Free Oscillation Solution for Fractional Differential System

2019 ◽  
Vol 14 (12) ◽  
Author(s):  
Masataka Fukunaga
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Indirect Evidence ◽  
Turing Instability ◽  
Mode Analysis ◽  
Free Standing ◽  
Special Cases ◽  
Fractional Differential

Abstract There is a type of fractional differential equation that admits asymptotically free standing oscillations (Fukunaga, M., 2019, “Mode Analysis on Onset of Turing Instability in Time-Fractional Reaction-Subdiffusion Equations by Two-Dimensional Numerical Simulations,” ASME J. Comput. Nonlinear Dyn., 14, p. 061005). In this paper, analytical solutions to fractional differential equation for free oscillations are derived for special cases. These analytical solutions are direct evidence for asymptotically standing oscillations, while numerical solutions give indirect evidence.

Invariant solutions of hyperbolic fuzzy fractional differential equations

2019 ◽  
Vol 34 (01) ◽  
pp. 2050015 ◽  
Author(s):  
C. Vinothkumar ◽  
J. J. Nieto ◽  
A. Deiveegan ◽  
P. Prakash
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Numerical Solutions ◽  
Banach Fixed Point Theorem ◽  
Invariant Solutions ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Fractional Differential ◽  
Fuzzy Fractional Differential Equations ◽  
Fuzzy Fractional Differential Equation

We consider the hyperbolic type fuzzy fractional differential equation and derive the second-order fuzzy fractional differential equation using scaling transformation. We present a theoretical and a numerical method to find the invariant solutions of such equations. Also, we prove the existence and uniqueness results using Banach fixed point theorem. Numerical solutions are approximated using finite difference method. Finally, numerical examples are given to illustrate the obtained results.

scholarly journals On Analytical Solutions of the Fractional Differential Equation with Uncertainty: Application to the Basset Problem

Entropy ◽  
2015 ◽  
Vol 17 (2) ◽  
pp. 885-902 ◽  
Author(s):  
Soheil Salahshour ◽  
Ali Ahmadian ◽  
Norazak Senu ◽  
Dumitru Baleanu ◽  
Praveen Agarwal
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Analytical Solutions ◽  
Fractional Differential ◽  
Basset Problem

scholarly journals Analytical solutions of incommensurate fractional differential equation systems with fractional order $ 1 < \alpha, \beta < 2 $ via bivariate Mittag-Leffler functions

2021 ◽  
Vol 7 (2) ◽  
pp. 2281-2317
Author(s):  
Yong Xian Ng ◽  
◽  
Chang Phang ◽  
Jian Rong Loh ◽  
Abdulnasir Isah ◽  
...  
Keyword(s):  
Differential Equation ◽  
Analytical Solution ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Successive Approximations ◽  
Explicit Analytical Solution ◽  
Special Cases ◽  
Fractional Differential ◽  
Alpha Beta ◽  
Published Paper

<abstract><p>In this paper, we derive the explicit analytical solution of incommensurate fractional differential equation systems with fractional order $ 1 &lt; \alpha, \beta &lt; 2 $. The derivation is extended from a recently published paper by Huseynov et al. in <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>, which is limited for incommensurate fractional order $ 0 &lt; \alpha, \beta &lt; 1 $. The incommensurate fractional differential equation systems were first converted to Volterra integral equations. Then, the Mittag-Leffler function and Picard's successive approximations were used to obtain the analytical solution of incommensurate fractional order systems with $ 1 &lt; \alpha, \beta &lt; 2 $. The solution will be simplified via some combinatorial concepts and bivariate Mittag-Leffler function. Some special cases will be discussed, while some examples will be given at the end of this paper.</p></abstract>

scholarly journals He’s fractional derivative for heat conduction in a fractal medium arising in silkworm cocoon hierarchy

2015 ◽  
Vol 19 (4) ◽  
pp. 1155-1159 ◽  
Author(s):  
Fu-Juan Liu ◽  
Zheng-Biao Li ◽  
Sheng Zhang ◽  
Hong-Yan Liu
Keyword(s):  
Differential Equation ◽  
Heat Conduction ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Natural Phenomenon ◽  
Fractal Medium ◽  
Complex Transformation ◽  
Special Cases ◽  
Fractional Differential ◽  
Silkworm Cocoon

He?s fractional derivative is adopted in this paper to study the heat conduction in fractal medium. The fractional complex transformation is applied to convert the fractional differential equation to ordinary different equation. Boltzmann transform and wave transform are used to further simplify the governing equation for some special cases. Silkworm cocoon are used as an example to elucidate its natural phenomenon.

