Research

26 pages, 403 KiB

Open AccessArticle

\bar{\partial}

and the

\bar{\partial}

-Neumann Operator on Pseudoconvex Manifolds

by Haroun Doud Soliman Adam, Khalid Ibrahim Adam Ahmed, Sayed Saber and Marin Marin

Mathematics 2023, 11(19), 4138; https://0-doi-org.brum.beds.ac.uk/10.3390/math11194138 - 30 Sep 2023

Cited by 1 | Viewed by 511

Abstract

Let D be a relatively compact domain in an n-dimensional Kähler manifold with a

C^{2}

smooth boundary that satisfies some “Hartogs-pseudoconvexity” condition. Assume that

Ξ

is a positive holomorphic line bundle over X whose curvature form

Θ

satisfies [...] Read more.

Let D be a relatively compact domain in an n-dimensional Kähler manifold with a

C^{2}

smooth boundary that satisfies some “Hartogs-pseudoconvexity” condition. Assume that

Ξ

is a positive holomorphic line bundle over X whose curvature form

Θ

satisfies

Θ \geq C ω

, where

C > 0

. Then, the

\bar{\partial}

-Neumann operator N and the Bergman projection

P

are exactly regular in the Sobolev space

W^{m} (D, Ξ)

for some

m

, as well as the operators

\bar{\partial} N

,

{\bar{\partial}}^{⋇} N

. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2023)

22 pages, 497 KiB

Open AccessArticle

Asymptotic Behavior of Certain Non-Autonomous Planar Competitive Systems of Difference Equations

by Mustafa R. S. Kulenović, Mehmed Nurkanović, Zehra Nurkanović and Susan Trolle

Mathematics 2023, 11(18), 3909; https://0-doi-org.brum.beds.ac.uk/10.3390/math11183909 - 14 Sep 2023

Viewed by 547

Abstract

This paper investigates the dynamics of non-autonomous competitive systems of difference equations with asymptotically constant coefficients. We are mainly interested in global attractivity results for such systems and the application of such results to the evolutionary population of competition models of two species. [...] Read more.

This paper investigates the dynamics of non-autonomous competitive systems of difference equations with asymptotically constant coefficients. We are mainly interested in global attractivity results for such systems and the application of such results to the evolutionary population of competition models of two species. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2023)

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19 pages, 444 KiB

Open AccessArticle

Ambrosetti–Prodi Alternative for Coupled and Independent Systems of Second-Order Differential Equations

by Feliz Minhós and Gracino Rodrigues

Mathematics 2023, 11(17), 3645; https://0-doi-org.brum.beds.ac.uk/10.3390/math11173645 - 23 Aug 2023

Viewed by 599

Abstract

This paper deals with two types of systems of second-order differential equations with parameters: coupled systems with the boundary conditions of the Sturm–Liouville type and classical systems with Dirichlet boundary conditions. We discuss an Ambosetti–Prodi alternative for each system. For the first type [...] Read more.

This paper deals with two types of systems of second-order differential equations with parameters: coupled systems with the boundary conditions of the Sturm–Liouville type and classical systems with Dirichlet boundary conditions. We discuss an Ambosetti–Prodi alternative for each system. For the first type of system, we present sufficient conditions for the existence and non-existence of its solutions, and for the second type of system, we present sufficient conditions for the existence and non-existence of a multiplicity of its solutions. Our arguments apply the lower and upper solutions method together with the properties of the Leary–Schauder topological degree theory. To the best of our knowledge, the present study is the first time that the Ambrosetti–Prodi alternative has been obtained for such systems with different parameters. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2023)

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13 pages, 843 KiB

Open AccessFeature PaperArticle

Conservation Laws and Exact Solutions for Time-Delayed Burgers–Fisher Equations

by Almudena P. Márquez, Rafael de la Rosa, Tamara M. Garrido and María L. Gandarias

Mathematics 2023, 11(17), 3640; https://0-doi-org.brum.beds.ac.uk/10.3390/math11173640 - 23 Aug 2023

Viewed by 789

Abstract

A generalization of the time-delayed Burgers–Fisher equation is studied. This partial differential equation appears in many physical and biological problems describing the interaction between reaction, diffusion, and convection. New travelling wave solutions are obtained. The solutions are derived in a systematic way by [...] Read more.

A generalization of the time-delayed Burgers–Fisher equation is studied. This partial differential equation appears in many physical and biological problems describing the interaction between reaction, diffusion, and convection. New travelling wave solutions are obtained. The solutions are derived in a systematic way by applying the multi-reduction method to the symmetry-invariant conservation laws. The translation-invariant conservation law yields a first integral, which is a first-order Chini equation. Under certain conditions on the coefficients of the equation, the Chini type equation obtained can be solved, yielding travelling wave solutions expressed in terms of the Lerch transcendent function. For a special case, the first integral becomes a Riccati equation, whose solutions are given in terms of Bessel functions, and for a special case of the parameters, the solutions are given in terms of exponential, trigonometric, and hyperbolic functions. Furthermore, a complete classification of the zeroth-order local conservation laws is obtained. To the best of our knowledge, our results include new solutions that have not been previously reported in the literature. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2023)

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19 pages, 905 KiB

Open AccessArticle

Dynamics of a Discrete Leslie–Gower Model with Harvesting and Holling-II Functional Response

by Chen Zhang and Xianyi Li

Mathematics 2023, 11(15), 3303; https://0-doi-org.brum.beds.ac.uk/10.3390/math11153303 - 27 Jul 2023

Cited by 2 | Viewed by 664

Abstract

Recently, Christian Cortés García proposed and studied a continuous modified Leslie–Gower model with harvesting and alternative food for predator and Holling-II functional response, and proved that the model undergoes transcritical bifurcation, saddle-node bifurcation and Hopf bifurcation. In this paper, we dedicate ourselves to [...] Read more.

Recently, Christian Cortés García proposed and studied a continuous modified Leslie–Gower model with harvesting and alternative food for predator and Holling-II functional response, and proved that the model undergoes transcritical bifurcation, saddle-node bifurcation and Hopf bifurcation. In this paper, we dedicate ourselves to investigating the bifurcation problems of the discrete version of the model by using the Center Manifold Theorem and bifurcation theory, and obtain sufficient conditions for the occurrences of the transcritical bifurcation and Neimark–Sacker bifurcation, and the stability of the closed orbits bifurcated. Our numerical simulations not only illustrate corresponding theoretical results, but also reveal new dynamic chaos occurring, which is an essential difference between the continuous system and its corresponding discrete version. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2023)

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13 pages, 297 KiB

Open AccessArticle

Neutral Differential Equations of Higher-Order in Canonical Form: Oscillation Criteria

by Abdulaziz Khalid Alsharidi, Ali Muhib and Sayed K. Elagan

Mathematics 2023, 11(15), 3300; https://0-doi-org.brum.beds.ac.uk/10.3390/math11153300 - 27 Jul 2023

Viewed by 538

Abstract

This paper aims to study a class of neutral differential equations of higher-order in canonical form. By using the comparison technique, we obtain sufficient conditions to ensure that the studied differential equations are oscillatory. The criteria that we obtained are to improve and [...] Read more.

This paper aims to study a class of neutral differential equations of higher-order in canonical form. By using the comparison technique, we obtain sufficient conditions to ensure that the studied differential equations are oscillatory. The criteria that we obtained are to improve and extend some of the results in previous literature. In addition, an example is given that shows the applicability of the results we obtained. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2023)

9 pages, 707 KiB

Open AccessArticle

Exploration of New Solitons for the Fractional Perturbed Radhakrishnan–Kundu–Lakshmanan Model

by Melike Kaplan and Rubayyi T. Alqahtani

Mathematics 2023, 11(11), 2562; https://0-doi-org.brum.beds.ac.uk/10.3390/math11112562 - 03 Jun 2023

Cited by 10 | Viewed by 1062

Abstract

The key objective of the current manuscript was to investigate the exact solutions of the fractional perturbed Radhakrishnan–Kundu–Lakshmanan model. For this purpose, we applied two reliable and efficient approaches; specifically, the modified simple equation (MSE) and exponential rational function (ERF) techniques. The methods [...] Read more.

