We applied multiple parameters method
(MPM) to obtain natural frequency of the nonlinear oscillator with rational
restoring force. A frequency analysis is carried out and the relationship
between the angular frequency and the initial amplitude is obtained in
analytical/numerical form. This equation is analyzed in three cases: the
relativistic harmonic oscillator, a mass attached of a stretched elastic wire
and oscillations of a punctual charge in the electric field of charged ring.
The three and four parameters solutions are obtained. The results obtained are
compared with the numerical solution, showing good agreement.

Abstract:
Unsteady flow of an incompressible Maxwell fluid with fractional derivative induced by a sudden moved plate has been studied, where the no-slip assumption between the wall and the fluid is no longer valid. The solutions obtained for the velocity field and shear stress, written in terms of Wright generalized hypergeometric functions , by using discrete Laplace transform of the sequential fractional derivatives, satisfy all imposed initial and boundary conditions. The no-slip contributions, that appeared in the general solutions, as expected, tend to zero when slip parameter is . Furthermore, the solutions for ordinary Maxwell and Newtonian fluids, performing the same motion, are obtained as special cases of general solutions. The solutions for fractional and ordinary Maxwell fluid for no-slip condition also obtained as limiting cases, and they are equivalent to the previously known results. Finally, the influence of the material, slip, and the fractional parameters on the fluid motion as well as a comparison among fractional Maxwell, ordinary Maxwell, and Newtonian fluids is also discussed by graphical illustrations. 1. Introduction There are many fluids in industry and technology whose behavior cannot be explained by the classical linearly viscous Newtonian model. The departure from the Newtonian behavior manifests itself in a variety of ways: non-Newtonian viscosity (shear thinning or shear thickening), stress relaxation, nonlinear creeping, development of normal stress differences, and yield stress [1]. The Navier-Stokes equations are inadequate to predicted the behavior of such type of fluids; therefore, many constitutive relations of non-Newtonian fluids are proposed [2]. These constitutive relations give rise to the differential equations, which, in general, are more complicated and higher order than the Navier-Stokes equations. Therefore, it is difficult to obtain exact analytical solutions for non-Newtonian fluids [3]. Modeling of the equation governing the behaviors of non-Newtonian fluids in different circumstance is important from many points of view. For examples, plastics and polymers are extensively handeled by the chemical industry, whereas biological and biomedical devices like hemodialyser make use of the rheological behavior [4]. In general, the analysis of the behavior of the fluid motion of non-Newtonian fluids tends to be much more complicated and subtle in comparison with that of the Newtonian fluids [5]. The fractional calculus, almost as old as the standard differential and integral one, is increasingly seen as an efficient tool

Abstract:
Using transformation method some exact solutions of equations of motion of a finitely conducting incompressible fluid of variable viscosity in the presence of a transverse magnetic field are determined. These solutions consist of flows for which the vorticity distribution is proportional to the stream function perturbed by a uniform stream. Streamline patterns for some of the solutions are also presented.

Abstract:
The unsteady flows of Burgers’ fluid with fractional derivatives model, through a circular cylinder, is studied by means of the Laplace and finite Hankel transforms. The motion is produced by the cylinder that at the initial moment begins to rotate around its axis with an angular velocity Ωt, and to slide along the same axis with linear velocity Ut. The solutions that have been obtained, presented in series form in terms of the generalized Ga,b,c( , t) functions, satisfy all imposed initial and boundary conditions. Moreover, the corresponding solutions for fractionalized Oldroyd-B, Maxwell and second grade fluids appear as special cases of the present results. Furthermore, the solutions for ordinary Burgers’, Oldroyd-B, Maxwell, second grade and Newtonian performing the same motion, are also obtained as special cases of general solutions by substituting fractional parameters α = β = 1. Finally, the influence of the pertinent parameters on the fluid motion, as well as a comparison among models, is shown by graphical illustrations.

Abstract:
An investigation is presented for the two-dimensional and axisymmetric stagnation flows of a couple stress fluids intrude on a moving plate under partial slip conditions. The governing partial differential equations are converted into ordinary differential equations by a similarity transformation. The important physical parameters of skin friction coefficients of the fluid are also obtained. The homotopy analysis method (HAM) is employed to obtain the analytical solution of the problem. Also, the convergence of the solutions is established by plotting graphs of convergence control parameter. The impacts of couple stresses and slip conditions on the flow and temperature of the fluid have been observed. The numerical comparison for the considered fluid is compared with previous solutions as special case. 1. Introduction The fluids exhibiting a boundary slip are important in industrial applications, for example, the polishing of artificial heart valves, rarefied fluid problems, and flow on multiple interfaces. There are many cases where no slip condition is replaced with Navier’s partial slip condition. Partial slip condition on solid boundary occurs in many problems such as oscillatory flow channel, transient flow, some coated surfaces, some rough or porous surfaces, and heat transfer on moving plate. The flow on a moving plate is termed as a basic content for convection processes. The partial slip condition on a moving plate was considered by Wang [1]; the steady, laminar, axis-symmetric flow of a Newtonian fluid due to a stretching sheet with partial slip was studied by Ariel [2], Nadeem et al. [3] investigated steady state rotating and MHD flow of a third grade fluid past a rigid plate with slip; flow and heat transfer of a non-Newtonian fluid past a stretching sheet with partial slip are considered by Sahoo [4], and Jamil and Khan [5] considered the slip effects on fractional viscoelastic fluids; the steady boundary layer flow past a moving horizontal flat plate with a slip effect is studied by Kumaran and Pop [6]. The theory of couple stresses, introduced by Stokes [7], explain the rheological behavior of various complex non-Newtonian fluids with body stresses and body couples which cannot be illustrated by the classical theory of continuum mechanics. Due to the rotational interaction of particles, the force-stress tensor is not symmetric and flow behaviors of such fluids are not similar to the Newtonian ones. It draws the researcher’s attention with the growing applications of such fluids in engineering, biomedical, and chemical industries. The

