d'Alembert Formula

[pdestud]

Use d'Alembert's solution of the wave equation to find the solution of

[MATH]\frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial t^2} \qquad (t \geq 0, x \geq 0) [/MATH]
[MATH] \frac{\partial u(0,t)}{\partial t} = \alpha \frac{\partial u(0,t)}{\partial x} \qquad (t \geq 0) [/MATH]
[MATH] u(x,0) = \phi_0 (x) \qquad (x \geq 0) [/MATH]
[MATH] \frac{\partial u}{\partial t} (x, 0) = \phi_1 (x) \qquad (x \geq 0) [/MATH]
where [MATH] \alpha \neq -1 [/MATH] is a constant and [MATH] \phi_0 (x) [/MATH] and [MATH] \phi_1 (x) [/MATH] are twice continuously differentiable for [MATH] x > 0 [/MATH] and vanish near [MATH] x = 0 [/MATH]. Show that in general no solution exists when [MATH] \alpha = -1 [/MATH].

 

[Deleted member 4993]

Use d'Alembert's solution of the wave equation to find the solution of

[MATH]\frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial t^2} \qquad (t \geq 0, x \geq 0) [/MATH]
[MATH] \frac{\partial u(0,t)}{\partial t} = \alpha \frac{\partial u(0,t)}{\partial x} \qquad (t \geq 0) [/MATH]
[MATH] u(x,0) = \phi_0 (x) \qquad (x \geq 0) [/MATH]
[MATH] \frac{\partial u}{\partial t} (x, 0) = \phi_1 (x) \qquad (x \geq 0) [/MATH]
where [MATH] \alpha \neq -1 [/MATH] is a constant and [MATH] \phi_0 (x) [/MATH] and [MATH] \phi_1 (x) [/MATH] are twice continuously differentiable for [MATH] x > 0 [/MATH] and vanish near [MATH] x = 0 [/MATH]. Show that in general no solution exists when [MATH] \alpha = -1 [/MATH].

Please share your work/thoughts about this assignment.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING

Please review your post to make sure that you do not have any "typos".

How is this problem different from the "almost" duplicate problem you posted at:

https://www.freemathhelp.com/forum/threads/dalemberts-solution-of-wave-equation.118385/

 

[pdestud]

Please share your work/thoughts about this assignment.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING

Please review your post to make sure that you do not have any "typos".

How is this problem different from the "almost" duplicate problem you posted at:

https://www.freemathhelp.com/forum/threads/dalemberts-solution-of-wave-equation.118385/

This is what I have so far. I just do not know what to do with the rest.

 

[nasi112]

$\displaystyle \frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial t^2} \qquad (t \geq 0, x \geq 0)$

$\displaystyle \frac{\partial u(0,t)}{\partial t} = \alpha \frac{\partial u(0,t)}{\partial x} \qquad (t \geq 0)$

$\displaystyle u(x,0) = \phi_0 (x) \qquad (x \geq 0)$

$\displaystyle \frac{\partial u}{\partial t} (x, 0) = \phi_1 (x) \qquad (x \geq 0)$

where $\displaystyle \alpha \neq -1$ is a constant and $\displaystyle \phi_0 (x)$ and $\displaystyle \phi_1 (x)$ are twice continuously differentiable for $\displaystyle x > 0$ and vanish near $\displaystyle x = 0$. Show that in general no solution exists when $\displaystyle \alpha = -1$.

Now I can read your code.