This paper applies the Polynomial Least Squares Method (PLSM) to the case of fractional Lane-Emden differential equations. PLSM offers an analytical approximate polynomial solution in a straightforward way. A comparison with previously obtained results proves how accurate the method is.

In this paper a new method called the least squares differential quadrature method (LSDQM) is introduced as a straightforward and efficient method to compute analytical approximate polynomial solutions for nonlinear partial differential equations with fractional time derivatives. LSDQM is a combination of the differential quadrature method and the least squares method and in this paper it is employed to find approximate solutions for a very general class of nonlinear partial differential equations, wherein the fractional derivatives are described in the Caputo sense. The paper contains a clear, step-by-step presentation of the method and a convergence theorem. In order to emphasize the accuracy of LSDQM we included two test problems previously solved by means of other, well-known methods, and observed that our solutions present not only a smaller error but also a much simpler expression. We also included a problem with no known exact solution and the solutions computed by LSDQM are in good agreement with previous ones.

This work emphasizes some geometrical properties of the Maxwell--Bloch equations. Based on these properties the closed-form solutions of their equations are established. Thus, the Maxwell-Bloch equations are reduced to a nonlinear differential equation depending on an auxiliary unknown function. The approximate analytical solutions were built using the Optimal Homotopy Asymptotic Method (OHAM). A good agreement between the analytical and corresponding numerical results was found. The accuracy of the obtained results is validated through the representative figures. This procedure could be successfully applied for more dynamical systems with geometrical properties.

This paper emphasizes some geometrical properties of the Maxwell–Bloch equations. Based on these properties, the closed-form solutions of their equations are established. Thus, the Maxwell–Bloch equations are reduced to a nonlinear differential equation depending on an auxiliary unknown function. The approximate analytical solutions were built using the optimal homotopy asymptotic method (OHAM). These represent the $\epsilon$-approximate OHAM solutions. A good agreement between the analytical and corresponding numerical results was found. The accuracy of the obtained results is validated through the representative figures. This procedure is suitable to be applied for dynamical systems with certain geometrical properties.

In this paper we apply the recently introduced Polynomial Least Squares Method (PLSM) to compute approximate analyticalpolynomial solutions for the Bagley-Torvik fractional equation with boundary conditions.The Bagley—Torvik equation may be used to model the motion of real physical systems such as the motion of a thin rigid plate immersed in a Newtonian fluid. In order to emphasize the accuracy of PLSM, we included a comparison with previous approximate solutions obtained for the Bagley-Torvik fractional equation by means of other approximation method.

Based on some geometrical properties (symmetries and global analytic first integrals) of the Rabinovich system the closed-form solutions of the equations have been established. The chaotic behaviors are excepted. Moreover, the Rabinovich system is reduced to a nonlinear differential equation depending on an auxiliary unknown function. The approximate analytical solutions are built using the Optimal Auxiliary Functions Method (OAFM). The advantage of this method is to obtain accurate solutions for special cases, with only an analytic first integral. An important output is the existence of complex eigenvalues, depending on the initial conditions and physical parameters of the system. This approach was not still analytically emphasized from our knowledge. A good agreement between the analytical and corresponding numerical results has been performed. The accuracy of the obtained results emphasizes that this procedure could be successfully applied to more dynamic systems with these geometrical properties.