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I have the following system of first order PDEs
[tex] \begin{array}{rcl}<br /> \frac{\partial v}{\partial t}+v\frac{\partial v}{\partial x} & = & -\varepsilon\gamma^{-3}(v)E \\<br /> \frac{\partial n}{\partial t}+\frac{\partial}{\partial x}(nv) & = & 0 \\<br /> \frac{\partial E}{\partial t}+E & = & nv<br /> \end{array}[/tex]
With inital conditions [itex]v(t,0)=\beta_{0},n(0,x)=1,E(x/\beta_{0},x)=0[/itex]. Now it is possible to solve for E explicitly to obtain:
[tex] E(t,x)=\int_{\frac{x}{\beta_{0}}}^{t}e^{s-t}n(s,x)v(s,x)ds[/tex]
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
[tex] \begin{array}{rcl}<br /> \frac{\partial v}{\partial t}+v\frac{\partial v}{\partial x} & = & -\varepsilon\gamma^{-3}(v)E \\<br /> \frac{\partial n}{\partial t}+\frac{\partial}{\partial x}(nv) & = & 0 \\<br /> \frac{\partial E}{\partial t}+E & = & nv<br /> \end{array}[/tex]
With inital conditions [itex]v(t,0)=\beta_{0},n(0,x)=1,E(x/\beta_{0},x)=0[/itex]. Now it is possible to solve for E explicitly to obtain:
[tex] E(t,x)=\int_{\frac{x}{\beta_{0}}}^{t}e^{s-t}n(s,x)v(s,x)ds[/tex]
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