Extending the D'Alembert Solution to Space-Time Modified Riemann-Liouville Fractional Wave Equations

In the realm of complexity, it is argued that adequate modeling of TeV-physics demands an approach based on fractal operators and fractional calculus (FC). Non-local theories and memory effects are connected to complexity and the FC. The non-differentiable nature of the microscopic dynamics may be connected with time scales. Based on the Modified Riemann-Liouville definition of fractional derivatives, we have worked out explicit solutions to a fractional wave equation with suitable initial conditions to carefully understand the time evolution of classical fields with a fractional dynamics. First, by considering space-time partial fractional derivatives of the same order in time and space, a generalized fractional D'Alembertian is introduced and by means of a transformation of variables to light-cone coordinates, an explicit analytical solution is obtained. To address the situation of different orders in the time and space derivatives, we adopt different approaches, as it will become clear throughout the paper. Aspects connected to Lorentz symmetry are analyzed in both approaches.Comment: 8 page

Fractional Subequation Method for Cahn-Hilliard and Klein-Gordon Equations

The fractional subequation method is applied to solve Cahn-Hilliard and Klein-Gordon equations of fractional order. The accuracy and efficiency of the scheme are discussed for these illustrative examples

Two Reliable Methods for Solving the (3 + 1)-Dimensional Space-Time Fractional Jimbo-Miwa Equation

We investigate methods for obtaining exact solutions of the (3 + 1)-dimensional nonlinear space-time fractional Jimbo-Miwa equation in the sense of the modified Riemann-Liouville derivative. The methods employed to analytically solve the equation are the G′/G,1/G-expansion method and the novel G′/G-expansion method. To the best of our knowledge, there are no researchers who have applied these methods to obtain exact solutions of the equation. The application of the methods is simple, elegant, efficient, and trustworthy. In particular, applying the novel G′/G-expansion method to the equation, we obtain more exact solutions than using other existing methods such as the G′/G-expansion method and the exp-Φ(ξ)-expansion method. The exact solutions of the equation, obtained using the two methods, can be categorized in terms of hyperbolic, trigonometric, and rational functions. Some of the results obtained by the two methods are new and reported here for the first time. In addition, the obtained exact explicit solutions of the equation characterize many physical meanings such as soliton solitary wave solutions, periodic wave solutions, and singular multiple-soliton solutions