Axioms

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Open AccessArticle

Operational Solution of Non-Integer Ordinary and Evolution-Type Partial Differential Equations

by Konstantin V. Zhukovsky and Hari M. Srivastava

Axioms 2016, 5(4), 29; https://doi.org/10.3390/axioms5040029 - 13 Dec 2016

Cited by 8 | Viewed by 4275

A method for the solution of linear differential equations (DE) of non-integer order and of partial differential equations (PDE) by means of inverse differential operators is proposed. The solutions of non-integer order ordinary differential equations are obtained with recourse to the integral transforms and the exponent operators. The generalized forms of Laguerre and Hermite orthogonal polynomials as members of more general Appèl polynomial family are used to find the solutions. Operational definitions of these polynomials are used in the context of the operational approach. Special functions are employed to write solutions of DE in convolution form. Some linear partial differential equations (PDE) are also explored by the operational method. The Schrödinger and the Black–Scholes-like evolution equations and solved with the help of the operational technique. Examples of the solution of DE of non-integer order and of PDE are considered with various initial functions, such as polynomial, exponential, and their combinations. Full article

(This article belongs to the Special Issue Special Functions: Fractional Calculus and the Pathway for Entropy Dedicated to Professor Dr. A.M. Mathai on the occasion of his 80th Birthday)

5434 KiB

Open AccessArticle

Operational Approach and Solutions of Hyperbolic Heat Conduction Equations

by Konstantin Zhukovsky

Axioms 2016, 5(4), 28; https://doi.org/10.3390/axioms5040028 - 12 Dec 2016

Cited by 36 | Viewed by 7147

Abstract

We studied physical problems related to heat transport and the corresponding differential equations, which describe a wider range of physical processes. The operational method was employed to construct particular solutions for them. Inverse differential operators and operational exponent as well as operational definitions and operational rules for generalized orthogonal polynomials were used together with integral transforms and special functions. Examples of an electric charge in a constant electric field passing under a potential barrier and of heat diffusion were compared and explored in two dimensions. Non-Fourier heat propagation models were studied and compared with each other and with Fourier heat transfer. Exact analytical solutions for the hyperbolic heat equation and for its extensions were explored. The exact analytical solution for the Guyer-Krumhansl type heat equation was derived. Using the latter, the heat surge propagation and relaxation was studied for the Guyer-Krumhansl heat transport model, for the Cattaneo and for the Fourier models. The comparison between them was drawn. Space-time propagation of a power–exponential function and of a periodic signal, obeying the Fourier law, the hyperbolic heat equation and its extended Guyer-Krumhansl form were studied by the operational technique. The role of various terms in the equations was explored and their influence on the solutions demonstrated. The accordance of the solutions with maximum principle is discussed. The application of our theoretical study for heat propagation in thin films is considered. The examples of the relaxation of the initial laser flash, the wide heat spot, and the harmonic function are considered and solved analytically. Full article

(This article belongs to the Special Issue Special Functions: Fractional Calculus and the Pathway for Entropy Dedicated to Professor Dr. A.M. Mathai on the occasion of his 80th Birthday)

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166 KiB

Open AccessEditorial

Discrete Geometry—From Theory to Applications: A Case Study

by David Gu and Emil Saucan

Axioms 2016, 5(4), 27; https://doi.org/10.3390/axioms5040027 - 9 Dec 2016

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Abstract

Science does not necessarily evolve along the lines that are taught to us in High School history classes and in popular films, that is, from simple to complex.[...] Full article

(This article belongs to the Special Issue Discrete Geometry and its Applications)

2282 KiB

Open AccessArticle

Forman-Ricci Flow for Change Detection in Large Dynamic Data Sets

by Melanie Weber, Jürgen Jost and Emil Saucan

Axioms 2016, 5(4), 26; https://doi.org/10.3390/axioms5040026 - 10 Nov 2016

Cited by 26 | Viewed by 8014

Abstract

We present a viable geometric solution for the detection of dynamic effects in complex networks. Building on Forman’s discretization of the classical notion of Ricci curvature, we introduce a novel geometric method to characterize different types of real-world networks with an emphasis on peer-to-peer networks. We study the classical Ricci-flow in a network-theoretic setting and introduce an analytic tool for characterizing dynamic effects. The formalism suggests a computational method for change detection and the identification of fast evolving network regions and yields insights into topological properties and the structure of the underlying data. Full article

(This article belongs to the Special Issue Discrete Geometry and its Applications)

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278 KiB

Open AccessArticle

Quantum Quasigroups and the Quantum Yang–Baxter Equation

by Jonathan Smith

Axioms 2016, 5(4), 25; https://doi.org/10.3390/axioms5040025 - 9 Nov 2016

Cited by 3 | Viewed by 4075

Abstract

Quantum quasigroups are algebraic structures providing a general self-dual framework for the nonassociative extension of Hopf algebra techniques. They also have one-sided analogues, which are not self-dual. The paper presents a survey of recent work on these structures, showing how they furnish various [...] Read more.

(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang-Baxter Equations 2016)

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Axioms, Volume 5, Issue 4 (December 2016) – 5 articles

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