2.5 Example: Different Boundary Conditions 167
Solve
Solve the eigenvalue problem forφ. That is, find the values ofλ^2 for which
the eigenvalue problem has nonzero solutions. Label the eigenfunctions and
eigenvaluesφn(x)andλ^2 n.
Solve the ordinary differential equation for the time factors,Tn(t).
Combine and Satisfy Remaining Condition
Form the general solution of the homogeneous problem as a sum of constant
multiples of the product solutions:
w(x,t)=∑
cnφn(x)Tn(t).Choose thecnso that the initial condition is satisfied. This may or may not
bearoutineFourierseriesproblem.Ifnot,anorthogonalityprinciplemust
be used to determine the coefficients. (We shall see the theory in Sections 7
and 8.)
Check
Form the solution of the original problem
u(x,t)=v(x)+w(x,t)and check that all conditions are satisfied.
EXERCISES
See Common Eigenvalue Problems on the CD.
- Find the steady-state solution of the problem stated in Eqs. (1)–(4).
- Determine whether 0 is an eigenvalue of the eigenvalue problem stated
in Eqs. (11) and (12). That is, takeλ=0 and see whether the solution is
nonzero. - Solve the problem stated in Eqs. (1)–(4), takingf(x)=Tx/a.
- Solve the problem stated in Eqs. (1)–(4) if
f(x)={
T 0 , 0 <x<a/2,
T 1 , a/ 2 <x<a.- Solve the nonhomogeneous problem
∂^2 u
∂x^2 =
1
k∂u
∂t−T
a^2 ,^0 <x<a,^0 <t,