New developments in Functional and Fractional Differential Equations and in Lie Symmetry

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (31 July 2020) | Viewed by 23587

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Guest Editor
Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece
Interests: differential equations with deviating arguments; difference equations; functional equations; partial differential equations; dynamics with delays; iterative methods

Special Issue Information

Dear Colleagues,

Delay, difference, functional, fractional and partial differential equations have many applications in science and engineering. During the last few decades various methods have been introduced in the study of the solutions to these equations.

In this Special Issue, we invite and welcome review, expository, and original research articles dealing with recent advances on the topics of “New Developments in Functional and Fractional Differential Equations and in Lie Symmetry”.

Prof. Dr. Ioannis Stavroulakis
Prof. Dr. H Jafari
Guest Editors

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Keywords

  • Delay Differential Equations
  • Difference Equations
  • Functional Equations
  • Lie Symmetry
  • Fractional Differential Equations
  • Local Fractional Differential Equations
  • Approximation Methods
  • Applied Functional Analysis
  • Partial Differential Equations

Published Papers (10 papers)

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Research

13 pages, 267 KiB  
Article
New Exact Solutions and Conservation Laws to the Fractional-Order Fokker–Planck Equations
by Nematollah Kadkhoda, Elham Lashkarian, Mustafa Inc, Mehmet Ali Akinlar and Yu-Ming Chu
Symmetry 2020, 12(8), 1282; https://doi.org/10.3390/sym12081282 - 3 Aug 2020
Cited by 16 | Viewed by 1937
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
15 pages, 1298 KiB  
Article
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 50 | Viewed by 2379
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
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13 pages, 289 KiB  
Article
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 - 8 Jun 2020
Cited by 2 | Viewed by 1453
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
15 pages, 1020 KiB  
Article
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 - 1 Jun 2020
Cited by 25 | Viewed by 2503
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
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17 pages, 448 KiB  
Article
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 - 2 May 2020
Cited by 7 | Viewed by 2101
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
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6 pages, 203 KiB  
Article
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 - 5 Apr 2020
Cited by 7 | Viewed by 1524
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
19 pages, 659 KiB  
Article
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 24 | Viewed by 3704
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
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10 pages, 242 KiB  
Article
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 1966
Abstract
We consider linear differential equations with variable delay of the form x ( t ) + p ( t ) x ( t τ ( t ) ) = 0 , t 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 t 0 , where p : [ t 0 , ) [ 0 , ) and τ : [ t 0 , ) ( 0 , ) are continuous functions, such that t τ ( t ) (as t ). It is well-known that, for the oscillation of all solutions, it is necessary that B : = lim sup t A ( t ) 1 e holds , where A : = ( t ) 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
10 pages, 247 KiB  
Article
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 9 | Viewed by 2276
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
22 pages, 310 KiB  
Article
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 - 6 Aug 2019
Cited by 17 | Viewed by 2977
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
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