How Does Changing Variables Simplify Nonlinear PDEs?

[aquarian11]

I need guidance regarding PDE.
If u have a nonlinear PDE as
Ut+Us+a*U*Us*b*Usss=0
where U is function of (s,t) and a,b are constants.
by introducing new variable x=s-t we will get
Ut+a*U*Ux+b*Uxxx=0
Ut means partial derivative w.r.t time
Us means partial derivative w.r.t s.

How can we get the second equation from the first one?

 

[HallsofIvy]

By using the chain rule.

[tex]U_s= U_t \frac{\partial t}{\partial s}+ U_x\frac{\partial x}{\partial s}[/tex]
Note: if you are going to use x= s- t to replace s only, you will need to think of s as a function of the other variable, t.
If x= s- t, then s= s+ t so both partial derivatives are 1:
[tex]U_s= U_t+ U_x[/itex]
[tex]U_ss= (U_t+ U_x)_s= (U_t+ U_x)_t + (U_t+ U_x)_x= U_tt+ 2U_tx+ U
_xx[/tex]

Similarly,
[tex]U_sss= U_ttt+ 3Uttx+ 3Utxx+ Uxxx[/tex]
Sustitute those into you equation.

 

[aquarian11]

Subsituting these , will not give me the desired equation.

 

[aquarian11]

I have solve it, same concept of chain rule but with different approach.