Foreword by Prof EE Rosinger
Partial
Differential Equations, or in short PDE-s, and especially the far more
difficult nonlinear ones, are ever since Newton, that is, for more than
three centuries, the precise mathematical models of the basic laws of
physics, chemistry, and lately, a good deal of biology as well.
Consequently, modern technology cannot be conceived without PDE-s. As
it happens, it is far more difficult to solve PDE-s than to establish,
or write them. Quite often, physicists, for instance, come up with a
PDE which formulates yet another fundamental law of nature, and for
their effort, they get a Nobel Prize. The far harder task of solving
those PDE-s, however, falls upon mathematicians. And due to some
strange reasons, mathematicians cannot get Nobel Prizes!
The
modern theory of solving PDE-s started in the 1930s, and has ever since
then been based on the branch of mathematics called functional
analysis. This method has managed to solve many of the simpler linear
PDE-s, and even some of the far more difficult nonlinear PDE-s.
However, its technical complications are considerable, and for solving
further PDE-s these complications grow very fast.
In
view of that, in the early 1990's, a radically new, much simpler, and
far more effective method was introduced, based on the order completion
of spaces of smooth functions defined on Euclidean spaces.
Our department has ever since been at the forefront of introducing and
then developing this new method based on order completion.
To
give a simple argument of the power of that new method, one can mention
that vast classes of linear and nonlinear PDE-s can be solved and can
be given rather classical solutions, classes of PDE-s which the usual
functional analytic method cannot even touch.