Solitary wave solutions of two nonlinear physical models by tanh–coth method

In this work, we established the exact solutions for some nonlinear physical models. The tanh–coth method was used to construct solitary wave solutions of nonlinear evolution equations. The tanh–coth method presents a wider applicability for handling nonlinear wave equations.     

Elastoplastic deformations of rotating parabolic solid disks using Tresca''s yield criterion

Analytical solutions for the stress distribution in rotating parabolic solid disks are obtained. The analysis is based on Tresca's yield criterion, its associated flow rule and linear strain hardening. It is shown that, the deformation behavior of the convex parabolic disk is similar to that of the uniform thickness disk, but in the case of concave parabolic solid disk, it is different. In the latter, the plastic core consists of three different plastic regions with different mathematical forms of the yield criteria. Accordingly, three different stages of elastic–plastic deformation occur. All these stages of elastic–plastic deformation are studied in detail. It is also shown mathematically that in the limiting case the parabolic disk solution reduces to the solution of rotating uniform thickness solid disk.     

New Solutions for (1+1)-Dimensional and (2+1)-Dimensional Kaup–Kupershmidt Equations

In this paper, using the exp-function method we obtain some new exact solutions for (1+1)-dimensional and (2+1)-dimensional Kaup–Kupershmidt (KK) equations. We show figures of some of the new solutions obtained here. We conclude that the exp-function method presents a wider applicability for handling nonlinear partial differential equations.     

Diffraction of sound waves by a rigid cylindrical cavity of finite length with an internal impedance surface

A rigorous solution is presented for the problem of diffraction of plane harmonic sound waves by a cavity formed by a terminated rigid cylindrical waveguide of finite length whose interior surface is lined by an acoustically absorbent material. The solution is obtained by a modification of the Wiener-Hopf technique and involve an infinite series of unknowns, which are determined from an infinite system of linear algebraic equations. Numerical solution of this system is obtained for various values of the parameters of the problem and their effects on the diffraction phenomenon are shown graphically.Received: December 12, 2001     

MHD mixed convection of a viscous dissipating fluid about a permeable vertical flat plate

The problem of steady laminar magnetohydrodynamic (MHD) mixed convection heat transfer about a vertical plate is studied numerically, taking into account the effects of Ohmic heating and viscous dissipation. A uniform magnetic field is applied perpendicular to the plate. The resulting governing equations are transformed into the non-similar boundary layer equations and solved using the Keller box method. Both the aiding-buoyancy mode and the opposing-buoyancy mode of the mixed convection are examined. The velocity and temperature profiles as well as the local skin friction and local heat transfer parameters are determined for different values of the governing parameters, mainly the magnetic parameter, the Richardson number, the Eckert number and the suction/injection parameter, f_w. For some specific values of the governing parameters, the results agree very well with those available in the literature. Generally, it is determined that the local skin friction coefficient and the local heat transfer coefficient increase owing to suction of fluid, increasing the Richardson number, Ri (i.e. the mixed convection parameter) or decreasing the Eckert number. This trend reverses for blowing of fluid and decreasing the Richardson number or decreasing the Eckert number. It is disclosed that the value of Ri determines the effect of the magnetic parameter on the momentum and heat transfer.     

He's homotopy perturbation method for solving the shock wave equation

In this article, we discuss the analytic solution of the fully developed shock waves. The homotopy perturbation method is used to solve the shock wave equation, which describes the flow of gases. Unlike the various numerical techniques, which are usually valid for short period of time, the solution of the presented equation is analytic for 0 < t < ∞. The results presented converge very rapidly, indicating that the method is reliable and accurate.     

Relaxed Hyperelastic Curves on a Non-Degenerate Surface

We study the variational problem belonging to a relaxed hyperelastic curve for non-null curve on a non-degenerate surface in Minkowski three-space \({E_{1}^{3}}\) . Firstly, we derive the intrinsic equations for a relaxed hyperelastic curve and we give the necessary condition for being relaxed hyperelastic curve of any non-null geodesic on the surface in \({E_{1}^{3}}\) . Then, we examine this formulation on non-null geodesics of pseudo-plane, pseudo-sphere \({S_{1}^{2}(r) }\) , hyperbolic space \({H_{0}^{2}(r)}\) and pseudo-cylinder \({C_{1}^{2}(r)}\) .     

Moderate deviations for martingale differences and applications to φ -mixing sequences

For a R d -valued sequence of martingale differences { m k } k S 1 , we obtain a moderate deviation principle for the sequence of partial sums { Z n ( t ) 1 ~ k =1 [ nt ] m k / b n , t ] [0,1]}, in the space of càdlàg functions equipped with the Skorohod topology, under the following conditions: a Chen-Ledoux type condition, an exponential convergence in probability of the associated quadratic variation process of the martingale, and a condition of "Lindeberg" type. For the small jumps of Z n (·), we apply the general result of Puhalskii [Puhalskii, A. (1994). "Large deviations of semimartingales via convergence of the predictable characteristics". Stoch. Stoch. Rep. , 49 , pp. 27-85]. Following the method of Ledoux [Ledoux, M. (1992). "Sur les déviations modérées des sommes de variables aléatoires vectorielles indépendantes de même loi". Ann. Inst. H. Poincaré , 28 , pp. 267-280] and Arcones [Arcones, A. (1999). "The large deviation principle for stochastic processes", Submitted for publication], we prove that the large jumps part of Z n (·) is negligible in the sense of the moderate deviations. One can regard our result as an extension to martingale differences, of the beautiful characterization of moderate deviations for i.i.d.r.v. case due to Chen [Chen, X. (1991). "The moderate deviations of independent vectors in Banach space". Chin. J. Appl. Probab. Stat. , 7 , pp. 124-32] and Ledoux [Ledoux, M. (1992). "Sur les déviations modérées des sommes de variables aléatoires vectorielles indépendantes de même loi". Ann. Inst. H. Poincaré , 28 , pp. 267-280]. Using the Gordin [Gordin, M.I. (1969). "The central limit theorem for stationary processes". Soviet Math. Dokl. , 10 , pp. 1174-1176] decomposition, the martingale result is applied to prove the moderate deviation principle for a wide class of stationary { -mixing sequences of random variables.