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Open AccessArticle

New Exact Solutions and Conservation Laws to the Fractional-Order Fokker–Planck Equations

by

,

,

,

and

Symmetry 2020, 12(8), 1282; https://doi.org/10.3390/sym12081282 - 03 Aug 2020

Cited by 3 | Viewed by 1389

Abstract

The main purpose of this paper is to present a new approach to achieving analytical solutions of parameter containing fractional-order differential equations. Using the nonlinear self-adjoint notion, approximate solutions, conservation laws and symmetries of these equations are also obtained via a new formulation [...] Read more.

The main purpose of this paper is to present a new approach to achieving analytical solutions of parameter containing fractional-order differential equations. Using the nonlinear self-adjoint notion, approximate solutions, conservation laws and symmetries of these equations are also obtained via a new formulation of an improved form of the Noether’s theorem. It is indicated that invariant solutions, reduced equations, perturbed or unperturbed symmetries and conservation laws can be obtained by applying a nonlinear self-adjoint notion. The method is applied to the time fractional-order Fokker–Planck equation. We obtained new results in a highly efficient and elegant manner. Full article

(This article belongs to the Special Issue New developments in Functional and Fractional Differential Equations and in Lie Symmetry)

Open AccessArticle

New Soliton Solutions of Fractional Jaulent-Miodek System with Symmetry Analysis

by

Subhadarshan Sahoo

,

Santanu Saha Ray

,

Mohamed Aly Mohamed Abdou

,

Mustafa Inc

and

Yu-Ming Chu

Symmetry 2020, 12(6), 1001; https://doi.org/10.3390/sym12061001 - 12 Jun 2020

Cited by 33 | Viewed by 1704

Abstract

New soliton solutions of fractional Jaulent-Miodek (JM) system are presented via symmetry analysis and fractional logistic function methods. Fractional Lie symmetry analysis is unified with symmetry analysis method. Conservation laws of the system are used to obtain new conserved vectors. Numerical simulations of [...] Read more.

New soliton solutions of fractional Jaulent-Miodek (JM) system are presented via symmetry analysis and fractional logistic function methods. Fractional Lie symmetry analysis is unified with symmetry analysis method. Conservation laws of the system are used to obtain new conserved vectors. Numerical simulations of the JM equations and efficiency of the methods are presented. These solutions might be imperative and significant for the explanation of some practical physical phenomena. The results show that present methods are powerful, competitive, reliable, and easy to implement for the nonlinear fractional differential equations. Full article

(This article belongs to the Special Issue New developments in Functional and Fractional Differential Equations and in Lie Symmetry)

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Open AccessArticle

On Nonlinear Fractional Difference Equation with Delay and Impulses

by

Rujira Ouncharoen

,

Saowaluck Chasreechai

and

Thanin Sitthiwirattham

Symmetry 2020, 12(6), 980; https://doi.org/10.3390/sym12060980 - 08 Jun 2020

Cited by 2 | Viewed by 975

Abstract

In this paper, we establish the existence results for a nonlinear fractional difference equation with delay and impulses. The Banach and Schauder’s fixed point theorems are employed as tools to study the existence of its solutions. We obtain the theorems showing the conditions [...] Read more.

In this paper, we establish the existence results for a nonlinear fractional difference equation with delay and impulses. The Banach and Schauder’s fixed point theorems are employed as tools to study the existence of its solutions. We obtain the theorems showing the conditions for existence results. Finally, we provide an example to show the applicability of our results. Full article

(This article belongs to the Special Issue New developments in Functional and Fractional Differential Equations and in Lie Symmetry)

Open AccessArticle

A Collocation Approach for Solving Time-Fractional Stochastic Heat Equation Driven by an Additive Noise

by

Afshin Babaei

,

Hossein Jafari

and

S. Banihashemi

Symmetry 2020, 12(6), 904; https://doi.org/10.3390/sym12060904 - 01 Jun 2020

Cited by 19 | Viewed by 1752

Abstract

A spectral collocation approach is constructed to solve a class of time-fractional stochastic heat equations (TFSHEs) driven by Brownian motion. Stochastic differential equations with additive noise have an important role in explaining some symmetry phenomena such as symmetry breaking in molecular vibrations. Finding [...] Read more.

