Techniques for solving coupled differential equations

Question

I am trying to solve a system of coupled differential equations to plot streamlines using Matlab.

The equations are these:

\begin{align} \frac{\mathrm dx}{\mathrm dt} &= -3x -5y \\ \frac{\mathrm dy}{\mathrm dt} &= 5x + 3y \end{align}

What method do you suggest for solving this system? I'd greatly appreciate any insight or suggestion. No need to solve the system :) as long as you tell me what literature I can refer to.

Thanks!

Answer 1

I think a simple finite difference scheme will do the trick here. http://en.wikipedia.org/wiki/Finite_difference_method

Put a mesh over your domain. Then a point $i$ has a certain position $(x_i,\ y_i)$. Then for the first equation you can write:

$\frac{dx_i}{dt} = -3x_i-5y_i$.

Similar for the second equation. Solving for all $i$ and combining both gives you the streamlines.

Answer 2

The equations may be rewritten $$ \left[\begin{array}{cc} x'(t)\\ y'(t)\end{array} \right]=\left[\begin{array}{cc} -3&-5\\ 5&3 \end{array} \right]\left[\begin{array}{cc} x(t) \\ y(t) \end{array} \right] $$ or $$ X'(t)=AX(t)$$

Do you see the analogy with ordinary differential equations?

Answer 3

Another approach:

Consider the following IVP problem:

\begin{align} \frac{\mathrm{d}x}{\mathrm{d}t} &= -3x - 5y \\ \frac{\mathrm{d}y}{\mathrm{d}t} &= 5x + 3y \end{align}

with $x(0)=x_0$ and $y(0)=y_0$.

Then, Laplace-transform both sides of both equations to get:

\begin{align} s X(s) - x_0 & = - 3 X(s) -5 Y(s) \\ s Y(s) - y_0 & = 5 X(s) + 3Y(s) , \end{align}

which is an algebraic system for $X(s) = \mathcal{L}_sx(t)$ and $Y(s) = \mathcal{L}_sy(t)$. Solve for the unknowns using (for example) Gauss elimination and compute the inverse Laplace transfrom to get the solution.

Cheers!