# How to solve a nonlinear differential equation of this type.

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

$$\frac{\partial u}{\partial t}=\frac{3}{x}\frac{\partial}{\partial x}\left(\sqrt{x}\frac{\partial}{\partial x}(\nu(T(u,x))u\sqrt{x})\right)$$

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:

$$\nu\propto\frac{T}{\Omega}$$

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

$$\Xi(T,\Omega)T^3=u^2\Omega$$

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:

$$\Xi(T,\Omega)T^3=u^2\Omega+\frac{T^4uH(T,\Omega)}{\Omega x^2}\left(\frac{d\log H(T,\Omega)}{d\log x}-\frac{d\log T}{d\log x}-\frac{d\log u}{d\log x}\right)$$

$$H^2\Omega^2-H\frac{T^4}{u}-T=0$$

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.