5.7 Finding the Green's functions
We can, if we want, write the solution for in other ways that may
be more efficient numerically. The solution was, rewritten more
concisely in terms of the eigenvalues and eigenfunctions:
The first part is due to the inhomogeneous term in the partial differential equation,
the second due to the initial condition
Look at the second term first, let's call it ,
We can substitute in the orthogonality relationship for :
and change the order of the terms to get:
We define a shorthand symbol for the term within the square brackets:
Since this does not depend on what function is, we can evaluate
once and for all. For any , the corresponding temperature
is then simply found by integration:
Function by itself is the temperature if is
a single spike of heat initially located at . Mathematically,
is the solution for if is the ``delta function''
Now look at the first term in , due to , let's call it :
We plug in the orthogonality expression for
We see that
where the function is exactly the same as it was before.
describes the temperature due to a spike
of heat added to the bar at a time and position ;
it is called the Green's function.
The fact that solving the initial value problem (), also
solves the inhomogeneous partial differential equation problem () is known as the Duhamel
principle. The idea behind this principle is that function can be ``sliced up'' as a cake. The contribution of each
of the cake to the solution can be
found as an initial value problem with
initial condition at time .