The benefit of finding particular exact and asymptotic solutions of non-integrable non-linear partial differential equations is considered. It is shown how explicit analytical solutions predict important features of the waves behavior even outside their formal applicability. In particular, an arbitrary initial pulse splits in a numerical solution into the train of localized waves each described...

(Received July 23, 2013; in nal form February 26, 2014) In this paper, the (G′/G, 1/G) and (1/G′)-expansion methods with the aid of Maple are used to obtain new exact traveling wave solutions of the Boussinesq equation and the system of variant Boussinesq equations. The travelling wave solutions are expressed by the hyperbolic functions, the trigonometric functions, and the rational functions. ...

In this paper, we study the nonlinear stability of travelling wave solutions with shock profile for a relaxation model with a nonconvex flux, which is proposed by Jin and Xin [Comm. Pure Appl. Math., 48 (1995), pp. 555–563] to approximate an original hyperbolic system numerically under the subcharacteristic condition introduced by T. P. Liu [Comm. Math. Phys., 108 (1987), pp. 153–175]. The trav...

Age-structured epidemic models have been used to describe either the age of individuals or the age of infection of certain diseases and to determine how these characteristics affect the outcomes and consequences of epidemiological processes. Most results on age-structured epidemicmodels focus on the existence, uniqueness, and convergence to disease equilibria of solutions. In this paper we inve...

Nonlinear operations such as multiplication of distributions are not allowed in the classical theory of distributions. As a result, some ambiguities arise when we want to solve nonlinear partial differential equations such as differential equations of elasticity andmultifluid flows, or some new cosmological models such as signature changing space-times. Colombeau’s new theory of generalized fun...

An e cient way of obtaining travelling waves in a periodic uid system is described and tested. We search for steady states in a reference frame travelling at the wave phase velocity using a rst-order pseudospectral semi-implicit time scheme adapted to carry out the Newton’s iterations. The method is compared to a standard Newton–Raphson solver and is shown to be highly e cient in performing thi...

The stability of travelling wave solutions of singularly perturbed, diffusive predator-prey systems is proved by showing that the linearized operator about such a solution has no unstable spectrum and that the translation eigenvalue at k 0 is simple. The proof illustrates the application of some recently developed geometric and topological methods for counting eigenvalues.

Nonlinear partial differential equations are more suitable to model many physical phenomena in science and engineering. In this paper, we consider three nonlinear partial differential equations such as Novikov equation, an equation for surface water waves and the Geng-Xue coupled equation which serves as a model for the unidirectional propagation of the shallow water waves over a flat bottom. T...

A similarity analysis of a nonlinear fin equation has been carried out by M. Pakdemirli and A.Z. Sahin [Similarity analysis of a nonlinear fin equation, Appl. Math. Lett. (2005) (in press)]. Here, we consider a further group theoretic analysis that leads to an alternative set of exact solutions or reduced equations with an emphasis on travelling wave solutions, steady state type solutions and s...

and Applied Analysis 3 Two papers in our special issue are concerned about the study of application to the symbolic computation method and special function. In the paper titled “New nonlinear systems admitting Virasoro-type symmetry algebra and groupinvariant solutions” by L. Wang et al., the authors extend the application of Virasoro-type symmetry prolongationmethod to coupled systems with two...