Symmetry

Research

25 pages, 393 KiB

Open AccessArticle

A Study on Solutions for a Class of Higher-Order System of Singular Boundary Value Problem

by Biswajit Pandit, Amit K. Verma and Ravi P. Agarwal

Symmetry 2023, 15(9), 1729; https://doi.org/10.3390/sym15091729 - 8 Sep 2023

Viewed by 568

In this article, we propose a fourth-order non-self-adjoint system of singular boundary value problems (SBVPs), which arise in the theory of epitaxial growth by considering hte equation [...] Read more.

In this article, we propose a fourth-order non-self-adjoint system of singular boundary value problems (SBVPs), which arise in the theory of epitaxial growth by considering hte equation

\frac{1}{r^{β}} {\{r^{β} {[\frac{1}{r^{β}} {(r^{β} Θ^{'})}^{^{'}}]}^{'}\}}^{^{'}} = \frac{1}{2 r^{β}} (K_{11} (μ^{'} Θ^{' 2} + 2 μ Θ^{^{'}} Θ^{^{″}}) + K_{12} (μ^{'} φ^{' 2} + 2 μ φ^{^{'}} φ^{^{″}})) + λ_{1} G_{1} (r),

\frac{1}{r^{β}} {\{r^{β} {[\frac{1}{r^{β}} {(r^{β} φ^{'})}^{^{'}}]}^{'}\}}^{^{'}} = \frac{1}{2 r^{β}} (K_{21} (μ^{'} Θ^{' 2} + 2 μ Θ^{^{'}} Θ^{^{″}}) + K_{22} (μ^{'} φ^{' 2} + 2 μ φ^{^{'}} φ^{^{″}})) + λ_{2} G_{2} (r),

where

λ_{1} \geq 0

and

λ_{2} \geq 0

are two parameters,

μ = p r^{2 β - 2}, p \in R^{+}

,

G_{1}, G_{2} \in L^{1} [0, 1]

such that

M_{1}^{*} \geq G_{1} (r) \geq M_{1} > 0, M_{2}^{*} \geq G_{2} (r) \geq M_{2} > 0

and

K_{12} > 0

,

K_{11} \geq 0

, and

K_{21} > 0

,

K_{22} \geq 0

are constants that are connected by the relation

(K_{12} + K_{22}) \geq (K_{11} + K_{21})

and

β > 1

. To study the governing equation, we consider three different types of homogeneous boundary conditions. We use the transformation

t = \frac{r^{1 + β}}{1 + β}

to deduce the second-order singular boundary value problem. Also, for

β = p = G_{1} (r) = G_{2} (r) = 1

, it admits dual solutions. We show the existence of at least one solution in continuous space. We derive a sign of solutions. Furthermore, we compute the approximate bound of the parameters to point out the region of nonexistence. We also conclude bounds are symmetric with respect to two different transformations. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear and Convex Analysis)

19 pages, 356 KiB

Open AccessFeature PaperArticle

Existence for Two-Point nth Order Boundary Value Problems under Barrier Strips

by Ravi P. Agarwal, Todor Z. Todorov and Petio S. Kelevedjiev

Symmetry 2023, 15(7), 1394; https://doi.org/10.3390/sym15071394 - 10 Jul 2023

Viewed by 624

Abstract

Using barrier strip conditions, we study the solvability of two-point boundary value problems for the equation

x^{(n)} = f (t, x, x^{'}, \dots, x^{(n - 1)}) .

In the case [...] Read more.

Using barrier strip conditions, we study the solvability of two-point boundary value problems for the equation

x^{(n)} = f (t, x, x^{'}, \dots, x^{(n - 1)}) .

In the case

n = 4,

we apply the used approach to obtain results guaranteeing positive or non-negative, monotone, convex solutions to boundary value problems with various boundary conditions. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear and Convex Analysis)

7 pages, 255 KiB

Open AccessArticle

Application of the New Iterative Method (NIM) to the Generalized Burgers–Huxley Equation

by Belal Batiha, Firas Ghanim and Khaled Batiha

Symmetry 2023, 15(3), 688; https://doi.org/10.3390/sym15030688 - 9 Mar 2023

Cited by 1 | Viewed by 1381

Abstract

In this paper, we propose the new iterative method (NIM) for solving the generalized Burgers–Huxley equation. NIM provides an approximate solution without the need for discretization and is based on a set of iterative equations. We compared the NIM with other established methods, [...] Read more.