Abstract:
We applied a new approach to obtain natural frequency of the nonlinear oscillator with discontinuity. He's Hamiltonian approach is modified for nonlinear oscillator with discontinuity for which the elastic force term is proportional to sgn(u). We employed this method for higher-order approximate solution of the nonlinear oscillator equation. This property is used to obtain approximate frequency-amplitude relationship of a nonlinear oscillator with high accuracy. Many numerical results are given to prove the efficiency of the suggested technique. 1. Introduction The study of nonlinear oscillator problems is of crucial importance not only in all areas of physics but also in engineering and other disciplines. It is of great importance to study analytically nonlinear oscillators to obtain approximate frequency-amplitude relationship because of their wide applications. Traditional perturbation method provides us with a simple approach to the determination of the frequency-amplitude relationship, but the results are valid only for special cases, that is, for weakly nonlinear systems or for the case when the amplitude is very small. In order to overcome the shortcomings arising in traditional perturbation methods, various alternative approaches have been proposed, for example, variational iteration method [1–3], homotopy perturbation method [4–7], Lindstedt-Poincare method [8], variational approach [9, 10], parameter-expanding method [11] and max-min approach [12], harmonic balance method [13], and Hamiltonian approach [14]. In the present study, the mentioned parameters are those undetermined values in the assumed solution. In the three-parameters technique, the motion is assumed as , where , , are the angular frequency of motion and Fourier coefficients, respectively. The three undetermined parameters are determined by using the governing equation of motion and the initial conditions imposed. The way for obtaining the parameters in He’s Hamiltonian technique is quite different from that in the harmonic balance method. Therefore, the present technique is not the same as the harmonic balance method. Finally, the paper provides a lot of higher accurate results for the angular frequency of the motion. 2. Analysis In this paper, we consider a general form of nonlinear oscillator with initial conditions and . The variational principle for (2.1) suggested by He [9] can be written as where is period of the nonlinear oscillator, . In the functional (2.2), is the kinetic energy, so that the functional (2.2) is the least Lagrangian action, from which we can write the

Abstract:
This paper suggests two component homotopy method to solve nonlinear fractional integrodifferential equations, namely, Volterra's population model. Padé approximation was effectively used in this method to capture the essential behavior of solutions for the mathematical model of accumulated effect of toxins on a population living in a closed system. The behavior of the solutions and the effects of different values of fractional-order are indicated graphically. The study outlines significant features of this method as well as sheds some light on advantages of the method over the other. The results show that this method is very efficient, convenient, and can be adapted to fit a larger class of problems. 1. Introduction Ecology is the study of different species in relation to their surroundings, competition for resources within and among the species, and predator-prey [1] relations among them. At times the surroundings may be infected by metabolic actions of the crowd [2, 3]. In all these situations, since time rates of changes of population sizes are concerned, it is natural that the mathematical modeling be given by differential equations or integrodifferential equations. Integrodifferential equations are usually difficult to solve especially analytically, so an effective method is required to analyze the mathematical model which provides solutions conforming to physical reality. Also, fractional-order models are more accurate than integer-order models, that is, there are more degrees of freedom in the fractional-order models. Furthermore, fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes due to the existence of a “memory” term in the model. This memory term insures the history and its impact to the present and future. Problems of this type have gained escalating importance in recent years and many interesting outcomes have been accumulated. In the living thing population, the accumulation of metabolic products may cause inconvenience to the whole population and may ultimately result in a fall of the birth rate while the death rate is increased. We assumed that the total toxic effect on birth and death rates be expressed by the following nonlinear fractional-order integrodifferential equation [4]: where is the birth rate coefficient, is the crowding coefficient, and is the toxicity coefficient. The coefficient indicates the essential behavior of the population evolution before its level falls to zero in the long term, is the initial population, denotes the

Abstract:
The present work is devoted to study the numerical simulation for unsteady MHD flow and heat transfer of a couple stress fluid over a rotating disk. A similarity transformation is employed to reduce the time dependent system of nonlinear partial differential equations (PDEs) to ordinary differential equations (ODEs). The Runge-Kutta method and shooting technique are employed for finding the numerical solution of the governing system. The influences of governing parameters viz. unsteadiness parameter, couple stress and various physical parameters on velocity, temperature and pressure profiles are analyzed graphically and discussed in detail.