The key objective of the current manuscript was to investigate the exact solutions of the fractional perturbed Radhakrishnan–Kundu–Lakshmanan model. For this purpose, we applied two reliable and efficient approaches; specifically, the modified simple equation (MSE) and exponential rational function (ERF) techniques. The methods considered in this paper offer solutions for problems in nonlinear theory and mathematical physics practice. We also present solutions obtained graphically with the Maple package program. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2023)

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16 pages, 416 KiB

Open AccessArticle

On Fractional-Order Discrete-Time Reaction Diffusion Systems

by Othman Abdullah Almatroud, Amel Hioual, Adel Ouannas and Giuseppe Grassi

Mathematics 2023, 11(11), 2447; https://0-doi-org.brum.beds.ac.uk/10.3390/math11112447 - 25 May 2023

Cited by 6 | Viewed by 906

Abstract

Reaction–diffusion systems have a broad variety of applications, particularly in biology, and it is well known that fractional calculus has been successfully used with this type of system. However, analyzing these systems using discrete fractional calculus is novel and requires significant research in [...] Read more.

Reaction–diffusion systems have a broad variety of applications, particularly in biology, and it is well known that fractional calculus has been successfully used with this type of system. However, analyzing these systems using discrete fractional calculus is novel and requires significant research in a diversity of disciplines. Thus, in this paper, we investigate the discrete-time fractional-order Lengyel–Epstein system as a model of the chlorite iodide malonic acid (CIMA) chemical reaction. With the help of the second order difference operator, we describe the fractional discrete model. Furthermore, using the linearization approach, we established acceptable requirements for the local asymptotic stability of the system’s unique equilibrium. Moreover, we employ a Lyapunov functional to show that when the iodide feeding rate is moderate, the constant equilibrium solution is globally asymptotically stable. Finally, numerical models are presented to validate the theoretical conclusions and demonstrate the impact of discretization and fractional-order on system dynamics. The continuous version of the fractional-order Lengyel–Epstein reaction–diffusion system is compared to the discrete-time system under consideration. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2023)

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14 pages, 1408 KiB

Open AccessArticle

Spectral Collocation Technique for Solving Two-Dimensional Multi-Term Time Fractional Viscoelastic Non-Newtonian Fluid Model

by Mohammed M. Al-Shomrani, Mohamed A. Abdelkawy and António M. Lopes

Mathematics 2023, 11(9), 2078; https://0-doi-org.brum.beds.ac.uk/10.3390/math11092078 - 27 Apr 2023

Cited by 1 | Viewed by 889

Abstract

Applications of non-Newtonian fluids have been widespread across industries, accompanied by theoretical developments in engineering and mathematics. This paper studies a two-dimensional multi-term time fractional viscoelastic non-Newtonian fluid model by using two autonomous consecutive spectral collocation strategies. A modification of the spectral approach [...] Read more.

Applications of non-Newtonian fluids have been widespread across industries, accompanied by theoretical developments in engineering and mathematics. This paper studies a two-dimensional multi-term time fractional viscoelastic non-Newtonian fluid model by using two autonomous consecutive spectral collocation strategies. A modification of the spectral approach is implemented, leading to an algebraic system of equations able to obtain an approximate symmetric solution for the model. Numerical examples illustrate the effectiveness of the technique in terms of accuracy and convergence. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2023)

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13 pages, 285 KiB

Open AccessArticle

On Some Solvable Systems of Some Rational Difference Equations of Third Order

by Khalil S. Al-Basyouni and Elsayed M. Elsayed

Mathematics 2023, 11(4), 1047; https://0-doi-org.brum.beds.ac.uk/10.3390/math11041047 - 19 Feb 2023

Cited by 3 | Viewed by 1061

Abstract

Our aim in this paper is to obtain formulas for solutions of rational difference equations such as

x_{n + 1} = 1 \pm (x_{n - 1} y_{n}) / (1 - y_{n}),

[...] Read more.

