Feynman parametrization and Mellin summation at finite temperature

We show that the Mellin summation technique (MST) is a well defined and useful tool to compute loop integrals at finite temperature in the imaginary-time formulation of thermal field theory, especially when interested in the infrared limit of such integrals. The method makes use of the Feynman parametrization which has been claimed to have problems when the analytical continuation from discrete to arbitrary complex values of the Matsubara frequency is performed. We show that without the use of the MST, such problems are not intrinsic to the Feynman parametrization but instead, they arise as a result of (a) not implementing the periodicity brought about by the possible values taken by the discrete Matsubara frequencies before the analytical continuation is made and (b) to the changing of the original domain of the Feynman parameter integration, which seemingly simplifies the expression but in practice introduces a spurious endpoint singularity. Using the MST, there are no problems related to the implementation of the periodicity but instead, care has to be taken when the sum of denominators of the original amplitude vanishes. We apply the method to the computation of loop integrals appearing when the effects of external weak magnetic fields on the propagation of scalar particles is considered.Comment: 16 pages, 1 figure. Discussion expanded. References added. Published versio

An Integro-Differential Equation of the Fractional Form: Cauchy Problem and Solution

Producción CientíficaWe solve the Cauchy problem defined by the fractional partial differential equation [∂tt − κD]u = 0, with D the pseudo-differential Riesz operator of first order, and certain initial conditions. The solution of the Cauchy problem resulting from the substitution of the Gaussian pulse u(x, 0) by the Dirac delta distribution ϕ(x) = μδ(x) is obtained as corollary.MINECO grant MTM2014-57129-C2-1-P

Universality in Systems with Power-Law Memory and Fractional Dynamics

There are a few different ways to extend regular nonlinear dynamical systems by introducing power-law memory or considering fractional differential/difference equations instead of integer ones. This extension allows the introduction of families of nonlinear dynamical systems converging to regular systems in the case of an integer power-law memory or an integer order of derivatives/differences. The examples considered in this review include the logistic family of maps (converging in the case of the first order difference to the regular logistic map), the universal family of maps, and the standard family of maps (the latter two converging, in the case of the second difference, to the regular universal and standard maps). Correspondingly, the phenomenon of transition to chaos through a period doubling cascade of bifurcations in regular nonlinear systems, known as "universality", can be extended to fractional maps, which are maps with power-/asymptotically power-law memory. The new features of universality, including cascades of bifurcations on single trajectories, which appear in fractional (with memory) nonlinear dynamical systems are the main subject of this review.Comment: 23 pages 7 Figures, to appear Oct 28 201

Positive solutions for singular nonlinear fractional differential equation with integral boundary conditions

© 2015, Li et al. In this article, we study the existence of positive solutions for a class of singular nonlinear fractional differential equations with Riemann-Stieltjes integral boundary conditions. Using the properties of the Green function and the fixed point theory in cones, we obtain some results on the existence of positive solutions. Our results extend and improve many known results including singular and nonsingular cases

One-Dimensional and Multi-Dimensional Integral Transforms of Buschman–Erdélyi Type with Legendre Functions in Kernels

This paper consists of two parts. In the first part we give a brief survey of results on Buschman–Erdélyi operators, which are transmutations for the Bessel singular operator. Main properties and applications of Buschman–Erdélyi operators are outlined. In the second part of the paper we consider multi-dimensional integral transforms of Buschman–Erdélyi type with Legendre functions in kernels. Complete proofs are given in this part, main tools are based on Mellin transform properties and usage of Fox H-functions

Monotone iterative procedure and systems of a finite number of nonlinear fractional differential equations

The aim of the paper is to present a nontrivial and natural extension of the comparison result and the monotone iterative procedure based on upper and lower solutions, which were recently established in (Wang et al. in Appl. Math. Lett. 25:1019-1024, 2012), to the case of any finite number of nonlinear fractional differential equations.The author is very grateful to the reviewers for the remarks, which improved the final version of the manuscript. This article was financially supported by University of Łódź as a part of donation for the research activities aimed at the development of young scientists, grant no. 545/1117