Well Posed PDEs for Modeling Damped Strings

A large class of well posed PDEs is given by [46]

$${\ddot y} + 2\sum_{k=0}^M q_k \frac{\partial^{2k+1}y}{\partial x^{2k}\partial t} + \sum_{k=0}^N r_k\frac{\partial^{2k}y}{\partial x^{2k}} \protect$$ (G.30)

Thus, to the ideal string wave equation Eq.$\,$(G.1) we add any number of even-order partial derivatives in $x$, plus any number of mixed odd-order partial derivatives in $x$ and $t$, where differentiation with respect to $t$ occurs only once. Because every member of this class of PDEs is only second-order in time, it is guaranteed to be well posed, as shown in §M.2.2.