The Cole-Hopf transformation for Burger equation

In summary: This is not the case, and the only way to find a solution for ##\phi## is to use the substitution φ.
  • #1
docnet
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Homework Statement
please see below
Relevant Equations
please see below
Screen Shot 2021-01-14 at 9.23.12 PM.png
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 ##
Screen Shot 2021-01-14 at 9.26.25 PM.png

so φ solves our PDE. To show u solves the Burger's equation with viscosity, we substitute u into the heat equation and simplify
Screen Shot 2021-01-14 at 9.27.20 PM.png

which is where I am stuck. It seems like this equation does not simplify to anything useful. What I am getting is the following.
Screen Shot 2021-01-14 at 9.27.57 PM.png
 

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  • #2
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.
 
  • #3
sysprog said:
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.
 
  • #4
docnet said:
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.
 
  • #5
docnet said:
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
 
  • #6
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##.
 
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  • #7
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
 
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  • #8
MathematicalPhysicist said:
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)##
 

Related to The Cole-Hopf transformation for Burger equation

What is the Cole-Hopf transformation for Burger equation?

The Cole-Hopf transformation is a mathematical technique used to solve the Burger equation, which is a nonlinear partial differential equation that describes the behavior of a fluid in which the flow speed depends on both the position and time.

Who discovered the Cole-Hopf transformation?

The Cole-Hopf transformation was independently discovered by mathematicians Sidney D. Cole and Emil Hopf in the 1950s.

What is the significance of the Cole-Hopf transformation?

The Cole-Hopf transformation allows for the Burger equation to be transformed into a linear equation, making it easier to solve. This transformation has been used in various applications, including fluid mechanics, quantum field theory, and statistical mechanics.

What are the limitations of the Cole-Hopf transformation?

While the Cole-Hopf transformation simplifies the Burger equation, it only works for certain initial and boundary conditions. It also does not provide a solution for all time, but rather for a finite time interval.

Are there any alternative methods for solving the Burger equation?

Yes, there are other methods for solving the Burger equation, such as the method of characteristics and the Hopf-Cole transformation. However, the Cole-Hopf transformation remains a widely used and important technique in the study of nonlinear partial differential equations.

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