The Witten–Kontsevich theorem states that a certain generating function for intersection numbers on the moduli space of stable curves is a tau-function for the KdV integrable hierarchy. This generating function can be recovered via the topological recursion applied to the Airy curve x = 1 2 y 2. In this paper, we consider the topological recursion applied to the irregular spectral curve xy2 = 2...

In this paper the Bagley-Torvik equation as a prototype fractional differential equation with two derivatives is investigated by means of homotopy perturbation method. The results reveal that the present method is very effective and accurate.

Fractional derivative can be defined as a fractional power of derivative. The commutator (i/h̄)[H, . ], which is used in the Heisenberg equation, is a derivation on a set of observables. A derivation is a map that satisfies the Leibnitz rule. In this paper, we consider a fractional derivative on a set of quantum observables as a fractional power of the commutator (i/h̄)[H, . ]. As a result, we ob...

The fractional Boltzmann equation for resonance radiation transport in plasma is proposed. We start from the standard Boltzmann equation, averaging over frequencies leads to appearance of fractional derivative. This fact is in accordance with the conception of latent variables leading to hereditary and non-local dynamics (in particular, fractional dynamics). The presence of the fractional mater...

We present a mathematical analysis of the finite-energy Airy beam with a sharply truncated spectrum, which can be generated by a uniformly illuminated, finite-sized spatial light modulator, or windowed cubic phase mask. The resulting "incomplete Airy beam" is tractable mathematically, and differs from an infinite-energy Airy beam by an additional oscillating modulation and the decay of its frin...

We consider the description of the fractal media that uses the fractional integrals. We derive the fractional generalizations of the equation that defines the medium mass. We prove that the fractional integrals can be used to describe the media with noninteger mass dimensions. The fractional equation of continuity is considered. PACS: 05.45.Df; 47.53.+n; 05.40.-a

We extend the Exp-function method to fractional partial differential equations in the sense of modified Riemann-Liouville derivative based on nonlinear fractional complex transformation. For illustrating the validity of this method, we apply it to the space-time fractional Fokas equation and the nonlinear fractional Sharma-Tasso-Olver (STO) equation. As a result, some new exact solutions for th...

We consider the fractional generalizations of the phase volume, volume element, and Poisson brackets. These generalizations lead us to the fractional analog of the phase space. We consider systems on this fractional phase space and fractional analogs of the Hamilton equations. The fractional generalization of the average value is suggested. The fractional analogs of the Bogoliubov hierarchy equ...

this paper presents a computational method for solving two types of integro-differential equations, system of nonlinear high order volterra-fredholm integro-differential equation(vfides) and nonlinear fractional order integro-differential equations. our tools for this aims is operational matrices of integration and fractional integration. by this method the given problems reduce to solve a syst...

We study a Dirichlet boundary value problem for Langevin equation involving two fractional orders. Langevin equation has been widely used to describe the evolution of physical phenomena in fluctuating environments. However, ordinary Langevin equation does not provide the correct description of the dynamics for systems in complex media. In order to overcome this problem and describe dynamical pr...