In this paper, we propose the new iterative method (NIM) for solving the generalized Burgers–Huxley equation. NIM provides an approximate solution without the need for discretization and is based on a set of iterative equations. We compared the NIM with other established methods, such as Variational Iteration Method (VIM), Adomian Decomposition Method (ADM), and the exact solution, and found that it is efficient and easy to use. NIM has the advantage of quick convergence, easy implementation, and handling of a wide range of initial conditions. The comparison of the present symmetrical results with the existing literature is satisfactory. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear and Convex Analysis)

9 pages, 253 KiB

Open AccessArticle

Toeplitz Determinants for a Certain Family of Analytic Functions Endowed with Borel Distribution

by Abbas Kareem Wanas, Fethiye Müge Sakar, Georgia Irina Oros and Luminiţa-Ioana Cotîrlă

Symmetry 2023, 15(2), 262; https://doi.org/10.3390/sym15020262 - 17 Jan 2023

Cited by 4 | Viewed by 1460

Abstract

In this work, we derive coefficient bounds for the symmetric Toeplitz matrices

T_{2} (2)

,

T_{2} (3)

,

T_{3} (1)

, and

T_{3} (2)

, which are the known first four [...] Read more.

In this work, we derive coefficient bounds for the symmetric Toeplitz matrices

T_{2} (2)

,

T_{2} (3)

,

T_{3} (1)

, and

T_{3} (2)

, which are the known first four determinants for a new family of analytic functions with Borel distribution series in the open unit disk U. Further, some special cases of results obtained are also pointed. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear and Convex Analysis)

14 pages, 7662 KiB

Open AccessArticle

Visual Analysis of Mixed Algorithms with Newton and Abbasbandy Methods Using Periodic Parameters

by Safeer Hussain Khan, Lateef Olakunle Jolaoso and Maggie Aphane

Symmetry 2022, 14(12), 2484; https://doi.org/10.3390/sym14122484 - 23 Nov 2022

Cited by 1 | Viewed by 946

Abstract

In this paper, we proposed two mixed algorithms of Newton’s and Abbasbandy’s methods using a known iteration scheme from fixed point theory in polynomiography. We numerically investigated some properties of the proposed algorithms using periodic sequence parameters instead of the constant parameters that [...] Read more.

In this paper, we proposed two mixed algorithms of Newton’s and Abbasbandy’s methods using a known iteration scheme from fixed point theory in polynomiography. We numerically investigated some properties of the proposed algorithms using periodic sequence parameters instead of the constant parameters that are mostly used by many authors. Two pseudo-Newton algorithms were introduced based on the mixed iterations for the purpose of generating polynomiographs. The properties of the obtained polynomiographs were studied with respect to their graphics, turning effects and computation time. Moreover, some of these polynomiographs exhibited symmetrical properties when the degree of the polynomial was even. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear and Convex Analysis)

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13 pages, 663 KiB

Open AccessArticle

Freelance Model with Atangana–Baleanu Caputo Fractional Derivative

by Fareeha Sami Khan, M. Khalid, Areej A. Al-moneef, Ali Hasan Ali and Omar Bazighifan

Symmetry 2022, 14(11), 2424; https://doi.org/10.3390/sym14112424 - 16 Nov 2022

Cited by 12 | Viewed by 2068

Abstract

As technology advances and the Internet makes our world a global village, it is important to understand the prospective career of freelancing. A novel symmetric fractional mathematical model is introduced in this study to describe the competitive market of freelancing and the significance [...] Read more.

As technology advances and the Internet makes our world a global village, it is important to understand the prospective career of freelancing. A novel symmetric fractional mathematical model is introduced in this study to describe the competitive market of freelancing and the significance of information in its acceptance. In this study, fixed point theory is applied to analyze the uniqueness and existence of the fractional freelance model. Its numerical solution is derived using the fractional Euler’s method, and each case has been presented graphically as well as tabular. Further, the results have been compared with the classic freelance model and real data to show the importance of this model. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear and Convex Analysis)

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15 pages, 318 KiB

Open AccessArticle

New Conditions for Testing the Oscillation of Third-Order Differential Equations with Distributed Arguments

by A. Al Themairi, Belgees Qaraad, Omar Bazighifan and Kamsing Nonlaopon

Symmetry 2022, 14(11), 2416; https://doi.org/10.3390/sym14112416 - 15 Nov 2022

Cited by 5 | Viewed by 903

Abstract

In this paper, we consider a certain class of third-order nonlinear delay differential equations with distributed arguments. By the principle of comparison, we obtain the conditions for the nonexistence of positive decreasing solutions as well as, and by using the Riccati transformation technique, [...] Read more.

