Linear Methods of Applied Mathematics

Evans M. Harrell II and James V. Herod*

*(c) Copyright 1994,1995,1996 by Evans M. Harrell II and James V. Herod. All rights reserved.

At Georgia Tech, students took 50 minutes (calculators allowed but no notes). Most students did reasonably well, but few finished early.

1. Find the normal modes (product solutions) for the damped wave equation

u_tt + u_t = u_xx

for 0 < x < 1, with boundary conditions:

u_x(t,0) = 0

u(t,1) = 0

Make sure to give the specific dependence on x and on t.

THE NORMAL MODES ARE:_____________________________________________

FORMULA OR KEY FACT (FOR POSSIBLE PARTIAL CREDIT):_________

________________________________________________________________

2. Solve the wave equation

u_tt = u_xx

for 0 < x < 1, with boundary conditions

u_x(t,0) = u_x(t,1) = 0

and initial conditions

u(0,x) = sin²( \pi x)

ut(0,x) = 0

THE GENERAL SOLUTION APPROPRIATE TO THIS PDE WITH THE BC IS:

____________________________________________________________________

SPECIFIC SOLUTION:

u(t,x) = ______________________________________________________________

FORMULA OR KEY FACT (FOR POSSIBLE PARTIAL CREDIT):_________

________________________________________________________________

> int(sin(a*x) * (sin(b*x))^2, x);

                cos(a x)       cos((a + 2 b) x)       cos((a - 2 b) x)
          - 1/2 -------- + 1/4 ---------------- + 1/4 ----------------
                    a               a + 2 b                a - 2 b

> int(cos(a*x) * (sin(b*x))^2, x);

               sin(a x)       sin((a - 2 b) x)       sin((a + 2 b) x)
           1/2 -------- - 1/4 ---------------- - 1/4 ----------------
                   a               a - 2 b                a + 2 b

> int(sin(2*b*x) * (sin(b*x))^2, x);

                             cos(2 b x)        cos(4 b x)
                       - 1/4 ---------- + 1/16 ----------
                                  b                 b

> int(cos(2*b*x) * (sin(b*x))^2, x);

                        sin(2 b x)        sin(4 b x)
                    1/4 ---------- - 1/16 ---------- - 1/4 x
                             b                 b



> ODE := diff(y(t), t$2) + diff(y(t), t) = - mu * y(t);

                        /   2      \
                        |  d       |   /  d      \
                 ODE := |----- y(t)| + |---- y(t)| = - mu y(t)
                        |   2      |   \ dt      /
                        \ dt       /

> dsolve( ODE, y(t) );

y(t) =

                                 1/2                                    1/2
    _C1 exp(1/2 (- 1 + (1 - 4 mu)   ) t) + _C2 exp(- 1/2 (1 + (1 - 4 mu))   t)

y(t) =
                                    1/2                           1/2
   exp(- t/2) (   C1 cos(((4 mu - 1)   /2) t) + C2 sin(((4 mu - 1)   /2) t)   )

See the solutions.

Back to Chapter VII

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