2 Better Modeling
We mentioned earlier that analytical solutions are exact solutions since they
do not involve any approximations. But this statement needs some clarification.
Distinction should be made between an actual real-world problem and
the mathematical model that is an idealized representation of it. The solutions
we get are the solutions of mathematical models, and the degree of applicability
of these solutions to the actual physical problems depends on the accuracy
of the model. An “approximate” solution of a realistic model of a
physical problem is usually more accurate than the “exact” solution of a crude
mathematical model (Fig. 5–3).
When attempting to get an analytical solution to a physical problem, there
is always the tendency to oversimplify the problem to make the mathematical
model sufficiently simple to warrant an analytical solution. Therefore, it is
common practice to ignore any effects that cause mathematical complications
such as nonlinearities in the differential equation or the boundary conditions.
So it comes as no surprise that nonlinearities such as temperature dependence
of thermal conductivity and the radiation boundary conditions are seldom considered
in analytical solutions. A mathematical model intended for a numerical
solution is likely to represent the actual problem better. Therefore, the
numerical solution of engineering problems has now become the norm rather
than the exception even when analytical solutions are available.