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

[jpcjr]

1. Homework Statement

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...

u_t = κ u_xx ; (-∞<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...

u_t = κ u_xx

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




2. Homework 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)

 

[jpcjr]

Any help on any part will be GREATLY appreciated!

 

[cepheid]

Welcome to PF,

1. Homework Statement

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...

u_t = κ u_xx ; (-∞<x<∞)

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 u_tt = κu_xx. This is different from the PDE that you gave. But anyway, taking what you wrote at face value:

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

Homework Equations

See above and below...

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)

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

 

[jpcjr]

To your first point...

I am sure it is:

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

to your second point...

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)

 

[cepheid]

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