In this paper, we consider a certain class of third-order nonlinear delay differential equations with distributed arguments. By the principle of comparison, we obtain the conditions for the nonexistence of positive decreasing solutions as well as, and by using the Riccati transformation technique, we obtain the conditions for the nonexistence of increasing solutions. Therefore, we get new sufficient criteria that ensure that every solution of the studied equation oscillates. Asymmetry plays an important role in describing the properties of solutions of differential equations. An example is given to illustrate the importance of our results. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear and Convex Analysis)

9 pages, 276 KiB

Open AccessArticle

by Luminiţa-Ioana Cotîrlǎ and Abbas Kareem Wanas

Symmetry 2022, 14(11), 2263; https://doi.org/10.3390/sym14112263 - 28 Oct 2022

Cited by 6 | Viewed by 1213

Abstract

In this paper, we define certain families

S_{E}^{*} (ϑ)

and

C_{E} (ϑ)

of holomorphic and bi-univalent functions which are defined in the open unit disk U. We establish upper bounds for the initial Taylor–Maclaurin coefficients [...] Read more.

In this paper, we define certain families

S_{E}^{*} (ϑ)

and

C_{E} (ϑ)

of holomorphic and bi-univalent functions which are defined in the open unit disk U. We establish upper bounds for the initial Taylor–Maclaurin coefficients and Fekete–Szegö type inequalities for functions in these families. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear and Convex Analysis)

13 pages, 352 KiB

Open AccessArticle

On τ-Pseudo-ν-Convex κ-Fold Symmetric Bi-Univalent Function Family

by Sondekola Rudra Swamy and Luminiţa-Ioana Cotîrlă

Symmetry 2022, 14(10), 1972; https://doi.org/10.3390/sym14101972 - 21 Sep 2022

Cited by 4 | Viewed by 855

Abstract

The object of this article is to explore a

τ

-pseudo-

ν

-convex

κ

-fold symmetric bi-univalent function family satisfying subordinations condition generalizing certain previously examined families. We originate the initial Taylor–Maclaurin coefficient estimates of functions in the defined family. The classical Fekete–Szegö [...] Read more.

The object of this article is to explore a

τ

-pseudo-

ν

-convex

κ

-fold symmetric bi-univalent function family satisfying subordinations condition generalizing certain previously examined families. We originate the initial Taylor–Maclaurin coefficient estimates of functions in the defined family. The classical Fekete–Szegö inequalities for functions in the defined

τ

-pseudo-

ν

-convex family is also estimated. Furthermore, we present some of the special cases of the main results. Relevant connections with those in several earlier works are also pointed out. Our study in this paper is also motivated by the symmetry nature of

κ

-fold symmetric bi-univalent functions in the defined class. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear and Convex Analysis)

13 pages, 289 KiB

Open AccessArticle

Preserving Classes of Meromorphic Functions through Integral Operators

by Elisabeta-Alina Totoi and Luminiţa-Ioana Cotîrlă

Symmetry 2022, 14(8), 1545; https://doi.org/10.3390/sym14081545 - 28 Jul 2022

Cited by 12 | Viewed by 1296

Abstract

We consider three new classes of meromorphic functions defined by an extension of the Wanas operator and two integral operators, in order to study some preservation properties of the classes. The purpose of the paper is to find the conditions such that, when [...] Read more.

We consider three new classes of meromorphic functions defined by an extension of the Wanas operator and two integral operators, in order to study some preservation properties of the classes. The purpose of the paper is to find the conditions such that, when we apply the integral operator

J_{p, γ}

to some function from the new defined classes

Σ S_{p, q}^{n} (α, δ)

, respectively

Σ S_{p, q}^{n} (α)

, we obtain also a function from the same class. We also define a new integral operator on the class of meromorphic functions, denoted by

J_{p, γ, h}

, where h is a normalized analytic function on the unit disc. We study some basic properties of this operator. Then we look for the conditions which allow this operator to preserve a particular subclass of the classes mentioned above. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear and Convex Analysis)

9 pages, 478 KiB

Open AccessArticle

Continuous Limit, Rational Solutions, and Asymptotic State Analysis for the Generalized Toda Lattice Equation Associated with 3 × 3 Lax Pair

by Xue-Ke Liu and Xiao-Yong Wen

Symmetry 2022, 14(5), 920; https://doi.org/10.3390/sym14050920 - 30 Apr 2022

Cited by 2 | Viewed by 1267

Abstract

Discrete integrable nonlinear differential difference equations (NDDEs) have various mathematical structures and properties, such as Lax pair, infinitely many conservation laws, Hamiltonian structure, and different kinds of symmetries, including Lie point symmetry, generalized Lie bäcklund symmetry, and master symmetry. Symmetry is one of [...] Read more.

Discrete integrable nonlinear differential difference equations (NDDEs) have various mathematical structures and properties, such as Lax pair, infinitely many conservation laws, Hamiltonian structure, and different kinds of symmetries, including Lie point symmetry, generalized Lie bäcklund symmetry, and master symmetry. Symmetry is one of the very effective methods used to study the exact solutions and integrability of NDDEs. The Toda lattice equation is a famous example of NDDEs, which may be used to simulate the motions of particles in lattices. In this paper, we investigated the generalized Toda lattice equation related to

3 \times 3

matrix linear spectral problem. This discrete equation is related to continuous linear and nonlinear partial differential equations under the continuous limit. Based on the known

3 \times 3

Lax pair of this equation, the discrete generalized

(m, 3 N - m)

-fold Darboux transformation was constructed for the first time and extended from the

2 \times 2

Lax pair to the

3 \times 3

Lax pair to give its rational solutions. Furthermore, the limit states of such rational solutions are discussed via the asymptotic analysis technique. Finally, the exponential–rational mixed solutions of the generalized Toda lattice equation are obtained in the form of determinants. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear and Convex Analysis)

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17 pages, 325 KiB

Open AccessArticle

A Qualitative Study on Second-Order Nonlinear Fractional Differential Evolution Equations with Generalized ABC Operator

by Mohammed A. Almalahi, Amani B. Ibrahim, Alanoud Almutairi, Omar Bazighifan, Tariq A. Aljaaidi and Jan Awrejcewicz

Symmetry 2022, 14(2), 207; https://doi.org/10.3390/sym14020207 - 21 Jan 2022

Cited by 8 | Viewed by 2108

Abstract

This research paper is dedicated to an investigation of an evolution problem under a new operator (

g

-Atangana–Baleanu–Caputo type fractional derivative)(for short,

g

-ABC). For the proposed problem, we construct sufficient conditions for some properties of the solution like existence, uniqueness and [...] Read more.

This research paper is dedicated to an investigation of an evolution problem under a new operator (

g

-Atangana–Baleanu–Caputo type fractional derivative)(for short,

g

-ABC). For the proposed problem, we construct sufficient conditions for some properties of the solution like existence, uniqueness and stability analysis. Existence and uniqueness results are proved based on some fixed point theorems such that Banach and Krasnoselskii. Furthermore, through mathematical analysis techniques, we analyze different types of stability results. The symmetric properties aid in identifying the best strategy for getting the correct solution of fractional differential equations. An illustrative example is discussed for the control problem. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear and Convex Analysis)

8 pages, 1116 KiB

Open AccessArticle

New Solution of the Sine-Gordon Equation by the Daftardar-Gejji and Jafari Method

by Belal Batiha

Symmetry 2022, 14(1), 57; https://doi.org/10.3390/sym14010057 - 2 Jan 2022

Cited by 13 | Viewed by 1411

Abstract

In this article, the Daftardar-Gejji and Jafari method (DJM) is used to obtain an approximate analytical solution of the sine-Gordon equation. Some examples are solved to demonstrate the accuracy of DJM. A comparison study between DJM, the variational iteration method (VIM), and the [...] Read more.

In this article, the Daftardar-Gejji and Jafari method (DJM) is used to obtain an approximate analytical solution of the sine-Gordon equation. Some examples are solved to demonstrate the accuracy of DJM. A comparison study between DJM, the variational iteration method (VIM), and the exact solution are presented. The comparison of the present symmetrical results with the existing literature is satisfactory. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear and Convex Analysis)

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