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Thread Subject:
how to solve a differential equation with Gaussian white noise?

Subject: how to solve a differential equation with Gaussian white noise?

From: Min Gan

Date: 6 Apr, 2010 13:09:04

Message: 1 of 10

the equation has the following form:
x'' + 0.25x' + 14.8x=g
where g is a contiunous Gaussian white noise?
It is very easy to solve this differential equation using ODE45 if g is a constant or a sin or cos function.
However, in the case g is a contiunous Gaussian white noise, how to deal with this problem?

Subject: how to solve a differential equation with Gaussian white noise?

From: Mark Shore

Date: 6 Apr, 2010 13:26:03

Message: 2 of 10

"Min Gan" <aganmin@yahoo.com.cn> wrote in message <hpfbpg$f1u$1@fred.mathworks.com>...
> the equation has the following form:
> x'' + 0.25x' + 14.8x=g
> where g is a contiunous Gaussian white noise?
> It is very easy to solve this differential equation using ODE45 if g is a constant or a sin or cos function.
> However, in the case g is a contiunous Gaussian white noise, how to deal with this problem?

I do not work with ODEs, but I believe the problem as you've framed it is not solvable. I have difficulty seeing its possible physical relevance as well, but that could just be me.

Subject: how to solve a differential equation with Gaussian white noise?

From: Min Gan

Date: 6 Apr, 2010 13:40:25

Message: 3 of 10

"Mark Shore" <mshore@magmageosciences.ca> wrote in message <hpfcpa$1f7$1@fred.mathworks.com>...
> "Min Gan" <aganmin@yahoo.com.cn> wrote in message <hpfbpg$f1u$1@fred.mathworks.com>...
> > the equation has the following form:
> > x'' + 0.25x' + 14.8x=g
> > where g is a contiunous Gaussian white noise?
> > It is very easy to solve this differential equation using ODE45 if g is a constant or a sin or cos function.
> > However, in the case g is a contiunous Gaussian white noise, how to deal with this problem?
>
> I do not work with ODEs, but I believe the problem as you've framed it is not solvable. I have difficulty seeing its possible physical relevance as well, but that could just be me.

This differential equation is obtained from the article "Modeling nonlinear random vibrations using an amplitude-dependent autoregressive time series model" by Hanggan and Ozaki. They produced analogue simulations of this equation.
Howerver, they did not show details. It seems that it is not a problem to them.

Subject: how to solve a differential equation with Gaussian white noise?

From: Torsten Hennig

Date: 6 Apr, 2010 13:57:47

Message: 4 of 10

> "Mark Shore" <mshore@magmageosciences.ca> wrote in
> message <hpfcpa$1f7$1@fred.mathworks.com>...
> > "Min Gan" <aganmin@yahoo.com.cn> wrote in message
> <hpfbpg$f1u$1@fred.mathworks.com>...
> > > the equation has the following form:
> > > x'' + 0.25x' + 14.8x=g
> > > where g is a contiunous Gaussian white noise?
> > > It is very easy to solve this differential
> equation using ODE45 if g is a constant or a sin or
> cos function.
> > > However, in the case g is a contiunous Gaussian
> white noise, how to deal with this problem?
> >
> > I do not work with ODEs, but I believe the problem
> as you've framed it is not solvable. I have
> difficulty seeing its possible physical relevance as
> well, but that could just be me.
>
> This differential equation is obtained from the
> article "Modeling nonlinear random vibrations using
> an amplitude-dependent autoregressive time series
> model" by Hanggan and Ozaki. They produced analogue
> simulations of this equation.
> Howerver, they did not show details. It seems that it
> is not a problem to them.

http://en.wikipedia.org/wiki/Stochastic_differential_equation

Best wishes
Torsten.

Subject: how to solve a differential equation with Gaussian white noise?

From: Mark Shore

Date: 6 Apr, 2010 14:01:22

Message: 5 of 10

>
> This differential equation is obtained from the article "Modeling nonlinear random vibrations using an amplitude-dependent autoregressive time series model" by Hanggan and Ozaki. They produced analogue simulations of this equation.
> Howerver, they did not show details. It seems that it is not a problem to them.

Can't help you myself (twenty years since I took a grad course in nonlinear dynamics and chaos theory, didn't get much out of it and squeaked through with a B-), except to suggest that the authors probably used some type of iterative algorithm. For anyone else interested in looking at this, the original reference is Haggan and Ozaki (1981), Biometrika 68(1) pp 189-196.

Subject: how to solve a differential equation with Gaussian white noise?

