Initial Boundary Value Problem for Wave Equation

Question

I'm trying to solve this initial boundary value problem. But I'm getting stuck, some help would be appreciated. $$u_{tt}-u_{xx}=0, x>0, t>0$$ $$u(x,0)=0, u_t(x,0)=0, x\ge0$$ $$u(0,t)=1-\cos(t), t\ge0$$

I think that $$u(t,x)=F(x+t)+G(x-t).$$

Answer 1

From Pinchover and Rubinstein:

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Thus, we have $$f(x) = g(x) = 0$$

and

$$c^2 = 1 \iff c = \pm 1$$

Thus, I think you should have $u(x,t) = F(x \pm t) + G(x \pm t)$

Anyway, by d'Alembert's formula, we would supposedly have

$$u(x,t) = \frac{0 + 0}{2} + \frac{1}{2c}\int_{x-ct}^{x+ct} 0 ds = 0$$

However, the initial condition is not satisfied:

$$u(0,t) = 1 - \cos(t) \ne 0$$

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P&R have this exercise:

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From the solutions manual:

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Hence, we have

$$u(x,t) = 0 \times 1_{t \le x} + [1 - \cos(t - x)]1_{t \ge x}$$

However since $1 = h''(0) \ne f''(0) = 0$, $u(x,t)$ is singular along $x = t$