The Cole-Hopf transformation for Burger equation

[docnet]

Homework Statement

please see below

Relevant Equations

please see below

Attempt at a solution
To show φ satisfies our PDE, we first solve the substitution for φ

$\mathrm{ln(\phi) = -\frac {1} {2} \int u dx}$

which gives

$\mathrm{\phi = e^{-\frac {1} {2} \int u dx} }$

and plug it into our PDE, which simplifies to

$\mathrm{\frac {\partial } {\partial t} - \frac {\partial} {\partial x} u } = 0$

we use the substitution φ to show the following

$\mathrm{\frac {\partial } {\partial t} (\frac {-2 \frac {\partial} {\partial x} \phi} {\phi}) - {\frac {\partial ^2} {\partial x^2} (\frac {-2 \frac {\partial} {\partial x} \phi} {\phi} = 0$

so φ solves our PDE. To show u solves the Burger's equation with viscosity, we substitute u into the heat equation and simplify

which is where I am stuck. It seems like this equation does not simplify to anything useful. What I am getting is the following.

 

[sysprog]

Please type the $\mathrm{\LaTeX}$ instead of just using images $-$ that better enables us to excerpt parts of what you've said when we're responding.

 

[docnet]

Please type the $\mathrm{\LaTeX}$ instead of just using images $-$ that better enables us to excerpt parts of what you've said when we're responding.

Sorry. I am learning LATEX and cannot make integrals or derivatives.

 

[sysprog]

Sorry. I am learning LATEX and cannot make integrals or derivatives.

At the lower left corner just below the reply box there's a link to a good brief guide.

 

[sysprog]

Sorry. I am learning LATEX and cannot make integrals or derivatives.

Sorry. I am learning how to type in LATEX and still have yet to learn how to make integrals or derivatives.

Very nice edit. You went from "cannot" to "have yet to learn how to".
Yay!

I am confident that you'll get there soon on typing the $\mathrm{\LaTeX}$ for integrals and derivatives.

Professor Donald Knuth, the creator of $\TeX$, didn't write The Art of Computer Programming in one day.

https://www-cs-faculty.stanford.edu/~knuth/taocp.html

 

[MathematicalPhysicist]

Plug the ansatz: $u = -1/2\frac{\partial_x \phi(x,t)}{\phi(x,t)}$ into Burger's equation and then you should get: the second PDE for $\phi$.
By this ansatz we can see that these two PDEs are equivalent, so a solution for $\phi$'s PDE is a solution for Burger's original PDE with $u$.

We can notice that for $\phi$ that satisfies the Heat equation also satisfies Burger's $\phi$'s PDE, since $\phi_{xx}-\phi_t=0$.

 

[docnet]

yikes... I was editing my first post with LATEX code while reading the how-to article. Something must have timed out and we cannot see the equations. I am exhausted hence my quitter talk.. I will return to this problem tomorrow morning after getting some sleep. thanks

 

[docnet]

Plug the ansatz: $u = -1/2\frac{\partial_x \phi(x,t)}{\phi(x,t)}$ into Burger's equation and then you should get: the second PDE for $\phi$.
By this ansatz we can see that these two PDEs are equivalent, so a solution for $\phi$'s PDE is a solution for Burger's original PDE with $u$.

We can notice that for $\phi$ that satisfies the Heat equation also satisfies Burger's $\phi$'s PDE, since $\phi_{xx}-\phi_t=0$.

Thank you.

So, after stating the two PDEs ($\phi$ and Burger's) are equivalent by using pluginology, we re-write the heat equation as $\phi_{xx}-\phi_t=0$ and use pluginology again in the $\phi$ PDE to give $\partial x (\frac {1} {Φ(t,x)} 0 ) =\partial x (o) = 0$. This shows the solution to the heat equation is a solution to our $\phi$ PDE. I was confusing myself by thinking there is a solution where $\partial_t \phi(t, x) - \partial^2 _x \phi(t,x) = \phi (t,x)$