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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.