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In this study, we present the idea of variational fractal transform method to find the analytical solution for the fractal Kawahara equation that arises in shallow water waves in the microgravity space or with the unsmooth boundary. The fractal semi-inverse method is used to establish a fractal variational formulation of the problem, which provides the analysis of the soliton solution and its physical properties. An example is considered to show that this new scheme is straightforward and efficient, and can be generalized to other nonlinear evolution equations with fractal derivatives performance. [ABSTRACT FROM AUTHOR]

Under a microgravity condition, the microgravity-induced motion of a rigid pendulum has many special properties, which cannot be successfully revealed by any known theory. This work suggests a new fractal model for this purpose to describe the microgravity-induced motion by using the fractal derivative, and its variational principle is successfully established according to the semi-inverse method. The variational principle is very helpful to construct conservation laws and to suggest solution structures of the new fractal model. The approximate analytical solution of the model is gained by the two-scale transform method and the homotopy perturbation method. [ABSTRACT FROM AUTHOR]

In this paper, the non-linear local fractional Bratu-type equation is described by the local fractional derivative in a fractal space, and its variational formulation is successfully established according to semi-inverse transform method. Finally, we find the approximate analytical solution of the local fractional Bratu-type equation by using Adomina decomposition method. [ABSTRACT FROM AUTHOR]

In this paper, a new solution process of (1 / G ′) -expansion and (G ′ / G , 1 / G) -expansion methods has been proposed for the analytic solution of the Zhiber-Shabat (Z-S) equation. Rather than the classical (G ′ / G , 1 / G) -expansion method, a solution function in different formats has been produced with the help of the proposed process. New complex rational, hyperbolic, rational and trigonometric types solutions of the Z-S equation have been constructed. By giving arbitrary values to the constants in the obtained solutions, it can help to add physical meaning to the traveling wave solutions, whereas traveling wave has an important place in applied sciences and illuminates many physical phenomena. 3D, 2D and contour graphs are displayed to show the stationary wave or the state of the wave at any moment with the values given to these constants. Conditions that guarantee the existence of traveling wave solutions are given. Comparison of (G ′ / G , 1 / G) -expansion method and (1 / G ′) -expansion method, which are important instruments in the analytical solution, has been made. In addition, the advantages and disadvantages of these two methods have been discussed. These methods are reliable and efficient methods to obtain analytic solutions of nonlinear evolution equations (NLEEs). [ABSTRACT FROM AUTHOR]