- #1

jpcjr

- 17

- 0

**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

u(x,0) = [itex]\phi[/itex](x)

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

u(x,0) = [itex]\phi[/itex](x)

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

u

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

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) = [itex]\phi[/itex](x)

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

u(x,0) = [itex]\phi[/itex](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...

See above and below...

**3. The Attempt at a Solution**

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

v'(x,t) = c u[itex]_{x}[/itex](x+ct,t) + u[itex]_{t}[/itex](x+ct,t)

v[itex]_{x}[/itex](x,t) = c u[itex]_{x}[/itex](x+ct,t)

v[itex]_{xx}[/itex](x,t) = c[itex]^{2}[/itex] u[itex]_{xx}[/itex](x+ct,t)

v[itex]_{t}[/itex](x,t) = u[itex]_{t}[/itex](x+ct,t)

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

v'(x,t) = c u[itex]_{x}[/itex](x+ct,t) + u[itex]_{t}[/itex](x+ct,t)

v[itex]_{x}[/itex](x,t) = c u[itex]_{x}[/itex](x+ct,t)

v[itex]_{xx}[/itex](x,t) = c[itex]^{2}[/itex] u[itex]_{xx}[/itex](x+ct,t)

v[itex]_{t}[/itex](x,t) = u[itex]_{t}[/itex](x+ct,t)