Coupled system of 1st order PDEs

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    Coupled Pdes System
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SUMMARY

This discussion focuses on solving a coupled system of first-order partial differential equations (PDEs) using MATLAB. The system includes equations for variables v, n, and E, with specific initial conditions. The user has successfully implemented a predictor-corrector method for v and n but seeks assistance in coding the integral for E, which is defined as E(t,x)=∫(x/β₀)^(t)e^(s-t)n(s,x)v(s,x)ds. The discussion highlights the challenges of numerical integration in the context of PDEs.

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  • Understanding of first-order partial differential equations (PDEs)
  • Familiarity with MATLAB programming
  • Knowledge of numerical methods, specifically predictor-corrector techniques
  • Experience with numerical integration techniques
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  • Research MATLAB's built-in functions for numerical integration, such as 'integral' and 'integral2'
  • Explore advanced numerical methods for solving PDEs, including finite difference and finite element methods
  • Learn about the implementation of predictor-corrector methods in MATLAB for systems of equations
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Mathematicians, engineers, and researchers working on numerical solutions of partial differential equations, particularly those using MATLAB for computational modeling and simulation.

hunt_mat
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I have the following system of first order PDEs
<br /> \begin{array}{rcl}<br /> \frac{\partial v}{\partial t}+v\frac{\partial v}{\partial x} &amp; = &amp; -\varepsilon\gamma^{-3}(v)E \\<br /> \frac{\partial n}{\partial t}+\frac{\partial}{\partial x}(nv) &amp; = &amp; 0 \\<br /> \frac{\partial E}{\partial t}+E &amp; = &amp; nv<br /> \end{array}<br />
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:
<br /> E(t,x)=\int_{\frac{x}{\beta_{0}}}^{t}e^{s-t}n(s,x)v(s,x)ds<br />
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
 

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