Hi,

i have a problem with the solving of nonlinear hyperbolic first oder system of pde. This look like following

http://farm3.static.flickr.com/2398/...bdc8e280_o.jpg

Can anyone help me to solve it?

Thank you in advance.

Best regards

Minh

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- Jun 19th 2008, 04:56 AMmr.mu_vnHelp me for nonlinear hyperbolic first order system of pde
Hi,

i have a problem with the solving of nonlinear hyperbolic first oder system of pde. This look like following

http://farm3.static.flickr.com/2398/...bdc8e280_o.jpg

Can anyone help me to solve it?

Thank you in advance.

Best regards

Minh - Jul 20th 2008, 08:21 PMRebesques
Well... that's just great (Rock)

This system is not trivial to solve -- at least in general. Start by scaling and then define $\displaystyle \underbar{u}=(p, q)$, to obtain $\displaystyle (P): \ \frac{\partial}{\partial t}\underbar{u}=A\frac{\partial}{\partial x}\underbar{u}$, where $\displaystyle A=\left[ \begin{matrix} -a_2/a_1 & -a_3/a_1 \\ -b_2/b_1 & -b_3/b_1\end{matrix} \right]$. By saying this is hyperbolic, I presume you mean the (constant!) matrix has no real eigenvalues. Then, prescribing differentiable Cauchy data $\displaystyle (C): \ u(0,x)=u_0(x)$ might__not__mean there is a solution at all! In fact, $\displaystyle u_0$ has to be at least analytic, and I have no idea on obtaining an actual solution. (Worried)

For the sake of argument, suppose...

- The matrix of the system has real and distinct eigenvalues. Then the Cauchy problem (P)&(C) has always a solution of the form

$\displaystyle \underbar{u}(t,x)=(p(t,x),q(t,x))=\left( \int\exp(c_1ts+c_2xs)u_0(s)ds,\int\exp(d_1ts+d_2xs )u_0(s)ds\right)$

where we have to pick the constants involved so that the integrals converge and $\displaystyle \underbar{u}$ is truly a solution, the latter part being algebraic manipulations.

- The matrix of the system is self adjoint. Then, for the Cauchy data (C) being a tempered distribution, there is a unique solution for (P)&(C). In fact, a stronger result on regularity is true: This solution must necessarily be $\displaystyle C^{\infty}!$

Hope this helps. - The matrix of the system has real and distinct eigenvalues. Then the Cauchy problem (P)&(C) has always a solution of the form