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In this article, we investigate the decay rate of the solutions of two water wave models with a nonlocal viscous term written in the KdV form $$u_t+u_x+\beta u_{xxx}+\frac{\sqrt v}{\sqrt \pi}\int^t_0\frac{u_t(s)}{\sqrt{t-s}}ds+uu_x=vu_{xx}$$ and $$u_t+u_x-\beta u_{txx}+\frac{\sqrt v}{\sqrt \pi}\int^t_0\frac{u_t(s)}{\sqrt{t-s}}ds+uu_x=vu_{xx}$$ in the BBM form. In order to realize this numerical study, a numerical scheme based on the $G^{\alpha}$-scheme is developed.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/571.html} }In this article, we investigate the decay rate of the solutions of two water wave models with a nonlocal viscous term written in the KdV form $$u_t+u_x+\beta u_{xxx}+\frac{\sqrt v}{\sqrt \pi}\int^t_0\frac{u_t(s)}{\sqrt{t-s}}ds+uu_x=vu_{xx}$$ and $$u_t+u_x-\beta u_{txx}+\frac{\sqrt v}{\sqrt \pi}\int^t_0\frac{u_t(s)}{\sqrt{t-s}}ds+uu_x=vu_{xx}$$ in the BBM form. In order to realize this numerical study, a numerical scheme based on the $G^{\alpha}$-scheme is developed.

*International Journal of Numerical Analysis and Modeling*.

*10*(2). 333-349. doi: