#### Milestone Year

### 1977

#### Are Partial Differential Equations of Various Types that Different?

The classification of partial differential equations into elliptic
(corresponding to stationary processes), hyperbolic (describing dynamics
and waves), and parabolic (diffusion processes) is well established.
However, in 1977, Weizmann
mathematicians discovered that it is possible to study properties of
equation of one type by applying known results concerning equations of a
different type, using so-called transmutation formulas (those were
well-known, but not applied substantially).

In this manner it was possible to derive the asymptotic behavior, away from
the diagonal, of fundamental solutions of diffusion equation, obtaining
delicate exponential error estimates directly from the well-known form of
wave kernels.

As shown by a WIS scientist and his collaborators,
it is also possible to go the other way and obtain representations for the
wave kernels of degenerate operators such as the Grushin operator or the
Heisenberg Laplacian from the diffusion kernels associated with such
operators. It is also possible to break a wave operator into simpler
components and use easily obtained information on the smaller elements so
as to build up the structure of the more complicated kernels.

The philosophy of using well-known or easily obtained knowledge of
operators of one class to solve seemingly different problems for operators
of an altogether different class was also used to gain understanding of
singular perturbations and boundary layers. Last but not least, analytic
continuation leads occasionally from one type to the other - this is an old
idea, but its power for studying degenerate wave kernels has been
manifested only in recent years by a WIS scientist and his collaborator.