This discussion focuses on solving quasi-linear partial differential equations (PDEs) with non-zero initial conditions, specifically using the characteristic method. The participants explore the feasibility of applying alternative initial conditions, such as ##u(x,y=c) = f(x)##, instead of the traditional ##u(x,0) = f(x)##. It is established that one can redefine the independent variables to facilitate the solution, using transformations like (x, v = y - c) or (x, c - y) to adapt to different initial conditions.
PREREQUISITESMathematicians, physicists, and engineers working with partial differential equations, particularly those interested in advanced methods for solving initial value problems.
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?