Our aim in this paper is to obtain formulas for solutions of rational difference equations such as

x_{n + 1} = 1 \pm (x_{n - 1} y_{n}) / (1 - y_{n}),

y_{n + 1} = 1 \pm (y_{n - 1} x_{n}) / (1 - x_{n}),

and

x_{n + 1} = 1 \pm (x_{n - 1} y_{n - 2}) / (1 - y_{n}),

y_{n + 1} = 1 \pm (y_{n - 1} x_{n - 2}) / (1 - x_{n}),

where the initial conditions

x_{- 2}

,

x_{- 1}

,

x_{0}

,

y_{- 2}

,

y_{- 1}

,

y_{0}

are non-zero real numbers. In addition, we show that the some of these systems are periodic with different periods. We also verify our theoretical outcomes at the end with some numerical applications and draw it by using some mathematical programs to illustrate the results. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2023)

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19 pages, 320 KiB

Open AccessArticle

Relative Controllability and Ulam–Hyers Stability of the Second-Order Linear Time-Delay Systems

by Kinda Abuasbeh, Nazim I. Mahmudov and Muath Awadalla

Mathematics 2023, 11(4), 806; https://0-doi-org.brum.beds.ac.uk/10.3390/math11040806 - 05 Feb 2023

Cited by 1 | Viewed by 1071

Abstract

We introduce the delayed sine/cosine-type matrix function and use the Laplace transform method to obtain a closed form solution to IVP for a second-order time-delayed linear system with noncommutative matrices A and

Ω

. We also introduce a delay Gramian matrix and examine [...] Read more.

We introduce the delayed sine/cosine-type matrix function and use the Laplace transform method to obtain a closed form solution to IVP for a second-order time-delayed linear system with noncommutative matrices A and

Ω

. We also introduce a delay Gramian matrix and examine a relative controllability linear/semi-linear time delay system. We have obtained the necessary and sufficient condition for the relative controllability of the linear time-delayed second-order system. In addition, we have obtained sufficient conditions for the relative controllability of the semi-linear second-order time-delay system. Finally, we investigate the Ulam–Hyers stability of a second-order semi-linear time-delayed system. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2023)

32 pages, 424 KiB

Open AccessArticle

Nonexistence and Existence of Solutions with Prescribed Norms for Nonlocal Elliptic Equations with Combined Nonlinearities

by Baoqiang Yan, Donal O’Regan and Ravi P. Agarwal

Mathematics 2023, 11(1), 75; https://0-doi-org.brum.beds.ac.uk/10.3390/math11010075 - 25 Dec 2022

Viewed by 849

Abstract

In this paper, we study the nonlocal equation

{(\int_{R^{N}} {| u (x) |}^{2} d x)}^{γ} Δ u = λ u + {μ | u |}^{q - 2} u + {| u |}^{p - 2} u

[...] Read more.

In this paper, we study the nonlocal equation

{(\int_{R^{N}} {| u (x) |}^{2} d x)}^{γ} Δ u = λ u + {μ | u |}^{q - 2} u + {| u |}^{p - 2} u

,

x in R^{N}

having a prescribed mass

\int_{R^{N}} {| u (x) |}^{2} d x = c^{2},

where

N \geq 3

,

μ

,

γ \in (0, + \infty)

,

q \in (2, 2^{*})

, c is a positive constant, p,

q \in (2, 2^{*})

with

p \neq q

and

2^{*} = \frac{2 N}{N - 2}

. This research is meaningful from a physical point of view. Using variational methods, we present some results on the nonexistence and existence of solutions under different cases p and q which improve upon the previous ones via topological theory. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2023)

11 pages, 275 KiB

Open AccessArticle

Convergence of the Boundary Parameter for the Three-Dimensional Viscous Primitive Equations of Large-Scale

by Zhanwei Guo, Jincheng Shi and Danping Ding

Mathematics 2022, 10(21), 4052; https://0-doi-org.brum.beds.ac.uk/10.3390/math10214052 - 01 Nov 2022

Viewed by 857

Abstract

The main objective of this paper is concerned with the convergence of the boundary parameter for the large-scale, three-dimensional, viscous primitive equations. Such equations are often used for weather prediction and climate change. Under the assumptions of some boundary conditions, we obtain a [...] Read more.

