Symmetry

Research

19 pages, 1150 KiB

Open AccessArticle

A Comparative Analysis of Fractional-Order Fokker–Planck Equation

by Fatemah Mofarreh, Asfandyar Khan, Rasool Shah and Alrazi Abdeljabbar

Symmetry 2023, 15(2), 430; https://doi.org/10.3390/sym15020430 - 6 Feb 2023

Cited by 6 | Viewed by 1613

The importance of partial differential equations in physics, mathematics and engineering cannot be emphasized enough. Partial differential equations are used to represent physical processes, which are then solved analytically or numerically to examine the dynamical behaviour of the system. The new iterative approach [...] Read more.

The importance of partial differential equations in physics, mathematics and engineering cannot be emphasized enough. Partial differential equations are used to represent physical processes, which are then solved analytically or numerically to examine the dynamical behaviour of the system. The new iterative approach and the Homotopy perturbation method are used in this article to solve the fractional order Fokker–Planck equation numerically. The Caputo sense is used to characterize the fractional derivatives. The suggested approach’s accuracy and applicability are demonstrated using illustrations. The proposed method’s accuracy is expressed in terms of absolute error. The proposed methods are found to be in good agreement with the exact solution of the problems using graphs and tables. The results acquired using the given approaches are also obtained at various fractional orders of the derivative. It is observed from the graphs and tables that fractional order solutions converge to an integer solution when the fractional orders approach the integer-order of the problems. The tabular and graphical view for the given problems is obtained through Maple. The presented approaches can be applied to existing non-linear fractional partial differential equations due to their accurate, simple and straightforward implementation. Full article

(This article belongs to the Special Issue Differential/Difference Equations and Its Application)

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15 pages, 6972 KiB

Open AccessArticle

Modified Exp-Function Method to Find Exact Solutions of Microtubules Nonlinear Dynamics Models

by Muhammad Shakeel, Attaullah, Nehad Ali Shah and Jae Dong Chung

Symmetry 2023, 15(2), 360; https://doi.org/10.3390/sym15020360 - 29 Jan 2023

Cited by 26 | Viewed by 1767

Abstract

In this paper, we use the modified exp

(- ψ (θ))

-function method to observe some of the solitary wave solutions for the microtubules (MTs). By treating the issues as nonlinear model partial differential equations describing microtubules, we were able to solve the [...] Read more.

In this paper, we use the modified exp

(- ψ (θ))

-function method to observe some of the solitary wave solutions for the microtubules (MTs). By treating the issues as nonlinear model partial differential equations describing microtubules, we were able to solve the problem. We then found specific solutions to the nonlinear evolution equation (NLEE) covering various parameters that are particularly significant in biophysics and nanobiosciences. In addition to the soliton-like pulse solutions, we also find the rational, trigonometric, hyperbolic, and exponential function characteristic solutions for this equation. The validity of the method we developed and the fact that it provides more solutions are demonstrated by comparison to other methods. We next use the software Mathematica 10 to generate 2D, 3D, and contour plots of the precise findings we observed using the suggested technique and the proper parameter values. Full article

(This article belongs to the Special Issue Differential/Difference Equations and Its Application)

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28 pages, 380 KiB

Open AccessArticle

Mild Solutions for the Time-Fractional Navier-Stokes Equations with MHD Effects

by Kinda Abuasbeh, Ramsha Shafqat, Azmat Ullah Khan Niazi and Muath Awadalla

Symmetry 2023, 15(2), 280; https://doi.org/10.3390/sym15020280 - 19 Jan 2023

Cited by 1 | Viewed by 1077

Abstract

Recently, various techniques and methods have been employed by mathematicians to solve specific types of fractional differential equations (FDEs) with symmetric properties. The study focuses on Navier-Stokes equations (NSEs) that involve MHD effects with time-fractional derivatives (FDs). The (NSEs) with time-FDs of order [...] Read more.

Recently, various techniques and methods have been employed by mathematicians to solve specific types of fractional differential equations (FDEs) with symmetric properties. The study focuses on Navier-Stokes equations (NSEs) that involve MHD effects with time-fractional derivatives (FDs). The (NSEs) with time-FDs of order

β \in (0, 1)

are investigated. To facilitate anomalous diffusion in fractal media, mild solutions and Mittag-Leffler functions are used. In

H^{δ, r}

, the existence, and uniqueness of local and global mild solutions are proved, as well as the symmetric structure created. Moderate local solutions are provided in

J_{r}

. Moreover, the regularity and existence of classical solutions to the equations in

J_{r}

. are established and presented. Full article

(This article belongs to the Special Issue Differential/Difference Equations and Its Application)

10 pages, 299 KiB

Open AccessArticle

Asymptotic Constancy for the Solutions of Caputo Fractional Differential Equations with Delay

by Halis Can Koyuncuoğlu, Youssef Raffoul and Nezihe Turhan

Symmetry 2023, 15(1), 88; https://doi.org/10.3390/sym15010088 - 29 Dec 2022

Viewed by 1168

Abstract

In this paper, we aim to study the neutral-type delayed Caputo fractional differential equations of the form [...] Read more.

In this paper, we aim to study the neutral-type delayed Caputo fractional differential equations of the form

^{C} D^{α} (x (t) - g (t, x_{t})) = f (t, x_{t}), t \in (t_{0}, \infty), t_{0} \geq 0

with order

0 < α < 1

, which can be used to describe the growth processes in real-life sciences at which the present growth depends on not only the past state but also the past growth rate. Our ultimate goal in this study is to concentrate on the convergence of the solutions to a predetermined constant by establishing a linkage between the delayed fractional differential equation and an integral equation. In our analysis, the sufficient conditions for the asymptotic results are obtained due to fixed point theory. The utilization of the contraction mapping principle is a convenient approach in obtaining technical conditions that guarantee the asymptotic constancy of the solutions. Full article

(This article belongs to the Special Issue Differential/Difference Equations and Its Application)

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14 pages, 309 KiB

Open AccessArticle

Multiple Existence Results of Nontrivial Solutions for a Class of Second-Order Partial Difference Equations

by Huan Zhang and Yuhua Long

Symmetry 2023, 15(1), 6; https://doi.org/10.3390/sym15010006 - 20 Dec 2022

Cited by 6 | Viewed by 1131

Abstract

In this paper, we consider the existence and multiplicity of nontrivial solutions for discrete elliptic Dirichlet problems [...] Read more.

In this paper, we consider the existence and multiplicity of nontrivial solutions for discrete elliptic Dirichlet problems

Δ_{1}^{2} u (i - 1, j) + Δ_{2}^{2} u (i, j - 1) = - f ((i, j), u (i, j)), (i, j) \in Ω, u (i, 0) = u (i, T_{2} + 1) = 0 i \in Z (1, T_{1}), u (0, j) = u (T_{1} + 1, j) = 0 j \in Z (1, T_{2}),

which have a symmetric structure. When the nonlinearity

f (\cdot, u)

is resonant at both zero and infinity, we construct a variational functional on a suitable function space and turn the problem of finding nontrivial solutions of discrete elliptic Dirichlet problems to seeking nontrivial critical points of the corresponding functional. We establish a series of results based on the existence of one, two or five nontrivial solutions under reasonable assumptions. Our results depend on the Morse theory and local linking. Full article

(This article belongs to the Special Issue Differential/Difference Equations and Its Application)

13 pages, 3815 KiB

Open AccessArticle

Solitons Solution of Riemann Wave Equation via Modified Exp Function Method

by Attaullah, Muhammad Shakeel, Bilal Ahmad, Nehad Ali Shah and Jae Dong Chung

Symmetry 2022, 14(12), 2574; https://doi.org/10.3390/sym14122574 - 6 Dec 2022

Cited by 14 | Viewed by 2077

Abstract

In the areas of tidal and tsunami waves in oceans, rivers, ion and magneto-sound waves in plasmas, electromagnetic waves in transmission lines, homogeneous and stationary media, etc., the Riemann wave equations are attractive nonlinear equations. The modified exp [...] Read more.

In the areas of tidal and tsunami waves in oceans, rivers, ion and magneto-sound waves in plasmas, electromagnetic waves in transmission lines, homogeneous and stationary media, etc., the Riemann wave equations are attractive nonlinear equations. The modified exp

(- Φ (η))

-function method is used in this article to show how well it can be applied to extract travelling and solitary wave solutions from higher-order nonlinear evolution equations (NLEEs) using the equations mentioned above. Trigonometric, hyperbolic, and exponential functions solitary wave solutions can be extracted using the above-mentioned technique. By changing specific values of the embedded parameters, we can obtain bell-form soliton, consolidated bell-shape soliton, compacton, singular kink soliton, flat kink shape soliton, smooth singular soliton, and other sorts of soliton solutions. The solutions are graphically illustrated in 3D and 2D for the accuracy of the outcome by using the Wolfram Mathematica 10. The verification of numerical solvers on the stability analysis of the solution is substantially aided by the analytic solutions. Full article

(This article belongs to the Special Issue Differential/Difference Equations and Its Application)

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16 pages, 487 KiB

Open AccessArticle

A New Approach to Approximate Solutions of Single Time-Delayed Stochastic Integral Equations via Orthogonal Functions

by Seyyedeh N. Kiaee, Morteza Khodabin, Reza Ezzati and António M. Lopes

Symmetry 2022, 14(10), 2085; https://doi.org/10.3390/sym14102085 - 7 Oct 2022

Cited by 2 | Viewed by 1363

Abstract

This paper proposes a new numerical method for solving single time-delayed stochastic differential equations via orthogonal functions. The basic principles of the technique are presented. The new method is applied to approximate two kinds of stochastic differential equations with additive and multiplicative noise. [...] Read more.

This paper proposes a new numerical method for solving single time-delayed stochastic differential equations via orthogonal functions. The basic principles of the technique are presented. The new method is applied to approximate two kinds of stochastic differential equations with additive and multiplicative noise. Excellence computational burden is achieved along with a

O (h^{2})

convergence rate, which is better than former methods. Two examples are examined to illustrate the validity and efficiency of the new technique. Full article

(This article belongs to the Special Issue Differential/Difference Equations and Its Application)

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14 pages, 1817 KiB

Open AccessArticle

Conservation Laws and Travelling Wave Solutions for a Negative-Order KdV-CBS Equation in 3+1 Dimensions

by Maria Luz Gandarias and Nauman Raza

Symmetry 2022, 14(9), 1861; https://doi.org/10.3390/sym14091861 - 6 Sep 2022

Cited by 7 | Viewed by 2019

Abstract

In this paper, we study a new negative-order KdV-CBS equation in

(3 + 1)

dimensions which is a combination of the Korteweg-de Vries (KdV) equation and Calogero–Bogoyavlenskii–Schiff (CBS) equation. Firstly, we determine the Lie point symmetries of the equation and conservation [...] Read more.

In this paper, we study a new negative-order KdV-CBS equation in

(3 + 1)

dimensions which is a combination of the Korteweg-de Vries (KdV) equation and Calogero–Bogoyavlenskii–Schiff (CBS) equation. Firstly, we determine the Lie point symmetries of the equation and conservation laws by using the multiplier method. The conservation laws will be used to obtain a triple reduction to a second order ordinary differential equation (ODE), which lead to line travelling waves and soliton solutions. Such solitons are obtained via the modified form of simple equation method and are displayed through three-dimensional plots at specific parameter values to lend physical meaning to nonlinear phenomena. It illustrates that these solutions might be extremely beneficial in understanding physical phenomena in a variety of applied mathematics areas. Full article

(This article belongs to the Special Issue Differential/Difference Equations and Its Application)

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11 pages, 776 KiB

Open AccessArticle

Extended Approach to the Asymptotic Behavior and Symmetric Solutions of Advanced Differential Equations

by Omar Bazighifan, Ali Hasan Ali, Fatemah Mofarreh and Youssef N. Raffoul

Symmetry 2022, 14(4), 686; https://doi.org/10.3390/sym14040686 - 26 Mar 2022

Cited by 28 | Viewed by 2193

Abstract

We studied the asymptotic behavior of fourth-order advanced differential equations of the form

{(a (υ) {(w^{'''} (υ))}^{β})}^{'} + q (υ) g (w (δ (υ))) = 0

. New results are presented for the oscillatory behavior of these equations [...] Read more.

We studied the asymptotic behavior of fourth-order advanced differential equations of the form

{(a (υ) {(w^{'''} (υ))}^{β})}^{'} + q (υ) g (w (δ (υ))) = 0

. New results are presented for the oscillatory behavior of these equations in the form of Philos-type and Hille–Nehari oscillation criteria. Some illustrative examples are presented. Full article

(This article belongs to the Special Issue Differential/Difference Equations and Its Application)

14 pages, 305 KiB

Open AccessArticle

Anticipated Generalized Backward Doubly Stochastic Differential Equations

by Tie Wang and Jiaxin Yu

Symmetry 2022, 14(1), 114; https://doi.org/10.3390/sym14010114 - 9 Jan 2022

Cited by 3 | Viewed by 1370

Abstract

In this paper, we explore a new class of stochastic differential equations called anticipated generalized backward doubly stochastic differential equations (AGBDSDEs), which not only involve two symmetric integrals related to two independent Brownian motions and an integral driven by a continuous increasing process [...] Read more.

In this paper, we explore a new class of stochastic differential equations called anticipated generalized backward doubly stochastic differential equations (AGBDSDEs), which not only involve two symmetric integrals related to two independent Brownian motions and an integral driven by a continuous increasing process but also include generators depending on the anticipated terms of the solution (Y, Z). Firstly, we prove the existence and uniqueness theorem for AGBDSDEs. Further, two comparison theorems are obtained after finding a new comparison theorem for GBDSDEs. Full article

(This article belongs to the Special Issue Differential/Difference Equations and Its Application)

17 pages, 338 KiB

Open AccessArticle

Nonlocal Neumann Boundary Value Problem for Fractional Symmetric Hahn Integrodifference Equations

by Thongchai Dumrongpokaphan, Nichaphat Patanarapeelert and Thanin Sitthiwirattham

Symmetry 2021, 13(12), 2303; https://doi.org/10.3390/sym13122303 - 2 Dec 2021

Cited by 1 | Viewed by 1211

Abstract

In this article, we present a nonlocal Neumann boundary value problems for separate sequential fractional symmetric Hahn integrodifference equation. The problem contains five fractional symmetric Hahn difference operators and one fractional symmetric Hahn integral of different orders. We employ Banach fixed point theorem [...] Read more.

In this article, we present a nonlocal Neumann boundary value problems for separate sequential fractional symmetric Hahn integrodifference equation. The problem contains five fractional symmetric Hahn difference operators and one fractional symmetric Hahn integral of different orders. We employ Banach fixed point theorem and Schauder’s fixed point theorem to study the existence results of the problem. Full article

(This article belongs to the Special Issue Differential/Difference Equations and Its Application)

12 pages, 754 KiB

Open AccessArticle

On a Conjecture for the One-Dimensional Perturbed Gelfand Problem for the Combustion Theory

by Huizeng Qin and Youmin Lu

Symmetry 2021, 13(11), 2137; https://doi.org/10.3390/sym13112137 - 10 Nov 2021

Cited by 1 | Viewed by 1490

Abstract

We investigate the well-known one-dimensional perturbed Gelfand boundary value problem and approximate the values of

α_{0}, λ_{*}

and

λ^{*}

such that this problem has a unique solution when

0 < α < α_{0}

and

λ > 0,

[...] Read more.

We investigate the well-known one-dimensional perturbed Gelfand boundary value problem and approximate the values of

α_{0}, λ_{*}

and

λ^{*}

such that this problem has a unique solution when

0 < α < α_{0}

and

λ > 0,

and has three solutions when

α > α_{0}

and

λ_{*} < λ < λ^{*} .

The solutions of this problem are always even functions due to its symmetric boundary values and autonomous characteristics. We use numerical computation to show that

4.0686722336 < α_{0} < 4.0686722344

. This result improves the existing result for

α_{0} \approx 4.069

and increases the accuracy of

α_{0}

to

10^{- 8} .

We developed an algorithm that reduces errors and increases efficiency in our computation. The interval of

λ

for this problem to have three solutions for given values of

α

is also computed with accuracy up to

10^{- 14} .

Full article

(This article belongs to the Special Issue Differential/Difference Equations and Its Application)

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9 pages, 245 KiB

Open AccessArticle

A Note on Solitons with Generalized Geodesic Vector Field

by Adara M. Blaga, Amira Ishan and Sharief Deshmukh

Symmetry 2021, 13(7), 1104; https://doi.org/10.3390/sym13071104 - 22 Jun 2021

Cited by 5 | Viewed by 1349

Abstract

We consider a general notion of an almost Ricci soliton and establish some curvature properties for the case in which the potential vector field of the soliton is a generalized geodesic or a 2-Killing vector field. In this vein, we characterize trivial generalized [...] Read more.

We consider a general notion of an almost Ricci soliton and establish some curvature properties for the case in which the potential vector field of the soliton is a generalized geodesic or a 2-Killing vector field. In this vein, we characterize trivial generalized Ricci solitons. Full article

(This article belongs to the Special Issue Differential/Difference Equations and Its Application)

12 pages, 307 KiB

Open AccessArticle

An Existence Result for a Class of p(x)—Anisotropic Type Equations

by Anass Ourraoui and Maria Alessandra Ragusa

Symmetry 2021, 13(4), 633; https://doi.org/10.3390/sym13040633 - 9 Apr 2021

Cited by 9 | Viewed by 1489

Abstract

In this paper, we study a class of anisotropic variable exponent problems involving the

\vec{p}

(.)-Laplacian. By using the variational method as our main tool, we present a result regarding the existence of solutions without the so-called Ambrosetti–Rabinowitz-type conditions. Full article

(This article belongs to the Special Issue Differential/Difference Equations and Its Application)

11 pages, 279 KiB

Open AccessArticle

Oscillation Results for Nonlinear Higher-Order Differential Equations with Delay Term

by Alanoud Almutairi, Omar Bazighifan and Youssef N. Raffoul

Symmetry 2021, 13(3), 446; https://doi.org/10.3390/sym13030446 - 10 Mar 2021

Cited by 3 | Viewed by 1690

Abstract

The aim of this work is to investigate the oscillation of solutions of higher-order nonlinear differential equations with a middle term. By using the integral averaging technique, Riccati transformation technique and comparison technique, several oscillatory properties are presented that unify the results obtained [...] Read more.

The aim of this work is to investigate the oscillation of solutions of higher-order nonlinear differential equations with a middle term. By using the integral averaging technique, Riccati transformation technique and comparison technique, several oscillatory properties are presented that unify the results obtained in the literature. Some examples are presented to demonstrate the main results. Full article

(This article belongs to the Special Issue Differential/Difference Equations and Its Application)

16 pages, 319 KiB

Open AccessArticle

On the Hybrid Fractional Differential Equations with Fractional Proportional Derivatives of a Function with Respect to a Certain Function

by Mohamed I. Abbas and Maria Alessandra Ragusa

Symmetry 2021, 13(2), 264; https://doi.org/10.3390/sym13020264 - 4 Feb 2021

Cited by 108 | Viewed by 3668

Abstract

This paper deals with a new class of hybrid fractional differential equations with fractional proportional derivatives of a function with respect to a certain continuously differentiable and increasing function

ϑ

. By means of a hybrid fixed point theorem for a product of [...] Read more.

This paper deals with a new class of hybrid fractional differential equations with fractional proportional derivatives of a function with respect to a certain continuously differentiable and increasing function

ϑ

. By means of a hybrid fixed point theorem for a product of two operators, an existence result is proved. Furthermore, the sufficient conditions of the continuous dependence on the given parameters are investigated. Finally, a simulative example is given to highlight the acquired outcomes. Full article

(This article belongs to the Special Issue Differential/Difference Equations and Its Application)

11 pages, 1315 KiB

Open AccessFeature PaperArticle

Multiscale Discrete Element Modeling

by Andrew A. Zhuravlev, Karine K. Abgaryan and Dmitry L. Reviznikov

Symmetry 2021, 13(2), 219; https://doi.org/10.3390/sym13020219 - 28 Jan 2021

Cited by 2 | Viewed by 1950

Abstract

A multiscale approach to discrete element modeling is presented. A distinctive feature of the method is that each macroscopic discrete element has an associated atomic sample representing the material’s atomic structure. The dynamics of the elements on macro and micro levels are described [...] Read more.

