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

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.

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

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.

Subharmonic Resonance of Duffing Oscillator With Fractional-Order Derivative

2016 ◽  
Vol 11 (5) ◽  
Author(s):  
Nguyen Van Khang ◽  
Truong Quoc Chien
Keyword(s):  
Analytical Solution ◽  
Fractional Order ◽  
Duffing Oscillator ◽  
Existence Condition ◽  
Lyapunov Theory ◽  
Subharmonic Resonance ◽  
System Stability ◽  
Order Derivative ◽  
Fractional Order Derivative ◽  
Approximately Analytical Solution

In this paper, the subharmonic resonance of Duffing oscillator with fractional-order derivative is investigated using the averaging method. First, the approximately analytical solution and the amplitude–frequency equation are obtained. The existence condition for subharmonic resonance based on the approximately analytical solution is then presented, and the corresponding stability condition based on Lyapunov theory is also obtained. Finally, a comparison between the fractional-order and the traditional integer-order of Duffing oscillators is made using numerical simulation. The influences of the parameters in fractional-order derivative on the steady-state amplitude, the amplitude–frequency curves, and the system stability are also investigated.

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

2017 ◽  
Vol 21 (suppl. 1) ◽  
pp. 317-326 ◽  
Author(s):  
Xiaojun Yang
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Integral Transforms ◽  
Kernel Functions ◽  
Order Derivative ◽  
Fractional Order Derivative ◽  
Laplace Integral ◽  
Fractional Calculus Operators ◽  
Anomalous Relaxation ◽  
First Time

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.

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

2019 ◽  
Vol 23 (6 Part B) ◽  
pp. 3711-3718 ◽  
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.

scholarly journals Numerical Solution of the Bagley-Torvik Equation Using the Integer-Order Derivatives Expansion

2018 ◽  
Vol 6 (2) ◽  
Author(s):  
Afrah Sadiq Hasan
Keyword(s):  
Differential Equation ◽  
Analytical Solution ◽  
Numerical Solution ◽  
Fractional Order ◽  
Infinite Series ◽  
Exact Analytical Solution ◽  
Second Order ◽  
Order Derivative ◽  
Fractional Order Derivative ◽  
Integer Order

Numerical solution of the well-known Bagley-Torvik equation is considered. The fractional-order derivative in the equation is converted, approximately, to ordinary-order derivatives up to second order. Approximated Bagley-Torvik equation is obtained using finite number of terms from the infinite series of integer-order derivatives expansion for the Riemann–Liouville fractional derivative. The Bagley-Torvik equation is a second-order differential equation with constant coefficients. The derived equation, by considering only the first three terms from the infinite series to become a second-order ordinary differential equation with variable coefficients, is numerically solved after it is transformed into a system of first-order ordinary differential equations. The approximation of fractional-order derivative and the order of the truncated error are illustrated through some examples. Comparison between our result and exact analytical solution are made by considering an example with known analytical solution to show the preciseness of our proposed approach.

scholarly journals Fractional Model for a Class of Diffusion-Reaction Equation Represented by the Fractional-Order Derivative

2020 ◽  
Vol 4 (2) ◽  
pp. 15 ◽  
Author(s):  
Ndolane Sene
Keyword(s):  
Analytical Solution ◽  
Diffusion Equation ◽  
Fractional Order ◽  
Integration Method ◽  
Diffusion Reaction ◽  
Order Derivative ◽  
Reaction Equation ◽  
Fractional Model ◽  
Fractional Order Derivative ◽  
The Laplace Transform

This paper proposes the analytical solution for a class of the fractional diffusion equation represented by the fractional-order derivative. We mainly use the Grunwald–Letnikov derivative in this paper. We are particularly interested in the application of the Laplace transform proposed for this fractional operator. We offer the analytical solution of the fractional model as the diffusion equation with a reaction term expressed by the Grunwald–Letnikov derivative by using a double integration method. To illustrate our findings in this paper, we represent the analytical solutions for different values of the used fractional-order derivative.

scholarly journals Novel Approaches for Getting the Solution of the Fractional Black–Scholes Equation Described by Mittag-Leffler Fractional Derivative

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Ndolane Sene ◽  
Babacar Sène ◽  
Seydou Nourou Ndiaye ◽  
Awa Traoré
Keyword(s):  
Analytical Solution ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Numerical Scheme ◽  
Fractional Derivatives ◽  
Fractional Integration ◽  
Order Derivative ◽  
Graphical Representations ◽  
Fractional Order Derivative ◽  
Black Scholes

The value of an option plays an important role in finance. In this paper, we use the Black–Scholes equation, which is described by the nonsingular fractional-order derivative, to determine the value of an option. We propose both a numerical scheme and an analytical solution. Recent studies in fractional calculus have included new fractional derivatives with exponential kernels and Mittag-Leffler kernels. These derivatives have been found to be applicable in many real-world problems. As fractional derivatives without nonsingular kernels, we use a Caputo–Fabrizio fractional derivative and a Mittag-Leffler fractional derivative. Furthermore, we use the Adams–Bashforth numerical scheme and fractional integration to obtain the numerical scheme and the analytical solution, and we provide graphical representations to illustrate these methods. The graphical representations prove that the Adams–Bashforth approach is helpful in getting the approximate solution for the fractional Black–Scholes equation. Finally, we investigate the volatility of the proposed model and discuss the use of the model in finance. We mainly notice in our results that the fractional-order derivative plays a regulator role in the diffusion process of the Black–Scholes equation.

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

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.

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

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.

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