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Differential Equation : Space Factor, Time Factor, Eigenvalues and Sturm-Liouville

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7. Consider the differential equation
ut =1/2 uxx + ux for 0 <x < pi, t > 0
with boundary conditions
u(0,t) = u(pi,t) = 0.
(a) Separate variables and write the ordinary differential equations that the space factor X(x) and the time factor T(t) must satisfy.
(b) Show that 0 is not an eigenvalue of the Sturm-Liouville problem for X.
(c) Show that for any integer n>1,
Xn(x) = e^-x sinnx
is an eigenfunction of the Sturm-Liouville problem for X and determine the corresponding eigenvalue.
(d) Assuming that these are all the eigenvalues, write down in series form the general solution of the boundary value problem above assuming a general initial condition u(x, 0) = f(x).

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A differential equation is investigated with regard to space factor, time factor, eigenvalues and sturm-liouville problem. The solution is detailed and well presented.

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Solution:

We have to solve the PDE:
( 1)
with initial and boundary conditions:
( 2)
This kind of problem is known as "mixed problem", because it contains initial conditions as well as boundary ones.
The easiest way to solve the above equation is to apply the Fourier method, that means separation of variables:
( 3)
By doing partial derivatives and introducing in (1), we will have:
( 4)
We've chosen a negative constant in order to provide a periodic solution for bilocal problem (the boundary condition). For a positive constant, we will see ...

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