Making use of a Fan's sub-equation method and the Mittag-Leffler function with aid of the symbolic computation system Maple we construct a family of traveling wave solutions of the Burgers and the KdV equations. The obtained results include periodic, rational and solitonlike solutions.

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In this paper, we investigate the problem of MHD free convection cooling of a low-heat-resistance sheet that moves downwards in a viscous fluid. The basic equations are converted into coupled ordinary differential equations via the similarity transformation, and solved analytically using homotopy analysis method (HAM). The obtained analytical solutions for both of the velocity and the temperature with different values of the Prandtl number Pr and the magnetic parameter M are plotted and discussed in detail.

A fractal advection-dispersion equation and a fractional spacetime advection-dispersion equation have been developed to improve the simulation of groundwater transport in fractured aquifers. The space-time fractional advection-dispersion simulation is limited due to complex algorithms and the computational power required; conversely, the fractal advection-dispersion equation can be solved simply, yet only considers the fractal derivative in space. These limitations lead to combining these methods, creating a fractional and fractal advection-dispersion equation to provide an efficient non-local, in both space and time, modeling tool. The fractional and fractal model has two parameters, fractional order (α) and fractal dimension (β), where simulations are valid for specific combinations. The range of valid combinations reduces with decreasing fractional order and fractal dimension, and a final recommendation of 0.7 ≤ α, β ≤ 1 is made. The fractional and fractal model provides a flexible tool to model anomalous diffusion, where the fractional order controls the breakthrough curve peak, and the fractal dimension controls the position of the peak and tailing effect. These two controls potentially provide tools to improve the representation of anomalous breakthrough curves that cannot be described by the classical model.