d'Alembert Formula

pdestud

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

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

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

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

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

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 \(\displaystyle \alpha = -1 \).
 

Subhotosh Khan

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

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

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

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

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

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 \(\displaystyle \alpha = -1 \).
Please share your work/thoughts about this assignment.

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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

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