From: Richard Crozier

Date: 6 Apr, 2010 14:05:19

Message: 6 of 10

"Min Gan" <aganmin@yahoo.com.cn> wrote in message <hpfdk9$e5q$1@fred.mathworks.com>...
> "Mark Shore" <mshore@magmageosciences.ca> wrote in message <hpfcpa$1f7$1@fred.mathworks.com>...
> > "Min Gan" <aganmin@yahoo.com.cn> wrote in message <hpfbpg$f1u$1@fred.mathworks.com>...
> > > the equation has the following form:
> > > x'' + 0.25x' + 14.8x=g
> > > where g is a contiunous Gaussian white noise?
> > > It is very easy to solve this differential equation using ODE45 if g is a constant or a sin or cos function.
> > > However, in the case g is a contiunous Gaussian white noise, how to deal with this problem?
> >
> > I do not work with ODEs, but I believe the problem as you've framed it is not solvable. I have difficulty seeing its possible physical relevance as well, but that could just be me.
>
> This differential equation is obtained from the article "Modeling nonlinear random vibrations using an amplitude-dependent autoregressive time series model" by Hanggan and Ozaki. They produced analogue simulations of this equation.
> Howerver, they did not show details. It seems that it is not a problem to them.


My guess is you need to pregenerate a time series of random noise values and interpolate this time series within your ode solution, e.g.

g = rand(1001)
gtimes = 0:1000;

[T,Y] = ode45(@(t,y) odefun(t, y, gtimes, g), tspan)

function dx = odefun(t,y, g)

    gTemp = interp1(gtimes, g, t)

    dx = blah blah blah ...

end


Someone might point out a flaw in this scheme though.

Subject: how to solve a differential equation with Gaussian white noise?

From: Min Gan

Date: 6 Apr, 2010 14:44:04

Message: 7 of 10

"Richard Crozier" <r.crozier@ed.ac.uk> wrote in message <hpff2v$8n8$1@fred.mathworks.com>...
> "Min Gan" <aganmin@yahoo.com.cn> wrote in message <hpfdk9$e5q$1@fred.mathworks.com>...
> > "Mark Shore" <mshore@magmageosciences.ca> wrote in message <hpfcpa$1f7$1@fred.mathworks.com>...
> > > "Min Gan" <aganmin@yahoo.com.cn> wrote in message <hpfbpg$f1u$1@fred.mathworks.com>...
> > > > the equation has the following form:
> > > > x'' + 0.25x' + 14.8x=g
> > > > where g is a contiunous Gaussian white noise?
> > > > It is very easy to solve this differential equation using ODE45 if g is a constant or a sin or cos function.
> > > > However, in the case g is a contiunous Gaussian white noise, how to deal with this problem?
> > >
> > > I do not work with ODEs, but I believe the problem as you've framed it is not solvable. I have difficulty seeing its possible physical relevance as well, but that could just be me.
> >
> > This differential equation is obtained from the article "Modeling nonlinear random vibrations using an amplitude-dependent autoregressive time series model" by Hanggan and Ozaki. They produced analogue simulations of this equation.
> > Howerver, they did not show details. It seems that it is not a problem to them.
>
>
> My guess is you need to pregenerate a time series of random noise values and interpolate this time series within your ode solution, e.g.
>
> g = rand(1001)
> gtimes = 0:1000;
>
> [T,Y] = ode45(@(t,y) odefun(t, y, gtimes, g), tspan)
>
> function dx = odefun(t,y, g)
>
> gTemp = interp1(gtimes, g, t)
>
> dx = blah blah blah ...
>
> end

I did it as you said.
However, the data obtained is NAN.
>
>
> Someone might point out a flaw in this scheme though.

Subject: how to solve a differential equation with Gaussian white noise?

From: Min Gan

Date: 7 Apr, 2010 01:14:05

Message: 8 of 10


Any other suggestion?
I think this is a problem how to produce continuous Gaussian white noises.

Subject: how to solve a differential equation with Gaussian white noise?

From: Min Gan

Date: 7 Apr, 2010 07:50:09

Message: 9 of 10

"Min Gan" <aganmin@yahoo.com.cn> wrote in message <hpgm8t$ijf$1@fred.mathworks.com>...
>
> Any other suggestion?
> I think this is a problem how to produce continuous Gaussian white noises.

Nobody can help me ??

Subject: how to solve a differential equation with Gaussian white noise?

From: Ehsan Negahbani

Date: 30 May, 2013 23:57:17

Message: 10 of 10

"Min Gan" <aganmin@yahoo.com.cn> wrote in message <hphdfh$84q$1@fred.mathworks.com>...
> "Min Gan" <aganmin@yahoo.com.cn> wrote in message <hpgm8t$ijf$1@fred.mathworks.com>...
> >
> > Any other suggestion?
> > I think this is a problem how to produce continuous Gaussian white noises.
>
> Nobody can help me ??

One way is to write your own ODE solver based on Euler or Runge-Kutta methods (latter is recommended) and embed the white noise in your code.

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