Computer Algebra Recipes

278 CHAPTER 6. LINEAR PDE MODELS. PART 2
To see Russell's animated di®usion equation solution, you will have to execute
the program and use the animation tool bar.
PROBLEMS:
Problem 6-23: Numerical Instability
In the text recipe, con¯rm that the solution becomes numerically unstable for
r>0:5. Numerical instability is signaled by the appearance of increasingly
wild oscillations in the solution as time increases.
Problem 6-24: Comparison with exact solution
Explore the change in the percentage error in the numerical mesh values at the
endoftheruncomparedwiththeexactvaluesasM is increased.
Problem 6-25: A di®erent pro¯le
Modify the text recipe to produce an animated numerical solution for the initial
temperature pro¯le T (x; 0) = 25 sin(¼x). Compare the numerical solution in
the center of the rod with the exact solution as a function of time. At what
time is the temperature in the middle equal to one-quarter of the initial value?
6.3.2 Enjoy the Klein{Gordon Vibes
The world is never quiet, even its silence eternally resounds with the
same notes, in vibrations which escape our ears.
Albert Camus, French-Algerian philosopher, writer (1913{1960)
After tackling the Excalibur heat-di®usion example, Russell decides to numerically
investigate the small transverse oscillations of a light stretched horizontal
string embedded in a stretched vertical elastic membrane. In the absence of
the membrane, the instantaneous displacement Ã(x; t) of the string satis¯es the
one-dimensional wave equation. The e®ect of the membrane is to add an additional
Hooke's law restoring force, proportional to Ã, on the string, which tends
to speed up the vibrations. The relevant transverse wave equation, called the
Klein{Gordon equation (KGE), then is
@2Ã @x2 ¡ @2Ã = aÃ; (6.21)
@t2 where a is the elastic coe±cient of the membrane and the wave speed has been
set equal to unity.
If the string is of unit length and ¯xed at both ends, the boundary conditions
are Ã(0;t)=Ã(1;t)=0fort ¸ 0. Supposing that the string has the initial
transverse pro¯le f ´ Ã(x; 0) = x (1 ¡ x) 5 and velocity g ´ _ Ã(x; 0) = x 3 (1¡ x),
Russell wishes to numerically solve the KGE and animate the oscillations.
At the internal mesh points, he uses the standard CDAs for the second
derivatives and approximates the inhomogeneous term with aÃi;j. Thenumer-