Abstract:
We consider a class of discontinuous Liénard systems and study the number of limit cycles bifurcated from the origin when parameters vary. We establish a method of studying cyclicity of the system at the origin. As an application, we discuss some discontinuous Liénard systems of special form and study the cyclicity near the origin.

Abstract:
A homotopy perturbation transformation method (HPTM) which is based on homotopy perturbation method and Laplace transform is first applied to solve the approximate solution of the fractional nonlinear equations. The nonlinear terms can be easily handled by the use of He's polynomials. Illustrative examples are included to demonstrate the high accuracy and fastconvergence of this new algorithm.

Abstract:
A variational homotopy perturbation method (VHPM) which is based onvariational iteration method and homotopy perturbation method is applied to solve the approximatesolution of the fractional initial boundary value problems. The nonlinear terms can be easily handledby the use of He's polynomials. It is observed that the variational iteration method is very efficient andeasier to implements; illustrative examples are included to demonstrate the high accuracy and fastconvergence of this new algorithm.

Abstract:
Combination frequencies are observed in the Fourier spectra of pulsating DA and DB white dwarfs. They appear at sums and differences of frequencies associated with the stellar gravity-modes. Brickhill (1992) proposed that the combination frequencies result from mixing of the eigenmode signals as the surface convection zone varying in depth when undergoing pulsation. This depth changes cause time-dependent thermal impedance, which mix different harmonic frequencies in the light curve. Following Brickhill's proposal, we developed analytical expressions to describe the amplitudes and phases of these combination frequencies. The parameters that appear in these expressions are: the depth of the stellar convection zone when at rest, the sensitivity of this depth towards changes in stellar effective temperature, the inclination angle of the stellar pulsation axis with respect to the line of sight, and lastly, the spherical degrees of the eigenmodes involved in the mixing. Adopting reasonable values for these parameters, we apply our expressions to a DA and a DB variable white dwarf. We find reasonable agreement between theory and observation, though some discrepancies remain unexplained. We show that it is possible to identify the spherical degrees of the pulsation modes using the combination frequencies.

Abstract:
In this paper we prove the existence and uniqueness of positive classical solution of the fractional Laplacian with singular nonlinearity in a smooth bounded domain with zero Drichlet boundary conditions. By the method of sub-supersolution, we derive the existence of positive classical solution to the approximation problems. In order to obtain the regularity, we first establish the existence of weak solution for the fraction Laplacian. Thanks to \cite{XY}, the regularity follows from the boundedness of weak solution.

Abstract:
We implement relatively new analytical technique, the Homotopy perturbation method, for solving nonlinear fractional partial differential equations arising in predator-prey biological population dynamics system. Numerical solutions are given, and some properties exhibit biologically reasonable dependence on the parameter values. And the fractional derivatives are described in the Caputo sense.

Abstract:
Location problems exist extensively in the real world and they mainly deal with finding optimal locations for facilities. However, the reverse location problem is also often met in practice, in which the facilities may already exist in a network and cannot be moved to a new place, the task is to improve the network within a given budget such that the improved network works as efficient as possible. This paper is dedicated to the problem of how to use a limited budget to modify the lengths of the edges on a cycle such that the overall sum of the weighted distances of the vertices to the respective closest facility of two prespecified vertices becomes as small as possible (shortly, R2MC problem). It has already been shown that the reverse 2-median problem with edge length modification on general graphs is strongly NP-hard. In this paper, we transform the R2MC problem to a reverse 3-median problem on a path and show that this problem can be solved efficiently by strongly polynomial algorithm.