# Coupled system of 1st order PDEs

• hunt_mat
In summary, the conversation is about solving a system of first order PDEs numerically using a predictor-corrector method. The solution involves using the built-in function "integral2" in MATLAB to solve an integral for E with given initial conditions.
hunt_mat
Homework Helper
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

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.

## What is a coupled system of 1st order PDEs?

A coupled system of 1st order partial differential equations (PDEs) is a set of two or more PDEs that are interconnected and affect each other's solutions. These equations involve multiple independent variables and their derivatives.

## What are some examples of coupled systems of 1st order PDEs?

Examples of coupled systems of 1st order PDEs include the Navier-Stokes equations for fluid flow, the Maxwell's equations for electromagnetism, and the reaction-diffusion equations for chemical kinetics.

## How are coupled systems of 1st order PDEs solved?

Coupled systems of 1st order PDEs are typically solved using numerical methods, such as finite difference or finite element methods. These methods involve discretizing the equations and solving them iteratively to obtain approximate solutions.

## What are some applications of coupled systems of 1st order PDEs?

Coupled systems of 1st order PDEs have various applications in physics, engineering, and other fields. They are often used to model complex physical phenomena, such as fluid dynamics, electromagnetism, and chemical reactions.

## What are the challenges in solving coupled systems of 1st order PDEs?

Solving coupled systems of 1st order PDEs can be challenging due to their complex nature and the need for accurate and efficient numerical methods. These systems also often have a large number of unknowns, which can make the solution process computationally demanding.

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