Many possible definitions have been proposed for fractional derivatives and integrals, starting from the classical Riemann–Liouville formula and its generalisations and modifying it by replacing the ...power function kernel with other kernel functions. We demonstrate, under some assumptions, how all of these modifications can be considered as special cases of a single, unifying, model of fractional calculus. We provide a fundamental connection with classical fractional calculus by writing these general fractional operators in terms of the original Riemann–Liouville fractional integral operator. We also consider inversion properties of the new operators, prove analogues of the Leibniz and chain rules in this model of fractional calculus, and solve some fractional differential equations using the new operators.
We provide a small C++ library with Mathematica and Python interfaces for computing thermal functions, defined JB/F(y2)≡ℜ∫0∞x2log1∓e−x2+y2dx,which appear in finite-temperature quantum field theory ...and play a role in phase-transitions in the early Universe, including baryogenesis, electroweak symmetry breaking and the Higgs mechanism. The code is available from https://github.com/andrewfowlie/thermal_funcs.
Program title: thermal_funcs
Program Files doi: http://dx.doi.org/10.17632/5nzrr2jjpd.1
Licensing provisions: BSD 3-Clause
Programming language: C++, C interface to Mathematica and SWIG interface to Python
Nature of problem: Thermal functions appear in finite-temperature quantum field theory and influence phase transitions in the early Universe. They have no closed-form solution. Studying phase transitions requires repeated evaluations of, inter alia, thermal functions. We provide several fast and accurate methods, and first and second derivatives.
Solution method: We implement numerical quadrature, a Bessel function representation, asymptotic solutions expressed in terms of zeta functions and polylogarithms, and approximations and limits.
In the present work, a numerical technique for solving a general form of nonlinear fractional order integro-differential equations (GNFIDEs) with linear functional arguments using Chebyshev series is ...presented. The recommended equation with its linear functional argument produces a general form of delay, proportional delay, and advanced non-linear arbitrary order Fredholm–Volterra integro-differential equations. Spectral collocation method is extended to study this problem as a matrix discretization scheme, where the fractional derivatives are characterized in the Caputo sense. The collocation method transforms the given equation and conditions to an algebraic nonlinear system of equations with unknown Chebyshev coefficients. Additionally, we present a general form of the operational matrix for derivatives. The introduced operational matrix of derivatives includes arbitrary order derivatives and the operational matrix of ordinary derivative as a special case. To the best of authors’ knowledge, there is no other work discussing this point. Numerical test examples are given, and the achieved results show that the recommended method is very effective and convenient.
In this paper, we establish the generalized Riemann–Liouville (RL) fractional integrals in the sense of another increasing, positive, monotone, and measurable function
Ψ
. We determine certain new ...double-weighted type fractional integral inequalities by utilizing the said integrals. We also give some of the new particular inequalities of the main result. Note that we can form various types of new inequalities of fractional integrals by employing conditions on the function
Ψ
given in the paper. We present some corollaries as particular cases of the main results.
In this article, we suggest a new notion of fractional derivative involving two singular kernels. Some properties related to this new operator are established and some examples are provided. We also ...present some applications to fractional differential equations and propose a numerical algorithm based on a Picard iteration for approximating the solutions. Finally, an application to a heat conduction problem is given.
In the present paper, we apply the n–ary aggregation functions on several special functions (Hypergeometric function, generalized exponential function, and Fox's H–function) to define a class of ...matrix–valued fuzzy controllers which help us to study the Ulam–Hyers stability for a (non) autonomous fractional differential system in the Hilfer sense, through the fixed point theorem, in a matrix valued fuzzy n–normed space. Next, by the properties of Mittag-Leffler functions, the Laplace transform and the non–standard Gronwall inequality, we propose some efficient conditions on the (asymptotic) stability of the governing model, in matrix fuzzy normed spaces.
Recently many authors are spending lot of time and efforts to evaluate various operators of fractional order integration and differentiation and their generalizations of classes of, or particular, ...special functions. The list of such works is rather long and yet growing daily, so we limit ourselves to mention here only a few, just to illustrate our general approach. As special functions present indeed a great variety, and the operators of fractional calculus do as well, the mentioned job produces a huge flood of publications. Many of them use same formal and standard procedures, and besides, often the results sound not of practical use, with except to increase authors’ publication activities.
In this survey, we point out on some few basic classical results, combined with author’s ideas and developments, that show how one can do the task at once, in the rather general case: for both operators of generalized fractional calculus and generalized hypergeometric functions. Thus, great part of the results in the mentioned publications are well predicted and fall just as special cases of the discussed general scheme.
In 3 André showed that the minimal differential equations of Э-functions and E-functions have a basis of holomorphic solutions at every point except for zero and infinity. In addition, in 5 he ...observed that if f1(z) is an E-function with rational coefficients such that f1(1)=0, then f1(z)/(z−1) is also an E-function. With this additional result André derived transcendence results for the values of E-functions. These results were further applied by Beukers 7 to obtain a strengthened Siegel-Shidlovskii theorem for E-functions. The arguments of André and Beukers should aid in obtaining a Siegel-Shidlovskii type theorem for Э-functions provided that one can show that if f−1(z) is a Э-function with rational coefficients such that the 1-summation F−1(z) vanishes at z=1, then f−1(z)/(z−1) is also a Э-function. In this paper, we investigate this problem and derive several criteria to check its validity.
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