The main objective of this paper is concerned with the convergence of the boundary parameter for the large-scale, three-dimensional, viscous primitive equations. Such equations are often used for weather prediction and climate change. Under the assumptions of some boundary conditions, we obtain a prior bounds for the solutions of the equations by using the differential inequality technology and method of the energy estimates, and the convergence of the equations on the boundary parameter is proved. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2023)

19 pages, 5298 KiB

Open AccessArticle

Computational Analysis of Fractional Diffusion Equations Occurring in Oil Pollution

by Jagdev Singh, Ahmed M. Alshehri, Shaher Momani, Samir Hadid and Devendra Kumar

Mathematics 2022, 10(20), 3827; https://0-doi-org.brum.beds.ac.uk/10.3390/math10203827 - 17 Oct 2022

Cited by 10 | Viewed by 1188

Abstract

The fractional model of diffusion equations is very important in the study of oil pollution in the water. The key objective of this article is to analyze a fractional modification of diffusion equations occurring in oil pollution associated with the Katugampola derivative in [...] Read more.

The fractional model of diffusion equations is very important in the study of oil pollution in the water. The key objective of this article is to analyze a fractional modification of diffusion equations occurring in oil pollution associated with the Katugampola derivative in the Caputo sense. An effective and reliable computational method q-homotopy analysis generalized transform method is suggested to obtain the solutions of fractional order diffusion equations. The results of this research are demonstrated in graphical and tabular descriptions. This study shows that the applied computational technique is very effective, accurate, and beneficial for managing such kind of fractional order nonlinear models occurring in oil pollution. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2023)

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21 pages, 5910 KiB

Open AccessArticle

Closed-Form Solutions in a Magneto-Electro-Elastic Circular Rod via Generalized Exp-Function Method

by Muhammad Shakeel, Attaullah, Mohammed Kbiri Alaoui, Ahmed M. Zidan, Nehad Ali Shah and Wajaree Weera

Mathematics 2022, 10(18), 3400; https://0-doi-org.brum.beds.ac.uk/10.3390/math10183400 - 19 Sep 2022

Cited by 13 | Viewed by 1391

Abstract

In this study, the dispersal caused by the transverse Poisson’s effect in a magneto-electro-elastic (MEE) circular rod is taken into consideration using the nonlinear longitudinal wave equation (LWE), a mathematical physics problem. Using the generalized exp-function method, we investigate the families of solitary [...] Read more.

In this study, the dispersal caused by the transverse Poisson’s effect in a magneto-electro-elastic (MEE) circular rod is taken into consideration using the nonlinear longitudinal wave equation (LWE), a mathematical physics problem. Using the generalized exp-function method, we investigate the families of solitary wave solutions of one-dimensional nonlinear LWE. Using the computer program Wolfram Mathematica 10, these new exact and solitary wave solutions of the LWE are derived as trigonometric function, periodic solitary wave, rational function, hyperbolic function, bright and dark solitons solutions, sinh, cosh, and sech² function solutions of the LWE. These solutions represent the electrostatic potential and pressure for LWE as well as the graphical representation of electrostatic potential and pressure. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2023)

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22 pages, 925 KiB

Open AccessArticle

Asymptotic Behavior for the Discrete in Time Heat Equation

by Luciano Abadias and Edgardo Alvarez

Mathematics 2022, 10(17), 3128; https://0-doi-org.brum.beds.ac.uk/10.3390/math10173128 - 01 Sep 2022

Cited by 3 | Viewed by 1253

Abstract

In this paper, we investigate the asymptotic behavior and decay of the solution of the discrete in time N-dimensional heat equation. We give a convergence rate with which the solution tends to the discrete fundamental solution, and the asymptotic decay, both in [...] Read more.

In this paper, we investigate the asymptotic behavior and decay of the solution of the discrete in time N-dimensional heat equation. We give a convergence rate with which the solution tends to the discrete fundamental solution, and the asymptotic decay, both in

L^{p} (R^{N}) .

