Computer Algebra Recipes

2.3. NUMERICAL SOLUTION OF ODES 95
closed loop. The incorrect behavior is due to cumulative error arising from use
of the Euler approximation (which has low accuracy) and the ¯nite step size.
Keeping in mind the danger of round-o® error for a ¯xed number of digits, the
growth of the spiral can in principle be reduced by reducing h. But in practice,
the CPU time goes up so rapidly that the Euler method is not used for serious
numerical calculations. In the next recipe, the classical fourth-order Runge{
Kutta method is introduced, a much more accurate numerical scheme than the
Euler method for a given step size.
Finally, the CPU time for the forward Euler dial-up method is determined,
> CPUtime2:=(time()-begin2)*seconds;
CPUtime2 := 0:330 seconds
and is actually slightly longer than for the ¯rst-principles calculation.
PROBLEMS:
Problem 2-21: Van der Pol equation
By setting _x = y, the second-order Van der Pol (VdP) equation
Äx ¡ ² (1 ¡ x 2 )_x + x =0
may be written as a coupled system of two ¯rst-order ODEs. Choosing ² =5
and x(0) = _x(0) = 0:1, solve the VdP system for h =0:01 and t =0to15using
the ¯rst principles Euler method. Rotate your plot to produce a phase-plane
portrait solution and compare the result with what would be obtained using
the dial-up Euler option of the dsolve command.
Problem 2-22: White dwarf equation
In his theory of white dwarf stars, Chandrasekhar [Cha39] introduced the nonlinear
equation
x (d 2 y=dx 2 )+2(dy=dx)+x (y 2 ¡ C) 3=2 =0;
with the boundary conditions y(0) = 1 and y 0 (0) = 0. Write the second-order
equation as two ¯rst-order equations and solve the system using the ¯rst principle's
Euler method. Take h =0:01 and 10-digit accuracy, and numerically
compute y(x) over the range 0 · x · 4withC =0:1 and plot the result. (Hint:
Start at x =0:01 to avoid any problem at the origin.)
Problem 2-23: Baleen whales
May [May80] has discussed the solution of the following normalized equation
describing the population of sexually mature adult baleen whales:
_x(t) =¡ax(t)+bx(t ¡ T )(1 ¡ (x(t ¡ T )) N ):
Here x(t) is the normalized population number at time t, a and b are the
mortality and reproduction coe±cients, T is the time lag necessary to achieve
sexual maturity, and N is a positive parameter. If the term 1 ¡ (x(t ¡ T )) N is
negative, then this term is to be set equal to zero. Taking a =1,b =2,T =2,
step size h =0:01, and 4000 time steps, use the ¯rst principles Euler method to
solve numerically for x(t ¡ T )versusx(t) andforx(t) for(a)N =3:0, and (b)
N =3:5. Plot your results. For (a) you should observe a period-one solution,