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.

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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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