A semi-discrete numerical method for convolution-type unidirectional wave equations

Numerical approximation of a general class nonlinear unidirectional wave equations with convolution-type nonlocality in space is considered. A semi-discrete numerical method based on both uniform discretization and the discrete convolution operator introduced to solve Cauchy problem. The proved be uniformly convergent as mesh size goes zero. order convergence for error linear or quadratic depending smoothness kernel. problem defined whole spatial domain then truncated finite domain. Restricting introduces localization it that this stays below given threshold if large enough. For two particular kernel functions, examples concerning solitary solutions illustrate expected accuracy method. Our nonlocal includes Benjamin–Bona–Mahony equation special case present work inspired by previous Bona, Pritchard Scott solution equation.