The discussion revolves around the application of the characteristic method to solve quasi-linear partial differential equations (PDEs) with non-standard initial conditions, specifically focusing on conditions defined at ##u(x,y=c) = f(x)## rather than the typical ##u(x,0) = f(x)##. The scope includes theoretical exploration and mathematical reasoning related to PDEs.
Participants have not reached a consensus on the best approach to take with non-zero initial conditions, and multiple competing views remain regarding the methods proposed.
The discussion includes assumptions about the applicability of the characteristic method and the transformations suggested, which may depend on specific properties of the PDEs in question.
I think that sounds reasonable but I want to make sure.BvU said:What do you think of shifting the ##y##-axis and solving two different initial value problems (one backwards equation) ?
Yes - you can use (x, v = y - c) as your independent variable instead of (x,y). (Or use (x, c - y) if you need to work back to y = 0.)LieToMe said:The way I was taught to solve many quasi-linear PDEs was by harnessing the initial condition in the characteristic method at ##u(x,0) = f(x)##. What if however I need use alternative initial conditions such as ##u(x,y=c) = f(x)## for some constant ##c##? Can the solution be propagated the same way?