General fractional calculus operators containing the generalized Mittag-Leffler functions applied to anomalous relaxation

In this paper, we address a family of the general fractional calculus operators of Wiman and Prabhakar types for the first time. The general Mittag-Leffler function to structure the kernel functions of the fractional order derivative operators and their Laplace integral transforms are considered in detail. The formulations are as the mathematical tools proposed to investigate the anomalous relaxation.

Download Full-text

A new general fractional derivative Goldstein-Kac-type telegraph equation

Thermal Science ◽

10.2298/tsci2006893c ◽

2020 ◽

Vol 24 (6 Part B) ◽

pp. 3893-3898

Author(s):

Ping Cui ◽

Yi-Ying Feng ◽

Jian-Gen Liu ◽

Lu-Lu Geng

Keyword(s):

Fractional Order ◽

Telegraph Equation ◽

Order Derivative ◽

Mathematical Tool ◽

Fractional Order Derivative ◽

Derivative Formula ◽

The Laplace Transform ◽

The One ◽

First Time ◽

Derivatives Of

In this paper, we consider the Riemann-Liouville-type general fractional derivatives of the non-singular kernel of the one-parametric Lorenzo-Hartley function. A new general fractional-order-derivative Goldstein-Kac-type telegraph equation is proposed for the first time. The analytical solution of the considered model with the graphs is obtained with the aid of the Laplace transform. The general fractional-order-derivative formula is as a new mathematical tool proposed to model the anomalous behaviors in complex and power-law phenomena.

Download Full-text

A new general fractional-order derivative with Rabotnov fractional-exponential kernel

Thermal Science ◽

10.2298/tsci180825254y ◽

2019 ◽

Vol 23 (6 Part B) ◽

pp. 3711-3718 ◽

Cited By ~ 7

Author(s):

Xiao-Jun Yang ◽

Minvydas Ragulskis ◽

Thiab Taha

Keyword(s):

Fractional Order ◽

Physical Models ◽

Order Derivative ◽

Mathematical Approach ◽

Liouville Type ◽

Derivative Operator ◽

Fractional Order Derivative ◽

Fractional Exponential Function ◽

Complex Phenomena ◽

First Time

In this article, a general fractional-order derivative of the Riemann-Liouville type with the non-singular kernel involving the Rabotnov fractional-exponential function is addressed for the first time. A new general fractional-order derivative model for the anomalous diffusion is discussed in detail. The general fractional-order derivative operator formula is as a novel and mathematical approach proposed to give the generalized presentation of the physical models in complex phenomena with power law.

Download Full-text

A new general fractional-order wave model involving Miller-Ross kernel

Thermal Science ◽

10.2298/tsci180923245d ◽

2019 ◽

Vol 23 (Suppl. 3) ◽

pp. 953-957

Author(s):

Linming Dou ◽

Xiao-Jun Yang ◽

Jiangen Liu

Keyword(s):

Analytical Solution ◽

Fractional Order ◽

Wave Model ◽

Order Derivative ◽

Fractional Order Derivative ◽

Complex Processes ◽

First Time

In the paper we consider a general fractional-order wave model with the general fractional-order derivative involving the Miller-Ross kernel for the first time. The analytical solution for the general fractional-order wave model is investigated in detail. The obtained result is given to explore the complex processes in the mining rock.

Download Full-text

GENERALIZED THEORY OF MAGNETO-THERMO-VISCOELASTIC SPHERICAL CAVITY PROBLEM UNDER FRACTIONAL ORDER DERIVATIVE: STATE SPACE APPROACH

Advances in Mathematics: Scientific Journal ◽

10.37418/amsj.9.11.86 ◽

2020 ◽

Vol 9 (11) ◽

pp. 9769-9780

Author(s):

S.G. Khavale ◽

K.R. Gaikwad

Keyword(s):

State Space ◽

Fractional Order ◽

Order Derivative ◽

Laplace Inversion ◽

State Space Approach ◽

Spherical Region ◽

Fractional Order Derivative ◽

Numerical Laplace Inversion ◽

Mathcad Software ◽

Space Approach

This paper is dealing the modified Ohm's law with the temperature gradient of generalized theory of magneto-thermo-viscoelastic for a thermally, isotropic and electrically infinite material with a spherical region using fractional order derivative. The general solution obtained from Laplace transform, numerical Laplace inversion and state space approach. The temperature, displacement and stresses are obtained and represented graphically with the help of Mathcad software.

Download Full-text

Mkhitar Djrbashian and his contribution to Fractional Calculus

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2020-0089 ◽

2020 ◽

Vol 23 (6) ◽

pp. 1797-1809

Author(s):

Sergei Rogosin ◽

Maryna Dubatovskaya

Keyword(s):

Cauchy Problem ◽

Fractional Calculus ◽

Fractional Order ◽

Approximation Theory ◽

Fractional Derivatives ◽

Complex Analysis ◽

Integral Transforms ◽

Modern Theory ◽

Survey Paper ◽

The Cauchy Problem

Abstract This survey paper is devoted to the description of the results by M.M. Djrbashian related to the modern theory of Fractional Calculus. M.M. Djrbashian (1918-1994) is a well-known expert in complex analysis, harmonic analysis and approximation theory. Anyway, his contributions to fractional calculus, to boundary value problems for fractional order operators, to the investigation of properties of the Queen function of Fractional Calculus (the Mittag-Leffler function), to integral transforms’ theory has to be understood on a better level. Unfortunately, most of his works are not enough popular as in that time were published in Russian. The aim of this survey is to fill in the gap in the clear recognition of M.M. Djrbashian’s results in these areas. For same purpose, we decided also to translate in English one of his basic papers [21] of 1968 (joint with A.B. Nersesian, “Fractional derivatives and the Cauchy problem for differential equations of fractional order”), and were invited by the “FCAA” editors to publish its re-edited version in this same issue of the journal.

Download Full-text

Novel hyperbolic and exponential ansatz methods to the fractional fifth-order Korteweg–de Vries equations

Advances in Difference Equations ◽

10.1186/s13662-020-03087-w ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

Choonkil Park ◽

R. I. Nuruddeen ◽

Khalid K. Ali ◽

Lawal Muhammad ◽

M. S. Osman ◽

...

Keyword(s):

Fractional Derivative ◽

Fractional Order ◽

Mathematical Physics ◽

Order Derivative ◽

Conformable Fractional Derivative ◽

Fractional Order Derivative ◽

De Vries ◽

Fifth Order ◽

2D And 3D

Abstract This paper aims to investigate the class of fifth-order Korteweg–de Vries equations by devising suitable novel hyperbolic and exponential ansatze. The class under consideration is endowed with a time-fractional order derivative defined in the conformable fractional derivative sense. We realize various solitons and solutions of these equations. The fractional behavior of the solutions is studied comprehensively by using 2D and 3D graphs. The results demonstrate that the methods mentioned here are more effective in solving problems in mathematical physics and other branches of science.

Download Full-text