scholarly journals NUMERICAL CALCULATION OF NONSTATIONARY FRACTIONAL DIFFERENTIAL EQUATION IN PROBLEMS OF MODELING TOXIC SUBSTANCES DISTRIBUTION IN GROUND WATERS

2019 ◽  
pp. 70-80
Author(s):  
Alexandra Alekseyevna Afanasyeva ◽  
Tatyana Nikolayevna Shvetsova-Shilovskaya ◽  
Dmitriy Evgenevich Ivanov ◽  
Denis Igorevich Nazarenko ◽  
Elena Victorovna Kazarezova
Keyword(s):  
Differential Equation ◽  
Analytical Solution ◽  
Differential Equations ◽  
Difference Scheme ◽  
Numerical Method ◽  
Fractional Differential Equation ◽  
Numerical Solutions ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Fractional Differential

At present, the theory of fractional calculus is widely used in many fields of science for modeling various processes. Differential equations with fractional derivatives are used to model the migration of pollutants in porous inhomogeneous media and allow a more correct description of the behavior of pollutants at large distances from the source. The analytical solution of differential equations with fractional order derivatives is often very complicated or even impossible. There has been proposed a numerical method for solving fractional differential equations in partial derivatives with respect to time to describe the migration of pollutants in groundwater. An implicit difference scheme is developed for the numerical solution of a non-stationary fractional differential equation, which is an analogue of the well-known implicit Crank-Nicholson difference scheme. The system of difference equations is presented in matrix form. The solution of the problem is reduced to the multiple solution of a tridiagonal system of linear algebraic equations by the tridiagonal matrix algorithm. The results of evaluating the spread of pollutant in groundwater based on the numerical method for model examples are presented. The concentrations of the substance obtained on the basis of the analytical and numerical solutions of the unsteady one-dimensional fractional differential equation are compared. The results obtained using the proposed method and on the basis of the well-known analytical solution of the fractional differential equation are in fairly good agreement with each other. The relative error is on average 9%. In contrast to the well-known analytical solution, the developed numerical method can be used to model the spread of pollutants in groundwater, taking into account their biodegradation.

scholarly journals Analytic Solution for theRLElectric Circuit Model in Fractional Order

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
P. V. Shah ◽  
A. D. Patel ◽  
I. A. Salehbhai ◽  
A. K. Shukla
Keyword(s):  
Differential Equation ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Analytic Solution ◽  
Electrical Circuit ◽  
Special Cases ◽  
The Laplace Transform ◽  
Fractional Differential

This paper provides an analytic solution ofRLelectrical circuit described by a fractional differential equation of the order0<α≤1. We use the Laplace transform of the fractional derivative in the Caputo sense. Some special cases for the different source terms have also been discussed.

scholarly journals Least Squares Differential Quadrature Method for the Generalized Bagley–Torvik Fractional Differential Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Constantin Bota ◽  
Bogdan Căruntu ◽  
Mădălina Sofia Paşca ◽  
Dumitru Ţucu ◽  
Marioara Lăpădat
Keyword(s):  
Differential Equation ◽  
Least Squares ◽  
Fractional Differential Equation ◽  
Analytical Solutions ◽  
Differential Quadrature Method ◽  
Accurate Method ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Approximate Analytical Solutions ◽  
Fractional Differential

In this paper, the least squares differential quadrature method for computing approximate analytical solutions for the generalized Bagley–Torvik fractional differential equation is presented. This new method is introduced as a straightforward and accurate method, fact proved by the examples included, containing a comparison with previous results obtained by using other methods.

Explicit analytical solutions of incommensurate fractional differential equation systems

2021 ◽  
Vol 390 ◽  
pp. 125590
Author(s):  
Ismail T. Huseynov ◽  
Arzu Ahmadova ◽  
Arran Fernandez ◽  
Nazim I. Mahmudov
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Analytical Solutions ◽  
Fractional Differential

scholarly journals Analytical solutions for conformable fractional Bratu-type equations

2018 ◽  
Vol 7 (1) ◽  
pp. 15 ◽  
Author(s):  
Mousa Ilie ◽  
Jafar Biazar ◽  
Zainab Ayati
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Fractional Differential Equations ◽  
Analytical Solutions ◽  
Analytical Method ◽  
Conformable Fractional Derivative ◽  
Fractional Differential

Solving fractional differential equations have a prominent function in different science such as physics and engineering. Therefore, are different definitions of the fractional derivative presented in recent years. The aim of the current paper is to solve the fractional differential equation by a semi-analytical method based on conformable fractional derivative. Fractional Bratu-type equations have been solved by the method and to show its capabilities. The obtained results have been compared with the exact solution.

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