# Change of variables in PDEs

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?

Last edited:

HallsofIvy
Homework Helper
By using the chain rule.

$$U_s= U_t \frac{\partial t}{\partial s}+ U_x\frac{\partial x}{\partial s}$$
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:
$$U_s= U_t+ U_x[/itex] [tex]U_ss= (U_t+ U_x)_s= (U_t+ U_x)_t + (U_t+ U_x)_x= U_tt+ 2U_tx+ U _xx$$

Similarly,
$$U_sss= U_ttt+ 3Uttx+ 3Utxx+ Uxxx$$
Sustitute those into you equation.

Subsituting these , will not give me the desired equation.

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