A multiscale approach to discrete element modeling is presented. A distinctive feature of the method is that each macroscopic discrete element has an associated atomic sample representing the material’s atomic structure. The dynamics of the elements on macro and micro levels are described by systems of ordinary differential equations, which are solved in a self-consistent manner. A full cycle of multiscale simulations is applied to polycrystalline silicon. Macroscale elastic properties of silicon were obtained only using data extracted from the quantum mechanical properties. The results of computational experiments correspond well to the reference data. Full article

(This article belongs to the Special Issue Differential/Difference Equations and Its Application)

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11 pages, 276 KiB

Open AccessArticle

Neutral Delay Differential Equations: Oscillation Conditions for the Solutions

by Omar Bazighifan, Hammad Alotaibi and Abd Allaah A. Mousa

Symmetry 2021, 13(1), 101; https://doi.org/10.3390/sym13010101 - 8 Jan 2021

Cited by 31 | Viewed by 3894

Abstract

The purpose of this article is to explore the asymptotic properties for a class of fourth-order neutral differential equations. Based on a comparison with the differential inequality of the first-order, we have provided new oscillation conditions for the solutions of fourth-order neutral differential [...] Read more.

The purpose of this article is to explore the asymptotic properties for a class of fourth-order neutral differential equations. Based on a comparison with the differential inequality of the first-order, we have provided new oscillation conditions for the solutions of fourth-order neutral differential equations. The obtained results can be used to develop and provide theoretical support for and to further develop the study of oscillation for a class of fourth-order neutral differential equations. Finally, we provide an illustrated example to demonstrate the effectiveness of our new criteria. Full article

(This article belongs to the Special Issue Differential/Difference Equations and Its Application)

12 pages, 254 KiB

Open AccessArticle

Oscillation Criteria for a Class of Third-Order Damped Neutral Differential Equations

by Elmetwally M. Elabbasy, Belgees Qaraad, Thabet Abdeljawad and Osama Moaaz

Symmetry 2020, 12(12), 1988; https://doi.org/10.3390/sym12121988 - 1 Dec 2020

Cited by 6 | Viewed by 1533

Abstract

In this paper, we study the asymptotic and oscillatory properties of a certain class of third-order neutral delay differential equations with middle term. We obtain new characterizations of oscillation of the third-order neutral equation in terms of oscillation of a related, well-studied, second-order [...] Read more.

In this paper, we study the asymptotic and oscillatory properties of a certain class of third-order neutral delay differential equations with middle term. We obtain new characterizations of oscillation of the third-order neutral equation in terms of oscillation of a related, well-studied, second-order linear equation without damping. An Example is provided to illustrate the main results. Full article

(This article belongs to the Special Issue Differential/Difference Equations and Its Application)

10 pages, 245 KiB

Open AccessArticle

On the Differential Equation Governing Torqued Vector Fields on a Riemannian Manifold

by Sharief Deshmukh, Nasser Bin Turki and Haila Alodan

Symmetry 2020, 12(12), 1941; https://doi.org/10.3390/sym12121941 - 25 Nov 2020

Cited by 3 | Viewed by 1335

Abstract

In this article, we show that the presence of a torqued vector field on a Riemannian manifold can be used to obtain rigidity results for Riemannian manifolds of constant curvature. More precisely, we show that there is no torqued vector field on n [...] Read more.

In this article, we show that the presence of a torqued vector field on a Riemannian manifold can be used to obtain rigidity results for Riemannian manifolds of constant curvature. More precisely, we show that there is no torqued vector field on n-sphere

S^{n} (c)

. A nontrivial example of torqued vector field is constructed on an open subset of the Euclidean space

E^{n}

whose torqued function and torqued form are nowhere zero. It is shown that owing to topology of the Euclidean space

E^{n}

, this type of torqued vector fields could not be extended globally to

E^{n}

. Finally, we find a necessary and sufficient condition for a torqued vector field on a compact Riemannian manifold to be a concircular vector field. Full article

(This article belongs to the Special Issue Differential/Difference Equations and Its Application)

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