Coupled system of 1st order PDEs

[hunt_mat]

I have the following system of first order PDEs
$$\begin{array}{rcl} \frac{\partial v}{\partial t}+v\frac{\partial v}{\partial x} & = & -\varepsilon\gamma^{-3}(v)E \\ \frac{\partial n}{\partial t}+\frac{\partial}{\partial x}(nv) & = & 0 \\ \frac{\partial E}{\partial t}+E & = & nv \end{array}$$
With inital conditions $v(t,0)=\beta_{0},n(0,x)=1,E(x/\beta_{0},x)=0$. Now it is possible to solve for E explicitly to obtain:
$$E(t,x)=\int_{\frac{x}{\beta_{0}}}^{t}e^{s-t}n(s,x)v(s,x)ds$$
Now I have decided to solve this system numerically using a predictor corrector method for v and n and the solution above to find E. Now I have got the predictor-corrector to work (such that it gives no errors when I run it) but I am having a little trouble coding up the integral for E, can anyone suggest something? I am working in MATLAB.

Cheers

Mat

 

[gtoguy97]

lab has a built-in function called integral2 that you can use to solve the integral. You can call it like this:E = integral2(@(t,x)exp(t-s).*n(t,x).*v(t,x), x/beta0, t, 0, x);Where x and t are the independent variables for the integral, and n and v are functions of both t and x. This should solve the integral for E.