9. 1. Solve the Dirichlet problem for the Heat Equation:

: ,

Initial condition:

Boundary conditions: , (Dirichlet)

(Dirichlet)

2. Sketch the graph of solution for L=3 and a=0.1 and initial conditions:

a)

b)

c) cc)

3. Observe the Maximum principle for the Dirichlet problem for the Heat Equation

Solution:

The Heat Equation. 1-D. Finite domain . Basic case: 2 b.c. are homogeneous, equatuion is homogeneous.

1) Separation of variables:

Sturm-Liouville Problem (equation with two homogeneous b.c.):

Second equation:

The solution:

Write the solution in the form of the infinite series:

Apply the initial condition to determine coefficients :

Treat it as the sine Fourier series expansion of the function , then coefficients are:

Then solution is:

2)

a) Case

b) Case

c-corrected) Case

then

for (orthogonality)

And solution becomes: