Consider the dimensionless boundary-value problem on with and

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Error, Chance and the Machine, Ch Ordinary differential equations: order, stability and stiffness

Problem

Consider the dimensionless boundary-value problem

y'' = \tfrac32y^2

0 \le x \le 1

with

y(0) = 4

and

y(1) = 1

. Solve it by shooting on the trial slope

s = y'(0)

, with miss

F(s) = y(1; s) - 1

. Integrate with an adaptive Runge-Kutta code (the Dormand-Prince pair, for example) with relative tolerance

10^{-10}

and absolute tolerance

10^{-12}

. ((a)) Verify that

y = 4/(1 + x)^2

satisfies the boundary-value problem, and give the value of

s

it corresponds to. ((b)) Compute

F(s)

for

s = -40, -30, -20, -10

and

-6

, and sketch

F

-40 \le s \le 0

. How many roots does the sketch suggest? ((c)) Run the secant iteration from

(s_0, s_1) = (-10, -6)

. Give the first iterate

s_2

and the root to which the iteration converges. ((d)) Run it again from

(-40, -30)

. Give the root reached and the minimum value of

y

0 \le x \le 1

for the solution it defines. Is this a solution of the same boundary-value problem as in (a)? ((e)) Start the secant iteration from

(-20, -15)

instead. Find the first iterate and explain why this pair is a poor start.

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