Theoretical analysis of a water wave model with a nonlocal viscous dispersive term using the diffusive approach

Abstract

In this paper, we study the following water wave model with a nonlocal viscous term: u_{t}+u_{x}+\beta u_{xxx}+\frac{\sqrt{\nu}}{\sqrt{\pi}}\frac{\partial}{% \partial t}\int_{0}^{t}\frac{u(s)}{\sqrt{t-s}}\,ds+uu_{x}=\nu u_{xx}, where {\frac{1}{\sqrt{\pi}}\frac{\partial}{\partial t}\int_{0}^{t}\frac{u(s)}{\sqrt{% t-s}}\,ds} is the Riemann–Liouville half-order derivative. We prove the well-posedness of this model using diffusive realization of the half-order derivative, and we discuss the asymptotic convergence of the solution. Also, we compare our mathematical results with those given in [5] and [14] for similar equations.

Numerical solutions to a BBM‐Burgers model with a nonlocal viscous term

“…In this article, we have constructed two numerical schemes to approximate the solutions and the decay rates to an asymptotical water wave model where the nonlocal viscous term is described by the Riemann‐Liouville half derivative. We compare our numerical results to those given in . We show that using the diffusive realization of the nonlocal operator supplemented with a splitting scheme leads to a very interesting gain of time computing of the numerical solution.…”

“…We compare between these schemes. Furthermore, we compare our numerical results with those given in .Remark We note that the well‐posedness of the model (6) may be proved mathematically for initial data using the diffusive realization of the half‐order derivative and following the same steps as presented at .…”

“…We compare between these schemes. Furthermore, we compare our numerical results with those given in [1,2,10].…”

“…In addition, in the recent work , the authors succeeded to remove the smallness condition on the initial data in Theorem 1.1. Moreover, they proved the weak convergence to zero of the solution.…”

“…Remark 1 We note that the well-posedness of the model (6) may be proved mathematically for initial data u 0 ∈ L 2 (R) using the diffusive realization of the half-order derivative and following the same steps as presented at [10].…”