On a quasi-linear parabolic equation occurring in aerodynamics

Subordinators: Examples and Applications

“…A most important feature of (6.1) is that it is one of the very few non-linear equations that can be solved explicitly. Specifically, Hopf [75] and Cole [37] observed that applying the transformation γ = 2ε log g to the potential function γ given by ∂ x γ = −u ε , yields the heat equation ∂ t g = ε∂ 2 xx g. This enables one to determine g and hence u ε .…”

On simple moving grid methods for one-dimensional evolutionary partial differential equations

“…Then the solution slowly decays to zero while the layer, for the values of t of interest here, remains at the same position. The exact solution is available in the form of an infinite series (Cole [5]) whose evaluation, for our value r, = 10-3, is not practical. Therefore, in this case, we have assessed the accuracy of our numerical solutions with the help of a reference solution computed on a very fine grid.…”

Envelope soliton resonances and broer-kaup-type non-madelung fluids

“…This nonlinear equation is C-integrable in the sense of Calogero [9], and using (1) as a complex version of the Cole-Hopf transformation [10], [11], we reduce it to the linear Schrödinger equation…”