A spectral collocation approach is constructed to solve a class of time-fractional stochastic heat equations (TFSHEs) driven by Brownian motion. Stochastic differential equations with additive noise have an important role in explaining some symmetry phenomena such as symmetry breaking in molecular vibrations. Finding the exact solution of such equations is difficult in many cases. Thus, a collocation method based on sixth-kind Chebyshev polynomials (SKCPs) is introduced to assess their numerical solutions. This collocation approach reduces the considered problem to a system of linear algebraic equations. The convergence and error analysis of the suggested scheme are investigated. In the end, numerical results and the order of convergence are evaluated for some numerical test problems to illustrate the efficiency and robustness of the presented method. Full article

(This article belongs to the Special Issue New developments in Functional and Fractional Differential Equations and in Lie Symmetry)

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Open AccessArticle

Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays

by

Emad R. Attia

,

Hassan A. El-Morshedy

and

Ioannis P. Stavroulakis

Symmetry 2020, 12(5), 718; https://doi.org/10.3390/sym12050718 - 02 May 2020

Cited by 6 | Viewed by 1498

Abstract

New sufficient criteria are obtained for the oscillation of a non-autonomous first order differential equation with non-monotone delays. Both recursive and lower-upper limit types criteria are given. The obtained results improve most recent published results. An example is given to illustrate the applicability [...] Read more.

New sufficient criteria are obtained for the oscillation of a non-autonomous first order differential equation with non-monotone delays. Both recursive and lower-upper limit types criteria are given. The obtained results improve most recent published results. An example is given to illustrate the applicability and strength of our results. Full article

(This article belongs to the Special Issue New developments in Functional and Fractional Differential Equations and in Lie Symmetry)

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Open AccessArticle

Noether Symmetries of a Generalized Coupled Lane-Emden-Klein-Gordon-Fock System with Central Symmetry

by

B. Muatjetjeja

,

S. O. Mbusi

and

A. R. Adem

Symmetry 2020, 12(4), 566; https://doi.org/10.3390/sym12040566 - 05 Apr 2020

Cited by 6 | Viewed by 1117

Abstract

In this paper we carry out a complete Noether symmetry analysis of a generalized coupled Lane-Emden-Klein-Gordon-Fock system with central symmetry. It is shown that several cases transpire for which the Noether symmetries exist. Moreover, we derive conservation laws connected with the admitted Noether [...] Read more.

In this paper we carry out a complete Noether symmetry analysis of a generalized coupled Lane-Emden-Klein-Gordon-Fock system with central symmetry. It is shown that several cases transpire for which the Noether symmetries exist. Moreover, we derive conservation laws connected with the admitted Noether symmetries. Furthermore, we fleetingly discuss the physical interpretation of the these conserved vectors. Full article

(This article belongs to the Special Issue New developments in Functional and Fractional Differential Equations and in Lie Symmetry)

Open AccessArticle

Finite Difference Approximation Method for a Space Fractional Convection–Diffusion Equation with Variable Coefficients

by

Eyaya Fekadie Anley

and

Zhoushun Zheng

Symmetry 2020, 12(3), 485; https://doi.org/10.3390/sym12030485 - 23 Mar 2020

Cited by 16 | Viewed by 2474

Abstract

Space non-integer order convection–diffusion descriptions are generalized form of integer order convection–diffusion problems expressing super diffusive and convective transport processes. In this article, we propose finite difference approximation for space fractional convection–diffusion model having space variable coefficients on the given bounded domain over [...] Read more.

