# PDE on the line

In my work I faced a system of PDE of the form \begin{align} \varphi^\prime_v(t,v)|_{v=X_t} = \sigma(t,X_t),\newline \varphi^\prime_t(t,v)|_{v=X_t} = b(t,X_t), \end{align}

where $t\in[0,T]$, $v\in\mathbf R$, $X_t$ is known continuous function of unbounded variation, $\varphi(t,v)$ is unknown, $\varphi^\prime_v(t,v) = \frac{\partial}{\partial v} \varphi(t,v)$, $\varphi^\prime_t(t,v) = \frac{\partial}{\partial t} \varphi(t,v)$ and there is an initial value $\varphi(0,X_0) = x_0$. Function $\varphi(t,X_t)$ is to be determined.

Roughly this is a system of PDE \begin{align} \varphi^\prime_v(t,v)= \sigma(t,v),\newline \varphi^\prime_t(t,v) = b(t,v), \end{align} defined on the line $(t,X_t)$.

I need to find a solution for this system, but yet haven't dealt with such equations and don't even know if they have any special name, so that I could search. I will be very happy if somebody gives me recommendations on textbooks which consider this kind of problems.

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Doesn't it reduce immediately to ODEs? Your 1st DE says $\phi(t,v) = \sigma(t,X_t)v+c_t$ which you then plug into your 2nd DE to solve for $c_t$. –  Ryan Budney Jan 22 '12 at 20:11
Which functions are given, and which are unknown? –  Robert Israel Jan 22 '12 at 20:14
Have edited the question. So now it's clear which functions are known and which are to be determined and it doesn't reduce to ODEs. –  niyazets Jan 22 '12 at 21:06
If $X_t$ is a fixed function of $t$, isn't your PDE too underdetermined? You're only describing the partial derivatives of $\phi$ along the curve $(t,X_t)$. So it look (to me) like the only thing going on is your PDE is an ODE for the function $f(t) = \phi(t,X_t)$ given by $f'(t) = \sigma(t,X_t)X'_t + b(t,X_t)$. And now this is integrable since on the right hand side you've got a function of $t$ alone. –  Ryan Budney Jan 22 '12 at 21:10
Are you really sure this is what you're studying? What you've written down is not much of a constraint on $\phi(t,v)$. –  Ryan Budney Jan 22 '12 at 21:29