Furthermore, we prove optimal

L^{2}

-decay of solutions. Since the technique of energy methods is not applicable, we follow the approach of estimates based on the discrete fundamental solution which is given by an original integral representation and also by MacDonald’s special functions. As a consequence, the analysis is different to the continuous in time heat equation and the calculations are rather involved. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2023)

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7 pages, 950 KiB

Open AccessArticle

Fractional Advection Diffusion Models for Radionuclide Migration in Multiple Barriers System of Deep Geological Repository

by Shuai Yang, Qing Wei and Lu An

Mathematics 2022, 10(14), 2491; https://0-doi-org.brum.beds.ac.uk/10.3390/math10142491 - 18 Jul 2022

Viewed by 1226

Abstract

Based on the multiple barriers concept of deep geological disposal of high-level waste, fractional advection diffusion equations for radionuclide migration in multiple layers low-permeability porous media are proposed in this work. The presented fractional advection diffusion models in terms of different definitions of [...] Read more.

Based on the multiple barriers concept of deep geological disposal of high-level waste, fractional advection diffusion equations for radionuclide migration in multiple layers low-permeability porous media are proposed in this work. The presented fractional advection diffusion models in terms of different definitions of fractional derivative are analytically addressed via the Laplace integral transform method. This work provides a theoretical foundation for further simulations of radionuclide migration in the multiple barriers system of the high-level waste repository. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2023)

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19 pages, 348 KiB

Open AccessArticle

Hilfer Fractional Neutral Stochastic Volterra Integro-Differential Inclusions via Almost Sectorial Operators

by Sivajiganesan Sivasankar and Ramalingam Udhayakumar

Mathematics 2022, 10(12), 2074; https://0-doi-org.brum.beds.ac.uk/10.3390/math10122074 - 15 Jun 2022

Cited by 14 | Viewed by 1253

Abstract

In our paper, we mainly concentrate on the existence of Hilfer fractional neutral stochastic Volterra integro-differential inclusions with almost sectorial operators. The facts related to fractional calculus, stochastic analysis theory, and the fixed point theorem for multivalued maps are used to prove the [...] Read more.

In our paper, we mainly concentrate on the existence of Hilfer fractional neutral stochastic Volterra integro-differential inclusions with almost sectorial operators. The facts related to fractional calculus, stochastic analysis theory, and the fixed point theorem for multivalued maps are used to prove the result. In addition, an illustration of the principle is provided. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2023)

26 pages, 5163 KiB

Open AccessArticle

An Analytical Approach for Fractional Hyperbolic Telegraph Equation Using Shehu Transform in One, Two and Three Dimensions

by Mamta Kapoor, Nehad Ali Shah, Salman Saleem and Wajaree Weera

Mathematics 2022, 10(12), 1961; https://0-doi-org.brum.beds.ac.uk/10.3390/math10121961 - 07 Jun 2022

Cited by 8 | Viewed by 1482

Abstract

In the present research paper, an iterative approach named the iterative Shehu transform method is implemented to solve time-fractional hyperbolic telegraph equations in one, two, and three dimensions, respectively. These equations are the prominent ones in the field of physics and in some [...] Read more.

In the present research paper, an iterative approach named the iterative Shehu transform method is implemented to solve time-fractional hyperbolic telegraph equations in one, two, and three dimensions, respectively. These equations are the prominent ones in the field of physics and in some other significant problems. The efficacy and authenticity of the proposed method are tested using a comparison of approximated and exact results in graphical form. Both 2D and 3D plots are provided to affirm the compatibility of approximated-exact results. The iterative Shehu transform method is a reliable and efficient tool to provide approximated and exact results to a vast class of ODEs, PDEs, and fractional PDEs in a simplified way, without any discretization or linearization, and is free of errors. A convergence analysis is also provided in this research. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2023)

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12 pages, 332 KiB

Open AccessArticle

On the Wavelet Collocation Method for Solving Fractional Fredholm Integro-Differential Equations

by Haifa Bin Jebreen and Ioannis Dassios

Mathematics 2022, 10(8), 1272; https://0-doi-org.brum.beds.ac.uk/10.3390/math10081272 - 12 Apr 2022

Cited by 5 | Viewed by 1789

Abstract

An efficient algorithm is proposed to find an approximate solution via the wavelet collocation method for the fractional Fredholm integro-differential equations (FFIDEs). To do this, we reduce the desired equation to an equivalent linear or nonlinear weakly singular Volterra–Fredholm integral equation. In order [...] Read more.

