# Question:Differential equation solving with a twist

## Question:Differential equation solving with a twist

Maple

I have a set of equations which I need to solve.  I have them in LaTex format (can anyone tell me how to post them here in a better format?) they are:

(\partial_t - i x) \textbf{b} = -\frac{3}{2 \bar{\omega}_A} b_x \hat{y} + i \textbf{u}

(\partial_t - i x) \textbf{u} = \frac{2}{\bar{\omega}_A} u_y \hat{x} - \frac{1}{2 \bar{\omega}_A} u_x \hat{y} - \frac{3}{2 \bar{\omega}_A} \partial_x Q \hat{x} - i Q \hat{y} - i \frac{1}{\kappa} Q \hat{z} + i \textbf{b}

\frac{3}{2} \partial_x u_x + i \bar{\omega}_A u_y + \frac{i \bar{\omega}_A}{\kappa} u_z = 0

The last equation is a constraint equation.  The Q that is found in the equations is found by taking the divergence of equation 2 and solving it, leading to a helmholtz equation.  The solution which, after a few integration tricks, is given by:

Q = \int_x^{\infty} (-\frac{2}{3} u_y - i \frac{2}{9 k}  \bar{\omega}_A u_x){e^{ k (x-x')}} +\int_{-\infty}^x (\frac{2}{3} u_y - i \frac{2}{9 k}  \bar{\omega}_A u_x){e^{ -k (x-x')}}

There is also a \partial_x Q term which can be found by using Leibniz's formula and is found to be:

\frac{4}{3}u_y +\int_x^\infty \left( -\frac{2}{3} u_y k - i \frac{2}{9} \bar{\omega}_A u_x \right) e^{k(x-x')} dx' + \int_{-\infty}^x \left( -\frac{2}{3} u_y k + i \frac{2}{9} \bar{\omega}_A u_x \right) e^{-k(x-x')} dx'

There are a few free parameters here, \bar{\omega}_A and \kappa which can both be set to 0.5.

The initial conditions are both and b set to zero with a gaussian set in the middle of bx centered around x=50.  x should range from 0 to infinity but it can be truncated to 100 or so without too much trouble, by the same token the above integrals can be from 0 to 100 as well instead of -infinity to infinity.

I've already tried to solve these using a RK-4 scheme written in Fortran but I'm having some difficulty.  I'm not that good with Maple so I could use some help setting up this problem in Maple since I would like to see if Maple can do it and save me some trouble.  Any help would be much appreciated. ﻿