I have a wave equation with fixed interval with the following initial and boundary conditions.
$$u_{tt} = u_{xx}$$ $$u(x, 0) = \phi(x) = x(1-x)$$ $$u_t(x, 0) = x^3 - x$$ $$u(0, t) = u(1, t) = 0$$
With $c = 1$ and $l = 1$, we can apply the general solutions stated by Walter A. Strauss in his Book "Partial Differential Equations: And Introduction"( 2nd edition, 2009. Chapter 3.2, "The Finite Interval" section, p.63 - p.65 ):
$$u(x, t) = \frac{1}{2} \phi(x - ct + 2l) - \frac{1}{2} \phi(4 - x -ct) +\int_{x - ct + 2l} ^{4 - x -ct} \psi(s) \frac{ds}{2}$$.
By substituting $c = 1$ and $l = 1$, I get that
$$\frac{1}{2} \phi(x - ct + 2l) - \frac{1}{2} \phi(4 - x -ct) = 2tx - 2t - 5x + 5$$
and that:
$$\int_{x - ct + 2l} ^{4 - x -ct} \psi(s) \frac{ds}{2} = \frac{1}{2} \left( -\frac{(4 - t - x)^2}{2} + \frac{(4 - t - x)^4}{4} + \frac{(2 - t + x)^2}{2} - \frac{(2 - t + x)^4}{4} \right)$$
If I sum up both parts I get $u(x, t)$, but, $u(0, t) \ne u(1, t)$, which contradicts the initial and boundary conditions. What am I doing wrong? Is it incorrect to apply this solution to this case?