Volume 18, Issue 2 -

In this paper, the Kuramoto-Sivashinsky equation is solved numerically by implementing a new differential quadrature technique that uses quintic B-spline as the basis functions for space integration. The derivatives are approximated using differential quadrature method. The weighting coefficients are obtained by semi-explicit algorithm including an algebraic system with penta-diagonal coefficient matrix that is solved using the five-band Thomas algorithm. Stability analysis of method has also been done. The accuracy of the proposed scheme is demonstrated by applying on five test problems. Some theoretical properties of KS equation like periodicity, monotonicity and dissipativity etc. have also been discussed. The results are also shown graphically to demonstrate the accuracy and capabilities of this method and comparative study is done with results available in literature. The computed results are found to be in good agreement with the analytical solutions.

In order to make sense of the dynamic response of brittle materials under the certain range of impact strength, the numerical simulation for two kinds of representative ones glass and ceramic are conducted, in which the elastic micro-crack damage model is used. The plane impact experiments of ceramic and glass are summarized, which are used to compare with the simulation results. The simulation results show that the dynamic responses of brittle materials, failure wave and the plastic-like response appeared in glass and ceramic respectively are depended on their micro-cracks distribution in meso-scale. And moreover, both of failure wave and the plastic-like response are controlled by the same mechanism, and the different phenomena are just influenced by the size and distribution of the micro-cracks.

The Harry Dym hierarchy is derived with the help of Lenard recursion equations and zero curvature equation. Based on the Lax matrix, an algebraic curve Kn$\mathcal{K}_{n}$ of arithmetic genus n$n$ is introduced, from which the corresponding meromorphic function ϕ$\phi$ and Dubrovin-type equations are given. Further, the divisor and asymptotic properties of ϕ$\phi$ are studied. Finally, algebro-geometric solutions for the entire hierarchy are obtained according to above results and the theory of algebraic curve.

By employing the generalized (G′/G)$(G'/G)$-expansion method and symbolic computation, we obtain new exact solutions of the (3 + 1) dimensional generalized B-type Kadomtsev–Petviashvili equation, which include the traveling wave exact solutions and the non-traveling wave exact solutions showed by the hyperbolic function and the trigonometric function. Meanwhile, some interesting physics structure are discussed.

The nonlinear vibrations of composite laminated plates with time-dependent base excitation and boundary conditions are investigated. According to the von Kármán nonlinear plate theory, the dynamic equations of motion of the laminated plates are established. The nonlinear partial differential equations are transformed to the nonlinear ordinary differential ones using the Bubnov-Galerkin’s  method. The primary resonance and the primary parametric resonance of the laminated plate with time-dependent boundary conditions are investigated by means of the method of multiple scales. The validity of the present theoretical method is verified by comparing the amplitude–frequency relationship curves acquired from the present theoretical method with those calculated from the numerical simulation. The amplitude–frequency characteristic curves and the displacement time histories for different ply angles of the composite laminated plate are analyzed. The effects of the viscous damping factor and the transverse displacement excitation on the amplitude–frequency relationship curves are also studied. The present results are helpful for the nonlinear dynamical analysis and design of the composite laminated plate with time-dependent boundary conditions.

We study the properties of an ODE description of schools of fish (B. Birnir, An ODE model of the motion of pelagic fish, J. Stat. Phys. 128 (1/2) (2007), 535–568.) and how they change in the presence of a random acceleration. The model can be reduced to one ODE for the direction of the velocity of a generic fish and another ODE for its speed. These equations contain the mean speed vˉ$\bar{v}$ and a Kuramoto order parameter r$r$ for the phases of the fish velocities. In this paper, we give a complete qualitative analysis of the system for large number of particles. We show that the stationary solutions of the ODEs consist of an incoherent unstable solution with r=vˉ=0$r\!\!=\!\!\bar{v}\!\!=\!\!0$ and a globally stable solution with r=1$r\!=\!1$ and a constant vˉ>0$\bar{v}\!>\!0$. In the latter solution, all the fish move uniformly in the same direction with vˉ$\bar{v}$ and the direction of motion determined by the initial configuration of the school. This is called the “migratory solution”. In the second part of the paper, the directional headings of the particles are perturbed, in two distinct ways, and the speeds accelerated in order to obtain two distinct classes of non-stationary, complex solutions. We show that the perturbed systems have similar behavior as the unperturbed one, and derive the resulting constant value of the average speed, verifying the numerical observations. Finally, we show that the system exhibits a similar bifurcation to that in Vicsek and Czirok (T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett. 75 (6) (Aug 1995), 1226–1229.) between phases of synchronization and disorder. Either increasing the variance of the Brownian angular noise, or decreasing the turning rate, or coupling between the particles, cause a similar phase transition. These perturbed models represent a more realistic view of schools of fish found in nature. We apply the theory to compute the order parameter for a simulation of the Chile-Peru anchovy fishery.