Related papers: Picard-Fuchs equations for Feynman integrals
Anomalous diffusion is a phenomenon that cannot be modeled accurately by second-order diffusion equations, but is better described by fractional diffusion models. The nonlocal nature of the fractional diffusion operators makes substantially…
Using model theory and differential algebra, we give necessary conditions for algebraic ordinary differential equations to have a complex Pfaffian solution on some complex domain. These tools also allow us to give many examples of algebraic…
Functional Schr\"{o}dinger equations for interacting fields are solved via rigorous non-perturbative Feynman type integrals.
We focus on Fuchsian equations with four accessory parameters and three singular points. We see that the Fuchsian equations admit a "degeneration scheme" in some sense, which is expected to give rise to a degeneration scheme of discrete…
It is shown how strictly four-dimensional integration by parts combined with differential renormalization and its infrared analogue can be applied for calculation of Feynman diagrams.
The Schwarzian equations satisfied by certain Hauptmoduls (i.e., uniformizing functions for Riemann surfaces of genus zero) are derived from the Picard-Fuchs equations for families of elliptic curves and associated surfaces. The…
A numerical method to solve the fractional diffusion equation, which could also be easily extended to many other fractional dynamics equations, is considered. These fractional equations have been proposed in order to describe anomalous…
In this paper, we propose two approaches to derive the discrete Poincar\'e inequality for the B-schemes, a family of finite volume discretization schemes, for the one-dimensional Fokker-Planck equation in full space. We study the properties…
We present an approach to analyze the scalar integrals of any Feynman diagrams in detail here. This method not only completely recovers some well-known results in the literature, but also produces some brand new results on the $C_{_0}$…
Solution of fractional differential equations is an emerging area of present day research because such equations arise in various applied fields. In this paper we have developed analytical method to solve the system of fractional…
In this paper we introduce and investigate a new kind of functional (including ordinary and evolutionary partial) differential equations. The main goal of this paper is to explore our new philosophy by some examples on functional ODEs and…
We determine closed and compact expressions for the epsilon-expansion of certain Gaussian hypergeometric functions expanded around half-integer values by explicitly solving for their recurrence relations. This epsilon-expansion is…
In this study the general formula for differential and integral operations of fractional calculus via fractal operators by the method of cumulative diminution and cumulative growth is obtained. The under lying mechanism in the success of…
We provide of a method to integrate first order non-linear systems of differential equations with variable coefficients. It determines approximate solutions given initial or boundary conditions or even for Sturm-Liouville problems. This…
In this note, a numerical method based on finite differences to solve a class of nonlinear advection-diffusion fractional differential equation is proposed. The fractional operator considered here is the fractional Riemann-Liouville…
In view of the usefulness and importance of the kinetic equation in certain physical problems, the authors derive the explicit solution of a fractional kinetic equation of general character, that unifies and extends earlier results.…
We use Vessiot theory and exterior calculus to solve partial differential equations(PDEs) of the type uyy = F(x, y,u,ux,uy,uxx,uxy) and associated evolution equations. These equations are represented by the Vessiot distribution of vector…
This paper detailedly discusses the locally one-dimensional numerical methods for efficiently solving the three-dimensional fractional partial differential equations, including fractional advection diffusion equation and Riesz fractional…
The classical Feynman-Kac identity represents solutions of linear partial differential equations in terms of stochastic differential euqations. This representation has been generalized to nonlinear partial differential equations on the one…
In this paper we give an explicit solution of Dzherbashyan-Caputo-fractional Cauchy problems related to equations with derivatives of order $\nu k$, for $k$ non-negative integer and $\nu>0$. The solution is obtained by connecting the…