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 [itex](x, v = y - c)[/itex] as your independent variable instead of [itex](x,y)[/itex]. (Or use [itex](x, c - y)[/itex] if you need to work back to [itex]y = 0[/itex].)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?
A quasilinear equation is a type of partial differential equation in which the dependent variable appears in a linear fashion, while the independent variables appear in a nonlinear fashion. A non-zero initial condition means that the equation is not equal to zero at the starting point, but rather has a specific value.
The main difference is the presence of a non-zero initial condition. This means that the equation has a specific value at the starting point, rather than starting at zero. This can significantly change the behavior and solutions of the equation.
Quasilinear equations with non-zero initial condition are commonly used in physics and engineering to model various physical phenomena such as heat diffusion, fluid flow, and electrical circuits. They are also used in economics and finance to model changing market conditions and stock prices.
The specific method for solving a quasilinear equation with non-zero initial condition depends on the specific equation and initial condition. In general, numerical methods such as finite difference or finite element methods are used to approximate the solutions. Analytical solutions may also be possible for simpler equations.
One major challenge is the nonlinearity of the equations, which can make it difficult to find exact solutions. Another challenge is the dependence of the solutions on the initial condition, which can lead to complex and unpredictable behavior. Additionally, numerical methods can be computationally intensive and may require a lot of computational resources.