# Let v(x,t) = u(x+ct) and show that

1. Mar 19, 2012

### jpcjr

1. The problem statement, all variables and given/known data

NOTE: I realize the following is a partial differential equation, but I believe the answer to my question is a straight forward multi-variate calculus question.

Let v(x,t) = u(x+ct,t) and show that v(x,t) solves the following...

ut = κ uxx ; (-∞<x<∞)

u(x,0) = $\phi$(x)

NOTE: You may disregard the following, if necessary for you to answer this question:

u(x,0) = $\phi$(x)

and simply show that v(x,t) solves the following...

ut = κ uxx

ALTERNATIVELY, you may help me by commenting on the correctness of my work below.

2. Relevant equations

See above and below...

3. The attempt at a solution

v(x,t) = u(x+ct,t)

v'(x,t) = c u$_{x}$(x+ct,t) + u$_{t}$(x+ct,t)

v$_{x}$(x,t) = c u$_{x}$(x+ct,t)

v$_{xx}$(x,t) = c$^{2}$ u$_{xx}$(x+ct,t)

v$_{t}$(x,t) = u$_{t}$(x+ct,t)

2. Mar 19, 2012

### jpcjr

Any help on any part will be GREATLY appreciated!!!

3. Mar 19, 2012

### cepheid

Staff Emeritus
Welcome to PF,

Just before we start, are you sure it was not meant to be simply v(x,t) = u(x+ct)? Because I know that a function in which the temporal and spatial dependence are related in this way will satisfy a wave equation, which is given by utt = κuxx. This is different from the PDE that you gave. But anyway, taking what you wrote at face value:

First of all, I don't think that v' (v "prime") is meaningful here. In single-variable calculus, a prime symbol typically means "derivative with respect to the argument." But this function has two arguments. Since it is a multi-variable function, any derivative is going to be a partial derivative, and you must explicitly specify which variable you are differentiating with respect to. As for your attempt at partial derivatives, my approach would be to write:

u = u(w,t) where w = x+ct

then,

$$\frac{\partial u}{\partial x} = \frac{\partial u}{\partial w}\frac{\partial w}{\partial x}$$$$= \frac{\partial u}{\partial w}$$

Et cetera for the other derivatives

4. Mar 19, 2012

### jpcjr

I am sure it is:

Let v(x,t) = u(x+ct,t)

That would mean the following is incorrect, right?

v.sub.x(x,t) = c u.sub.x(x+ct,t)

and should have been...

v.sub.x(x,t) = u.sub.x(x+ct,t)

5. Mar 19, 2012

### cepheid

Staff Emeritus
What I was saying was that v_x = u_w, with underscores denoting subscripts.