Related papers: Fractional integration and differentiation
In this paper we describe a method of calculation of master integrals based on the solution of systems of difference equations in one variable. Various explicit examples are given, as well as the generalization to arbitrary diagrams.
We provide an algorithm for computing semi-Fourier sequences for expressions constructed from arithmetic operations, exponentiations and integrations. The semi-Fourier sequence is a relaxed version of Fourier sequence for polynomials…
It is well known that for every $f\in C^m$ there exists a polynomial $p_n$ such that $p^{(k)}_n\rightarrow f^{(k)}$, $k=0,\ldots,m$. Here we prove such a result for fractional (non-integer) derivatives. Moreover, a numerical method is…
In this paper, we propose a numerical method of computing a Hadamard finite-part integral with a non-integral power singularity at an endpoint, that is, a finite part of a divergent integral as a limiting procedure. In the proposed method,…
Applying proper orthogonal decomposition to a usual finite element (FE) formulation for space fractional partial differential equation, we get a reduced FE model, which greatly reduces the complexity of computation. Then, the stability…
This work is an analytical and numerical study of the composition of several fractals into one and of the relation between the composite dimension and the dimensions of the component fractals. In the case of composition of standard IFS with…
We present a spectral method for one-sided linear fractional integral equations on a closed interval that achieves exponentially fast convergence for a variety of equations, including ones with irrational order, multiple fractional orders,…
The theory of fractional calculus in the complex plane was not built with a specific application in mind. The main obstacle to application was the difficulty with obtaining analytic continuations of fractional derivatives and integrals. It…
A fractional Adomian decomposition method for fractional nonlinear differential equations is proposed. The iteration procedure is based on Jumarie's fractional derivative. An example is given to elucidate the solution procedure, and the…
We propose a new class of semi-implicit methods for solving nonlinear fractional differential equations and study their stability. Several versions of our new schemes are proved to be unconditionally stable by choosing suitable parameters.…
It is argued that the evolution of complex phenomena ought to be described by fractional, differential, stochastic equations whose solutions have scaling properties and are therefore random, fractal functions. To support this argument we…
It is well known that using high-order numerical algorithms to solve fractional differential equations leads to almost the same computational cost with low-order ones but the accuracy (or convergence order) is greatly improved, due to the…
In this manuscript, we generalize F-calculus to apply it on fractal Tartan spaces. The generalized standard F-calculus is used to obtain the integral and derivative of the functions on the fractal Tartan with different dimensions. The…
New methods for obtaining functional equations for Feynman integrals are presented. Application of these methods for finding functional equations for various one- and two- loop integrals described in detail. It is shown that with the aid of…
Fractional $q$-extensions of some classical $q$-orthogonal polynomials are introduced and some of the main properties of the new defined functions are given. Next, a fractional $q$-difference equation of Gauss type is introduced and solved…
We introduce a new numerical method, based on Bernoulli polynomials, for solving multiterm variable-order fractional differential equations. The variable-order fractional derivative was considered in the Caputo sense, while the…
In this paper we provide a definition of fractional gradient operators, related to directional derivatives. We develop a fractional vector calculus, providing a probabilistic interpretation and mathematical tools to treat multidimensional…
Fractional integral operators connected with real-valued scalar functions of matrix argument are applied in problems of mathematics, statistics and natural sciences. In this article we start considering the case of a Gauss hypergeometric…
The idea of generating integrals analogous to generating functions is first introduced in this paper. A new proof of the well-known Finite Harmonic Series Theorem in Analysis and Analytical Number Theory is then obtained by the method of…
New families of fourth-order composition methods for the numerical integration of initial value problems defined by ordinary differential equations are proposed. They are designed when the problem can be separated into three parts in such a…