Multidimensional exact solutions to a quasilinear parabolic equation with anisotropic heat conductivity.
The four natural boundary problems for the weighted form Laplacians acting on polynomial differential forms in the -dimensional Euclidean ball are solved explicitly. Moreover, an algebraic algorithm for generating a solution from the boundary data is given in each case.
This talk gives a brief review of some recent progress in the asymptotic analysis of short pulse solutions of nonlinear hyperbolic partial differential equations. This includes descriptions on the scales of geometric optics and diffractive geometric optics, and also studies of special situations where pulses passing through focal points can be analysed.
The existence of a solution of the two - dimensional heat conduction equation in a semi-infinite strip, under mixed boundary condition, is discussed.