MATLAB Central Newsreader - How to solve a nonlinear differential equation of this type

How to solve a nonlinear differential equation of this type

Nicolas — Thu, 10 May 2012 20:38:29 +0000

I don't know how to solve this kind of equations, can you give me a hand please.

I need to find T, which is a function of T=T(R).

And the equation is:

T^4 H(T) abs(d logH(T)/d logR?d logT/d logR?d logI(R)/d logR)?T^3 F(T)+G(R)=0
and
H=A T^4?B sqrt(CT^8+D T)

Please help. Ask me whatever you want to know about this equation.

Re: How to solve a nonlinear differential equation of this type

Star Strider — Fri, 11 May 2012 17:23:12 +0000

"Nicolas " <nikkolazo@gmail.com> wrote in message <joh905$da2$1@newscl01ah.mathworks.com>...
> I don't know how to solve this kind of equations, can you give me a hand please.
>
> I need to find T, which is a function of T=T(R).
>
> And the equation is:
>
> T^4 H(T) abs(d logH(T)/d logR?d logT/d logR?d logI(R)/d logR)?T^3 F(T)+G(R)=0
> and
> H=A T^4?B sqrt(CT^8+D T)
>
>
> Please help. Ask me whatever you want to know about this equation.

-----------------------------------------------------
What are you using the equation to do? Where did you get it?

Do you also have expressions for I(R), F(T) and G(R)?

Also, you may have typed a symbol the Newsreader doesn't recognize. What did you intend for what appeared in your equation as question marks ('?')?

Re: How to solve a nonlinear differential equation of this type

Nicolas — Fri, 11 May 2012 21:34:28 +0000

I am sorry I was trying to fix the typos but I can't find where to do it. I will explain better the problem.

Re: How to solve a nonlinear differential equation of this type

Nicolas — Sat, 12 May 2012 18:05:12 +0000

Here is the same info in latex.
http://s13.postimage.org/6qagm6brb/matlab_png.png

Trying to explain my problem better, I need to start with the diffusion Equation, which is the one that I am trying to solve.

The Diffusion Equation

\begin{equation}
\dfrac{\partial u}{\partial t}=\dfrac 3 x\dfrac \partial{\partial x}\left(\sqrt x \frac \partial {\partial x}(\nu(T(u,x))u\sqrt x)\right)
\end{equation}

Here $u$ is the surface density of a gas disk. $T$, is the temperature of that disk. My aim is calculate the evolution of that disk using an initial condition.
After treat this equation with MOL (Method of lines) in Matlab I am able to solve this equation. But for that I need to use another equations that will relate some of the variables in my problem, so finally I have:

\begin{equation}
\nu\propto \frac{T}{\Omega }
\end{equation}

$\Omega$ is just the keplarian velocity of the disk, therefore is a function that depends on $x$, and also I have

\begin{equation}
\Xi(T,\Omega)T^3=u^2\Omega
\end{equation}

and $\Xi$ is another function which depend also in $T$ and $\Omega$

Then, since I have an initial condition for $u$ and of course I define my dominion in $x$ ($x=1..10^3$ for instance). I just need to solve the last equation and find the value for T given some position and a value of $u$. I do this with a method to find zeros.

NOW MY PROBLEM:

I need to add advection to my problem. In terms of equations I have this now:

\begin{equation}
\Xi(T,\Omega)T^3=u^2\Omega + \frac{T^4\;u\;H(T,\Omega)}{\Omega x^2}\left|\frac{d\;\log\;H(T,\Omega)}{d\;\log\;x} - \frac{d\;\log\;T}{d\;\log\;x} - \frac{d\;\log\;u}{d\;\log\;x} \right|
\end{equation}

\begin{equation}
H^2\Omega^2-H\frac{T^4}{u}-T=0
\end{equation}

But I only need one root (the real and positive one), and of course the method to solve this (the one that I'm trying to find) will be the same using any root of H.

My problem here is the derivatives, without them, I can use the same method that before to find T. But now I can't do that. Because I need to calculate those derivates.

