Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

What is the necessary conditions we need to transform second order PDE to a system of ODE? e.g. If I have $$a^2*u_{tt}- u_{xx}+ u*u_{x}=0 $$ what conditions I needed to transform it to a system of ODE?

share|cite|improve this question

If we use a travelling wave solution we have $z=x-ct$ $$ c^{2}a^{2}\frac{\mathrm{d}^{2}\Phi}{\mathrm{dz}^{2}}-\frac{\mathrm{d}^{2}\Phi}{\mathrm{dz}^{2}}+\Phi\frac{\mathrm{d}\Phi}{\mathrm{dz}}=0 $$ or $$ \lambda\frac{\mathrm{d}^{2}\Phi}{\mathrm{dz}^{2}}+\Phi\frac{\mathrm{d}\Phi}{\mathrm{dz}}=0 $$ where $\lambda=a^{2}c^{2}-1$.

we can change of variables to reduce to a first order ode

$$ \lambda p\frac{\mathrm{d}p}{\mathrm{d\Phi}}+\Phi p=0\\ p=\frac{\mathrm{d}\Phi}{\mathrm{dz}} $$ This is separable and can be solved to yield $$ \frac{\mathrm{d}\Phi}{\mathrm{dz}} = -\frac{\Phi^{2}}{2\lambda}+D. $$ Then solving this equation for $\Phi$ involves computing the following integral $$ \int\frac{1}{\Phi^{2}-2\lambda D}\mathrm{d}\Phi = -\frac{t}{2\lambda}+C_{1}. $$ now we have to split the cases for the integration constant $D$ ( I leave this for you to do: But the hint is consider $2\lambda D = \pm k^{2}$ and zero.).

The above is assuming we can have a constant reference frame where the evolution can be considered to have a constant $c$. Also, this is not solved with any conditions on boundaries or initial value problem. This is just a travelling wave solution but the end solutions should solve the equation as required.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.