11. Superposition Principle for Non-Homogeneous Heat Equation with Non-Homogeneous Boundary Conditions:

: ,

Initial condition:

Boundary conditions: (Dirichlet)

(Neumann)

Supplemental problems: a) steady state solution:

:

b) transient solution:

: ,

The first supplemental problem is a BVP for ODE;

The second supplemental problem is an IBVP problem for a homogeneous Heat Equation with

homogeneous boundary conditions (basic case).

Show that is a solution of the non-homogeneous IBVP.

Solve the problem with , and a) ; b) . Sketch the graph.

Solution:

The Heat Equation. 1-D. Finite domain . B.c. are not homogeneous; Equation is non-homogeneous.

Reduce to basic case by subdivision into supplemental problems: steady state and transient solutions.

1) Steady state solution:

2) Transient solution:

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:

3) Then solution is: