In this study, some new hypotheses and techniques are presented to obtain some new analytical solutions to the generalized Kawahara equation. As a particular case, some traveling wave solutions to both Kawahara equation and modified Kawahara equation are derived in detail. Periodic and soliton solutions to this family are obtained. The periodic solutions are expressed in terms of Weierstrass elliptic functions and Jacobian elliptic functions. For KE, some direct and indirect approaches are carried out to derive the periodic and localized (...) solutions. For mKE, two different hypotheses in the form of WSEFs are used to derive the periodic and localized solutions. Also, the cnoidal wave solutions in the form of JEFs are obtained. As a realistic physical application, the solutions obtained can be dedicated to studying many nonlinear waves that propagate in plasma. (shrink)

The nonlinear differential equation governing the periodic motion of the one-dimensional, undamped, and unforced cubic-quintic Duffing oscillator is solved exactly by obtaining the period and the solution. The period is given in terms of the complete elliptic integral of the first kind and the solution involves Jacobian elliptic functions. We solve the cubic-quintic Duffing equation under arbitrary initial conditions. Physical applications are provided. The solution to the mixed parity Duffing oscillator is also formally derived. We illustrate the obtained results with (...) concrete examples. We give high accurate trigonometric approximations to the Jacobian function cn. (shrink)

In this paper, some novel analytical and numerical techniques are introduced for solving and analyzing nonlinear second-order ordinary differential equations that are associated to some strongly nonlinear oscillators such as a quadratically damped pendulum equation. Two different analytical approximations are obtained: for the first approximation, the ansatz method with the help of Chebyshev approximate polynomial is employed to derive an approximation in the form of trigonometric functions. For the second analytical approximation, a novel hybrid homotopy with Krylov–Bogoliubov–Mitropolsky method is introduced (...) for the first time for analyzing the evolution equation. For the numerical approximation, both the finite difference method and Galerkin method are presented for analyzing the strong nonlinear quadratically damped pendulum equation that arises in real life, such as nonlinear phenomena in plasma physics, engineering, and so on. Several examples are discussed and compared to the Runge–Kutta numerical approximation to investigate and examine the accuracy of the obtained approximations. Moreover, the accuracy of all obtained approximations is checked by estimating the maximum residual and distance errors. (shrink)