Please ask me anything about the problem.

Re: How to solve a nonlinear differential equation of this type

Nasser M. Abbasi — Sat, 12 May 2012 18:16:24 +0000

On 5/12/2012 1:05 PM, Nicolas wrote:
> Here is the same info in latex.
> http://s13.postimage.org/6qagm6brb/matlab_png.png
>
>
>
>

>my dominion in $x$ ($x=1..10^3$ for instance).

ps. I think the above will look better if you use $x=1 \cdots 10^3$

--Nasser

Re: How to solve a nonlinear differential equation of this type

Star Strider — Sun, 13 May 2012 14:35:07 +0000

"Nicolas " <nikkolazo@gmail.com> wrote in message <jom8oo$n5p$1@newscl01ah.mathworks.com>...

>
> My problem here is the derivatives, without them, I can use the same method that before to find T. But now I can't do that. Because I need to calculate those derivates.
>

-------------------------------------------------------

The Latex version helped. It still seems to me you need more information about H(T,Omega).

Someone much more knowledgable in maths than I am may have a better suggestion, but for a start I'd consider approximating the derivatives with a series, if that's appropriate. You also might want to do a bit more literature research to see if someone else has come up with a better approach.

Re: How to solve a nonlinear differential equation of this type

Nicolas — Sun, 13 May 2012 16:39:08 +0000

"Star Strider" wrote in message <joogqr$piq$1@newscl01ah.mathworks.com>...
> "Nicolas " <nikkolazo@gmail.com> wrote in message <jom8oo$n5p$1@newscl01ah.mathworks.com>...
>
> >
> > My problem here is the derivatives, without them, I can use the same method that before to find T. But now I can't do that. Because I need to calculate those derivates.
> >
>
> -------------------------------------------------------
>
> The Latex version helped. It still seems to me you need more information about H(T,Omega).
>
> Someone much more knowledgable in maths than I am may have a better suggestion, but for a start I'd consider approximating the derivatives with a series, if that's appropriate. You also might want to do a bit more literature research to see if someone else has come up with a better approach.

Ok, for one second make H=0. How would you solve the equation now?

Re: How to solve a nonlinear differential equation of this type

TideMan — Thu, 10 May 2012 22:45:32 +0000

On Friday, May 11, 2012 8:38:29 AM UTC+12, Nicolas wrote:
> I don't know how to solve this kind of equations, can you give me a hand please.
>
> I need to find T, which is a function of T=T(R).
>
> And the equation is:
>
> T^4 H(T) abs(d logH(T)/d logR?d logT/d logR?d logI(R)/d logR)?T^3 F(T)+G(R)=0
> and
> H=A T^4?B sqrt(CT^8+D T)
>
>
> Please help. Ask me whatever you want to know about this equation.

What do the question marks mean?
You need to use brackets to show us more clearly where the differentiation is.
Is it an ODE or PDE?
Is I(R) a known function?

Looks intractable to me.

Re: How to solve a nonlinear differential equation of this type

Nicolas — Wed, 16 May 2012 16:44:08 +0000

I already fix this =)
TideMan <mulgor@gmail.com> wrote in message <2274248.12.1336689932327.JavaMail.geo-discussion-forums@pbcst1>...
> On Friday, May 11, 2012 8:38:29 AM UTC+12, Nicolas wrote:
> > I don't know how to solve this kind of equations, can you give me a hand please.
> >
> > I need to find T, which is a function of T=T(R).
> >
> > And the equation is:
> >
> > T^4 H(T) abs(d logH(T)/d logR?d logT/d logR?d logI(R)/d logR)?T^3 F(T)+G(R)=0
> > and
> > H=A T^4?B sqrt(CT^8+D T)
> >
> >
> > Please help. Ask me whatever you want to know about this equation.
>
> What do the question marks mean?
> You need to use brackets to show us more clearly where the differentiation is.
> Is it an ODE or PDE?
> Is I(R) a known function?
>
> Looks intractable to me.