MHB Solving Transport Eq. for Level Curves: x=X(t)

[simo1]

I have this equation
u_x(x,t) + c(x)u_x(x,t) = 0 x>0

I want to obtain information on how the initial input u_o(x)=u(x,o) would deform when the sound speed is not constant. c(x) is the sound speedi wanted to start this by finding a DE for the level curves x=X(t) so that i can solve in terms the initial point x₀=X(0)(greater than or equal to) 0

but how can i solve for these level curves

 

[chisigma]

I have this equation...

$\displaystyle u_{t}\ (x,t) + c(x)\ u_{x}\ (x,t) = 0,\ x>0$

I want to obtain information on how the initial input u_o(x)=u(x,o) would deform when the sound speed is not constant. c(x) is the sound speedi wanted to start this by finding a DE for the level curves x=X(t) so that i can solve in terms the initial point x₀=X(0)(greater than or equal to) 0

but how can i solve for these level curves

The standard approach for a PDE like...

$\displaystyle u_{t} + c(x,t)\ u_{x} = 0\ (1)$

... is to find curves along which u is constant. If we introduce a new variable r for which is $t=t(r)$ and $x=x(r)$ , then for chaining rule is...

$\displaystyle \frac{d u}{d r} = u_{t}\ \frac {d t}{d r} + u_{x}\ \frac{d x}{d r}\ (2)$

... and combining (1) and (2) we arrive to the ODE pair...

$\displaystyle \frac{d t}{d r} = 1,\ \frac{d x}{d r}= c(x,r)\ (3)$

In Your case c(*) is function of the x alone so that is...

$\displaystyle \int \frac{d x}{c(x)} = r + \gamma\ (4)$

... where $\gamma$ is an arbitrary constant...

Kind regards

$\chi$ $\sigma$

 

[simo1]

The standard approach for a PDE like...

$\displaystyle u_{t} + c(x,t)\ u_{x} = 0\ (1)$

... is to find curves along which u is constant. If we introduce a new variable r for which is $t=t(r)$ and $x=x(r)$ , then for chaining rule is...

$\displaystyle \frac{d u}{d r} = u_{t}\ \frac {d t}{d r} + u_{x}\ \frac{d x}{d r}\ (2)$

... and combining (1) and (2) we arrive to the ODE pair...

$\displaystyle \frac{d t}{d r} = 1,\ \frac{d x}{d r}= c(x,r)\ (3)$

In Your case c(*) is function of the x alone so that is...

$\displaystyle \int \frac{d x}{c(x)} = r + \gamma\ (4)$

... where $\gamma$ is an arbitrary constant...Kind regards

$\chi$ $\sigma$

I solved the equation and I had x(t) = sinh^-1(sinhx₀e^c(0)t) is it possible to draw the characteristic curve of this function

 

[chisigma]

I solved the equation and I had x(t) = sinh^-1(sinhx₀e^c(0)t)...

Ehm!... what's the reason why You don't write simply $\displaystyle x= x_{0}\ e^{\ c(0)\ t}$?...

Kind regards

$\chi$ $\sigma$