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## Main Question or Discussion Point

I've got a set of coupled Ingegro-Differential equations that I've been working on for a while. Here they are:

[tex] \frac{\partial b_x}{\partial t} = i x b_x + i u_x [/tex]

[tex] \frac{\partial b_y}{\partial t} = i x b_y + i u_y [/tex]

[tex] \frac{\partial b_z}{\partial t} = i x b_z + i u_z - \frac{3}{2 \omega} b_x [/tex]

[tex] \frac{\partial u_x}{\partial t} = i x u_x + i b_x + \frac{2}{\omega} u_x -\frac{3}{2 \omega} \frac{\partial Q}{\partial x} [/tex]

[tex] \frac{\partial u_y}{\partial t} = i x u_y + i b_y - \frac{1}{2 \omega} u_x -i Q[/tex]

[tex] \frac{\partial u_z}{\partial t} = i x u_z + i b_z -\frac{i}{\kappa} Q[/tex]

where Q is defined as follows:

[tex] Q = e^{-K x} \int_x^{\infty} \left( \frac{2 u_y(x')}{3} + \frac{2 i \omega u_x(x')}{9 K} \right) e^{K x'} dx' + e^{K x} \int_{-\infty}^x \left( \frac{2 u_y(x')}{3} - \frac{2 i \omega u_x(x')}{9 K} \right) e^{-K x'} dx' [/tex]

Here I've included the dependence on x' in the integral to make it clear that the integral depends on ux and uy.

and [tex] K = \frac{2}{3} \omega \sqrt{1+\frac{1}{\kappa^2}} [/tex]

[tex] \kappa [/tex] and [tex] \omega [/tex] are parameters and if need be can be set to 0.0001 and 0.01 respectively.

To date I've written a code in Fortran 90 to solve these equations using a 4rth order Runge-Kutta method along with an adaptive Simpson integration. The code runs and spits out results but I question the validity. Thus, I would like to try and get Maple/Matlab/Mathematica to run through at least one time step to compare too (mainly the result of the integration).

I'm confident that Maple just can't do it, and I'm unsure of the exact syntax to make Mathematica work. As for Matlab, I've been messing around with the pdepe function with an integration call plopped into it but I've had little sucess.

If anyone can help with this it would be much appreciated.

[tex] \frac{\partial b_x}{\partial t} = i x b_x + i u_x [/tex]

[tex] \frac{\partial b_y}{\partial t} = i x b_y + i u_y [/tex]

[tex] \frac{\partial b_z}{\partial t} = i x b_z + i u_z - \frac{3}{2 \omega} b_x [/tex]

[tex] \frac{\partial u_x}{\partial t} = i x u_x + i b_x + \frac{2}{\omega} u_x -\frac{3}{2 \omega} \frac{\partial Q}{\partial x} [/tex]

[tex] \frac{\partial u_y}{\partial t} = i x u_y + i b_y - \frac{1}{2 \omega} u_x -i Q[/tex]

[tex] \frac{\partial u_z}{\partial t} = i x u_z + i b_z -\frac{i}{\kappa} Q[/tex]

where Q is defined as follows:

[tex] Q = e^{-K x} \int_x^{\infty} \left( \frac{2 u_y(x')}{3} + \frac{2 i \omega u_x(x')}{9 K} \right) e^{K x'} dx' + e^{K x} \int_{-\infty}^x \left( \frac{2 u_y(x')}{3} - \frac{2 i \omega u_x(x')}{9 K} \right) e^{-K x'} dx' [/tex]

Here I've included the dependence on x' in the integral to make it clear that the integral depends on ux and uy.

and [tex] K = \frac{2}{3} \omega \sqrt{1+\frac{1}{\kappa^2}} [/tex]

[tex] \kappa [/tex] and [tex] \omega [/tex] are parameters and if need be can be set to 0.0001 and 0.01 respectively.

To date I've written a code in Fortran 90 to solve these equations using a 4rth order Runge-Kutta method along with an adaptive Simpson integration. The code runs and spits out results but I question the validity. Thus, I would like to try and get Maple/Matlab/Mathematica to run through at least one time step to compare too (mainly the result of the integration).

I'm confident that Maple just can't do it, and I'm unsure of the exact syntax to make Mathematica work. As for Matlab, I've been messing around with the pdepe function with an integration call plopped into it but I've had little sucess.

If anyone can help with this it would be much appreciated.