Information Technology Reference
In-Depth Information
8.1
Properties of the Solution
The problem (
8.1
)-(
8.3
) is not trivial. We are seeking a function
u
of the space
variable x and time t , i.e.,
u
D
u
.x; t /, such that the first order derivative of
u
with
respect to t equals the second order derivative of
u
with respect to x. In addition,
this function must satisfy the homogeneous boundary condition (
8.2
) and the initial
condition (
8.3
). It is by no means obvious
1
that such a function exists. Furthermore,
if a solution exists, is it unique?
This situation is more the rule than the exception in science. We are faced with
a challenging problem and we do not know if we can solve it. Questions such as
existence and uniqueness of a solution and the solvability
2
of the problem are very
often open. Our ultimate goal is of course to compute the solution, provided that it
exists, of the problem. However, instead of aiming at this goal straight away we will
follow the methods of science: Start by deriving simple properties of the problem,
slowly and stepwise increase our knowledge and finally, if in the position to do so,
determine the problem's solution.
We will now derive a series of properties that a solution of (
8.1
)-(
8.3
)must
satisfy. These properties will do the following:
-
Increase our knowledge in a stepwise manner.
-
Help us determine whether or not this is a good model for the physical process
that it is supposed to describe.
-
Give us a set of properties that the numerical approximations of
u
should satisfy.
This is useful information for designing appropriate numerical schemes for the
diffusion equation and in the debugging process of a computer code developed
to solve it.
-
Provide us with a set of tools suitable for analyzing the diffusion equation. This
methodology will hopefully be applicable for understanding a broader class of
PDEs, that is, a class of problems related to (
8.1
)-(
8.3
) and which may include
equations for which it is impossible to derive an explicit solution formula.
Throughout this section we will assume that
u
is a smooth
3
solution of (
8.1
)-
(
8.3
).
8.1.1
Energy Arguments
Consider the diffusion equation (
8.1
), the boundary condition (
8.2
) and the initial
condition (
8.3
). Recall that this is a prototype of a model for, e.g., the heat evolution
1
Except for well-trained mathematicians.
2
The question of whether or not we are capable of computing the solution (or at least an
approximation of it).
3
The partial derivatives of all orders of
u
with respect to x and t are assumed to be continuous.
