1 Correction to: J Sci Comput (2018) 75:1721–1756 https://doi.org/10.1007/s10915-017-0605-6

The author would like to correct an error in calculation of the Eq. (A.2.8) in Appendix A.2 of the original article.

The correct equation is as follows:

$$ \begin{aligned} {\text{U}}_{xt}^{(4)} \left( b \right)(x,t) & = \frac{{\partial^{2} }}{\partial x\partial t}{\text{U}}_{{}}^{(4)} \left( b \right)(x,t) = \frac{\partial }{\partial t}\left\{ {{\text{U}}_{x}^{\left( 4 \right)} \left( b \right)} \right\} \\ & = \frac{1}{\pi }\frac{\partial }{\partial t}\int_{0}^{\infty } {\int_{0}^{t} {\int_{ - \infty }^{\infty } {\frac{k}{{1 + h_{0}^{2} k^{2} /3}} \cdot \cos \left[ {\omega_{\text{B}} \left( {t - \tau } \right)} \right] \cdot \sin \left[ {k\left( {\xi - x} \right)} \right] \cdot b(\xi ,\tau )d\xi d\tau dk} } } \\ & = \frac{1}{\pi }\int_{0}^{\infty } {\int_{0}^{t} {\int_{ - \infty }^{\infty } {\frac{k}{{1 + h_{0}^{2} k^{2} /3}} \cdot \frac{\partial }{\partial t}\cos \left[ {\omega_{\text{B}} \left( {t - \tau } \right)} \right] \cdot \sin \left[ {k\left( {\xi - x} \right)} \right] \cdot b\left( {\xi ,\tau } \right)d\xi d\tau dk} } } \\ & \quad + \frac{1}{\pi }\int_{0}^{\infty } {\left[ {\left\{ {\int_{ - \infty }^{\infty } {\frac{k}{{1 + h_{0}^{2} k^{2} /3}} \cdot \cos \left[ {\omega_{\text{B}} \left( {t - t} \right)} \right] \cdot \sin \left[ {k\left( {\xi - x} \right)} \right] \cdot b\left( {\xi ,\tau = t} \right)d\xi } } \right\}\frac{dt}{dt}} \right]dk} \\ & = - \frac{1}{\pi }\int_{0}^{t} {\int_{0}^{\infty } {\int_{ - \infty }^{\infty } {\frac{{k\omega_{\text{B}} }}{{1 + h_{0}^{2} k^{2} /3}} \cdot \sin \left[ {\omega_{\text{B}} \left( {t - \tau } \right)} \right] \cdot \sin \left[ {k\left( {\xi - x} \right)} \right] \cdot b(\xi ,\tau )d\xi dkd\tau } } } \\ & \quad + \frac{1}{\pi }\int_{0}^{\infty } {\int_{ - \infty }^{\infty } {\frac{k}{{1 + h_{0}^{2} k^{2} /3}} \cdot \sin \left[ {k\left( {\xi - x} \right)} \right] \cdot b(\xi ,t)d\xi dk} } . \\ \end{aligned} $$
(A.2.8)