Space non-integer order convection–diffusion descriptions are generalized form of integer order convection–diffusion problems expressing super diffusive and convective transport processes. In this article, we propose finite difference approximation for space fractional convection–diffusion model having space variable coefficients on the given bounded domain over time and space. It is shown that the Crank–Nicolson difference scheme based on the right shifted Grünwald–Letnikov difference formula is unconditionally stable and it is also of second order consistency both in temporal and spatial terms with extrapolation to the limit approach. Numerical experiments are tested to verify the efficiency of our theoretical analysis and confirm order of convergence. Full article

(This article belongs to the Special Issue New developments in Functional and Fractional Differential Equations and in Lie Symmetry)

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Open AccessArticle

A Sharp Oscillation Criterion for a Linear Differential Equation with Variable Delay

by

Ábel Garab

Symmetry 2019, 11(11), 1332; https://doi.org/10.3390/sym11111332 - 24 Oct 2019

Cited by 8 | Viewed by 1572

Abstract

We consider linear differential equations with variable delay of the form

x^{'} (t) + p (t) x (t - τ (t)) = 0, t \geq t_{0},

where [...] Read more.

We consider linear differential equations with variable delay of the form

x^{'} (t) + p (t) x (t - τ (t)) = 0, t \geq t_{0},

where

p : [t_{0}, \infty) \to [0, \infty)

and

τ : [t_{0}, \infty) \to (0, \infty)

are continuous functions, such that

t - τ (t) \to \infty

(as

t \to \infty

). It is well-known that, for the oscillation of all solutions, it is necessary that

B : = \underset{t \to \infty}{lim sup} A (t) \geq \frac{1}{e} holds, where A : = (t) \int_{t - τ (t)}^{t} p (s) d s .

Our main result shows that, if the function A is slowly varying at infinity (in additive form), then under mild additional assumptions on p and

τ

, condition

B > 1 / e

implies that all solutions of the above delay differential equation are oscillatory. Full article

(This article belongs to the Special Issue New developments in Functional and Fractional Differential Equations and in Lie Symmetry)

Open AccessArticle

Approximation of a Linear Autonomous Differential Equation with Small Delay

by

Áron Fehér

,

Lorinc Márton

and

Mihály Pituk

Symmetry 2019, 11(10), 1299; https://doi.org/10.3390/sym11101299 - 15 Oct 2019

Cited by 7 | Viewed by 1800

Abstract

A linear autonomous differential equation with small delay is considered in this paper. It is shown that under a smallness condition the delay differential equation is asymptotically equivalent to a linear ordinary differential equation with constant coefficients. The coefficient matrix of the ordinary [...] Read more.

A linear autonomous differential equation with small delay is considered in this paper. It is shown that under a smallness condition the delay differential equation is asymptotically equivalent to a linear ordinary differential equation with constant coefficients. The coefficient matrix of the ordinary differential equation is a solution of an associated matrix equation and it can be written as a limit of a sequence of matrices obtained by successive approximations. The eigenvalues of the approximating matrices converge exponentially to the dominant characteristic roots of the delay differential equation and an explicit estimate for the approximation error is given. Full article

(This article belongs to the Special Issue New developments in Functional and Fractional Differential Equations and in Lie Symmetry)

Open AccessEditor’s ChoiceArticle

Around the Model of Infection Disease: The Cauchy Matrix and Its Properties

by

Alexander Domoshnitsky

,

Irina Volinsky

and

Marina Bershadsky

Symmetry 2019, 11(8), 1016; https://doi.org/10.3390/sym11081016 - 06 Aug 2019

Cited by 12 | Viewed by 2423

Abstract

In this paper the model of infection diseases by Marchuk is considered. Mathematical questions which are important in its study are discussed. Among them there are stability of stationary points, construction of the Cauchy matrices of linearized models, estimates of solutions. The novelty [...] Read more.

In this paper the model of infection diseases by Marchuk is considered. Mathematical questions which are important in its study are discussed. Among them there are stability of stationary points, construction of the Cauchy matrices of linearized models, estimates of solutions. The novelty we propose is in a distributed feedback control which affects the antibody concentration. We use this control in the form of an integral term and come to the analysis of nonlinear integro-differential systems. New methods for the study of stability of linearized integro–differential systems describing the model of infection diseases are proposed. Explicit conditions of the exponential stability of the stationary points characterizing the state of the healthy body are obtained. The method of the paper is based on the symmetry properties of the Cauchy matrices which allow us their construction. Full article

(This article belongs to the Special Issue New developments in Functional and Fractional Differential Equations and in Lie Symmetry)

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