d'Alembert Formula

pdestud

New member
Joined
Sep 14, 2019
Messages
5
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].
 
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/
 
\(\displaystyle \frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial t^2} \qquad (t \geq 0, x \geq 0) \)

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

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

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

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.
 
Top