The main aim of this article is to use a new and simple algorithm namely the modified variational iteration algorithm-II (MVIA-II) to obtain numerical solutions of different types of fifth-order Korteweg-de Vries (KdV) equations. In order to assess the precision, stability and accuracy of the solutions, five test problems are offered for different types of fifth-order KdV equations. Numerical results are compared with the Adomian decomposition method, Laplace decomposition method, modified Adomian decomposition method and the homotopy perturbation transform method, which reveals that the MVIA-II exceptionally productive, computationally attractive and has more accuracy than the others.

Snow is of porous structure and good thermal insulation property. A fractal derivative model is established to reveal its thermal property, it is extremely high thermal-stable, the whole snow will not be affected much by the sudden environmental temperature change. A simple experiment is carried out to verify the theoretical finding, and the result is helpful to design advanced materials mimicking the snow structure.

He’s multiple scales method is a couple of the homotopy perturbation method and the multiple scales technology in the classic perturbation method. This method has been proved to be a powerful mathematical tool to various nonlinear equations, and it is extremely effective for forced nonlinear oscillators. This paper shows that the method can be further improved by incorporating some known technologies, e.g., the parameter-expanding technology, the enhanced perturbation method and the homotopy perturbation method with an auxiliary term. Due to the wide application of the homotopy perturbation method, He’s multiple scales method cleans solutions of nonlinear equations while the classic perturbation method fails.

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The study of nonlinear phenomena associated with physical phenomena is a hot topic in the present era. The fundamental aim of this paper is to find the iterative solution for generalized quintic complex Ginzburg–Landau (GCGL) equation using fractional natural decomposition method (FNDM) within the frame of fractional calculus. We consider the projected equations by incorporating the Caputo fractional operator and investigate two examples for different initial values to present the efficiency and applicability of the FNDM. We presented the nature of the obtained results defined in three distinct cases and illustrated with the help of surfaces and contour plots for the particular value with respect to fractional order. Moreover, to present the accuracy and capture the nature of the obtained results, we present plots with different fractional order, and these plots show the essence of incorporating the fractional concept into the system exemplifying nonlinear complex phenomena. The present investigation confirms the efficiency and applicability of the considered method and fractional operators while analyzing phenomena in science and technology.

In this paper, a fractional Zakharov-Kuznetsov equation with He's fractional derivative is studied by the fractional complex transform and He's homotopy perturbation method. The solution process is elucidated step by step to show its simplicity and effectiveness of the proposed method.

The homotopy perturbation method is extended to solve nonlinear oscillators with damping terms, and an explicit relationship between the frequency and amplitude is obtained, the main factor affecting the periodic property is revealed.