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METHODS OF APPLIED MATHEMATICS COMPREHENSIVE

EXAMINATION AUGUST 2012

Work on as many of the following problems as possible. Turn in all your work.

(1) Apply iterated homogenization (average fastest scale first and repeat), on thefollowing initial value problem to derive a leading order effective equation as 0+, assuming the functions K(x), M(x) are periodic sharing the sameperiod, and positive:

ut =

K(x/) + M(x/2)

uxx

u(x, t = 0) = u0(x)

(2) Consider the following quarter-plane boundary value problem for the diffu-sion equation with constant diffusivity ,

ct = cxx , c(x, 0) = 0 , c(0, t) = t

(a) Solve for all time t > 0 in the half line x 0.(b) Discuss the limit x 0+ (verify the boundary condition is satisfied).

(3) Consider the function F(s; x) dependent on complex argument s and the realparameter x > 0,

F(s; x) =ex

s

s2.

(a) Classify its singularities and define domains where it is single valued.(b) Choose the domains in such a way that the integral depending on the

parameter c > 0,

I(x, t) =1

2i

c+i

ciestF(s; x) ds ,

is well defined, and in particular discuss its convergence.(c) Evaluate the integral along appropriate contours in the s-complex plane

and discuss the first two terms of the asymptotic expansion of I(x, t) ast +.

(d) Discuss the limit t 0+.

(4) Find a two term asymptotic expansion for all the (complex) roots of the fol-

lowing as 0,(x 1)2 + x = 0 .

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(5) Compute the leading order term in an asymptotic expansion of the solutiony(x) of the following third order differential equation

y = xy + 2y

as x . Classify the different forms of the solutions with respect to theirasymptotic behavior, and discuss their linear (in)dependency.

(6) (a) Compute the Green function G(x, y) for the operator d2

dx2on the interval

[0, 1], that is

2G

x2= (x y) , x [0, 1], y [0, 1] ,

( Dirac-delta) subject to the boundary conditions

G(0, y) = 0 =G

x(1, y).

(b) For a solution (x) of the (trivial) differential equation

d2

dx2= 0 ,

derive Greens identity

(x) =

(y)

G

y(x, y) (y) G(x, y)

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where the square brackets at the right hand side denote the differencef(x, 1) f(x, 0) for any function f(x, y) of the arguments x, y.

(c) Hence compute the solution of the two-point boundary value problem onthe interval [0, 1],

= 0, (0) = 1 , (1) = 0 .

How does the solution obtained with Greens identity compare with adirect solution of this two point boundary value problem?

(7) (a) Explain the difference between pointwise convergence and asymptoticconvergence. Illustrate with the particular example of power series.

(b) Are asymptotic series unique? Explain

(c) Does an asympotic relation carry over to its integration? Discuss in thecontext of the following example: does

f(x) g(x) as x imply, for some x0,

x

x0

f(y) dy

x

x0

g(y) dy as x ?

Make sure to clarify your assumptions on f and x for a definitive, butnon trivial, statement.