− 1)e−1)t − (α− 1)e−1ε)+ O(ε2 )
using Van Dyke’s matching condition with m = n = 2.
5. Using Prandtl’s matching condition to show that the uniformly valid expansion for the following problem
εx.. + cos(t) x. + sin(2t)x = 0 , 0 < ε << 1 , 0 ≤ t ,
x(0; ε) = α , x(π2 ; ε) = β ; α ≠ βe2 ,
is
xcomp (t ; ε) = βe2cost + (α− βe2 )e− tε + O(ε) .