Related papers: The Symmetric Stable L\'{e}vy Flights and the Feyn…
We describe a formal procedure to obtain and specify the general form of a marginal distribution for the Lagrangian acceleration of fluid particle in developed turbulent flow using Langevin type equation and the assumption that velocity…
The time-fractional diffusion equation is considered, where the time derivative is either of Caputo or Riemann-Liouville type. The solution of a general initial-boundary value problem with time-dependent boundary conditions over bounded and…
We show that, for a class of systems described by a Lagrangian L(x,\dot{x},t) = 1/2\dot{x}^{2} - V(x,t) the propagator can be reduced via Noether's Theorem to a standard path integral multiplied by a phase factor. Using Henstock's…
In the present work, we investigate the potential of fractional derivatives to model atmospheric dispersion of pollutants. We propose simple fractional differential equation models for the steady state spatial distribution of concentration…
Modeling dispersed solid phases in fluids still represents a computational challenge when considering a small-scale coupling in wide systems, such as the atmosphere or industrial processes at high Reynolds numbers. A numerical method is…
We present the stochastic solution to a generalized fractional partial differential equation involving a regularized operator related to the so-called Prabhakar operator and admitting, amongst others, as specific cases the fractional…
The challenge to fruitfully merge state-of-the-art techniques from mathematical finance and numerical analysis has inspired researchers to develop fast deterministic option pricing methods. As a result, highly efficient algorithms to…
We study the spectral stability of travelling and stationary front and pulse solutions in a class of degenerate reaction-diffusion systems. We characterise the essential spectrum of the linearised operator in full generality and identify…
Applying the method of moving planes in integral forms, we establish radial symmetry for positive solutions to a class of semilinear equations involving the fractional Laplacian in the unit ball and obtain Liouville type theorems concerning…
The deduction of a constant of motion, a Lagrangian, and a Hamiltonian for relativistic particle moving in a dissipative medium characterized by a force which depends on the square of the velocity of the particle is done. It is shown that…
We study a semilinear PDE generalizing the Fujita equation whose evolution operator is the sum of a fractional power of the Laplacian and a convex non-linearity. Using the Feynman-Kac representation we prove criteria for asymptotic…
Feynman's Lagrangian path integral was an outgrowth of Dirac's vague surmise that Lagrangians have a role in quantum mechanics. Lagrangians implicitly incorporate Hamilton's first equation of motion, so their use contravenes the uncertainty…
We experimentally investigate the transmission of light by dense atomic vapor. The light propagating in dense atomic vapor can be modeled as a L\'evy flight random walk. For such system, the step-length distribution can be modeled as…
We study the convergence in $L^2$ of the time slicing approximation of Feynman path integrals under low regularity assumptions on the potential. Inspired by the custom in Physics and Chemistry, the approximate propagators considered here…
We study the spatial dependence of the quantum friction effect for an atom moving at a constant velocity, in a parallel direction to a material plane. In particular, we determine the probability per unit time and unit area, for exciting…
The Feynman-Kac equations are a type of partial differential equations describing the distribution of functionals of diffusive motion. The probability density function (PDF) of Brownian functionals satisfies the Feynman-Kac formula, being a…
Random walk simulation of the Levy flight shows a linear relation between the mean square displacement <r2> and time. We have analyzed different aspects of this linearity. It is shown that the restriction of jump length to a maximum value…
We analyze two different confining mechanisms for L\'{e}vy flights in the presence of external potentials. One of them is due to a conservative force in the corresponding Langevin equation. Another is implemented by Levy-Schroedinger…
L\'{e}vy walk is a practical model and has wide applications in various fields. Here we focus on the effect of an external constant force on the L\'{e}vy walk with the exponent of the power-law distributed flight time $\alpha\in(0,2)$. We…
We present a framework for the paraxial wave equation based on propagation-dependent unitary transformations closely related to the Lewis-Ermakov invariant. This approach establishes a formal equivalence between free-space propagation and…