An efficient algorithm is proposed to find an approximate solution via the wavelet collocation method for the fractional Fredholm integro-differential equations (FFIDEs). To do this, we reduce the desired equation to an equivalent linear or nonlinear weakly singular Volterra–Fredholm integral equation. In order to solve this integral equation, after a brief introduction of Müntz–Legendre wavelets, and representing the fractional integral operator as a matrix, we apply the wavelet collocation method to obtain a system of nonlinear or linear algebraic equations. An a posteriori error estimate for the method is investigated. The numerical results confirm our theoretical analysis, and comparing the method with existing ones demonstrates its ability and accuracy. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2023)

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16 pages, 284 KiB

Open AccessArticle

Various Types of q-Differential Equations of Higher Order for q-Euler and q-Genocchi Polynomials

by Cheon-Seoung Ryoo and Jung-Yoog Kang

Mathematics 2022, 10(7), 1181; https://0-doi-org.brum.beds.ac.uk/10.3390/math10071181 - 05 Apr 2022

Cited by 6 | Viewed by 1445

Abstract

One finds several q-differential equations of a higher order for q-Euler polynomials and q-Genocchi polynomials. Additionally, we have a few q-differential equations of a higher order, which are mixed with q-Euler numbers and q-Genocchi polynomials. Moreover, we [...] Read more.

One finds several q-differential equations of a higher order for q-Euler polynomials and q-Genocchi polynomials. Additionally, we have a few q-differential equations of a higher order, which are mixed with q-Euler numbers and q-Genocchi polynomials. Moreover, we investigate some symmetric q-differential equations of a higher order by applying symmetric properties of q-Euler polynomials and q-Genocchi polynomials. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2023)

17 pages, 603 KiB

Open AccessArticle

Analysis of the Influences of Parameters in the Fractional Second-Grade Fluid Dynamics

by Mehmet Yavuz, Ndolane Sene and Mustafa Yıldız

Mathematics 2022, 10(7), 1125; https://0-doi-org.brum.beds.ac.uk/10.3390/math10071125 - 01 Apr 2022

Cited by 27 | Viewed by 2220

Abstract

This work proposes a qualitative study for the fractional second-grade fluid described by a fractional operator. The classical Caputo fractional operator is used in the investigations. The exact analytical solutions of the constructed problems for the proposed model are determined by using the [...] Read more.

This work proposes a qualitative study for the fractional second-grade fluid described by a fractional operator. The classical Caputo fractional operator is used in the investigations. The exact analytical solutions of the constructed problems for the proposed model are determined by using the Laplace transform method, which particularly includes the Laplace transform of the Caputo derivative. The impact of the used fractional operator is presented; especially, the acceleration effect is noticed in the paper. The parameters’ influences are focused on the dynamics such as the Prandtl number

(P r)

, the Grashof numbers

(G r)

, and the parameter

η

when the fractional-order derivative is used in modeling the second-grade fluid model. Their impacts are also analyzed from a physical point of view besides mathematical calculations. The impact of the fractional parameter

α

is also provided. Finally, it is concluded that the graphical representations support the theoretical observations of the paper. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2023)

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22 pages, 1043 KiB

Open AccessArticle

Fractional Dynamics of Vector-Borne Infection with Sexual Transmission Rate and Vaccination

by Shah Hussain, Elissa Nadia Madi, Naveed Iqbal, Thongchai Botmart, Yeliz Karaca and Wael W. Mohammed

Mathematics 2021, 9(23), 3118; https://0-doi-org.brum.beds.ac.uk/10.3390/math9233118 - 03 Dec 2021

Cited by 11 | Viewed by 1749

Abstract

New fractional operators have the aim of attracting nonlocal problems that display fractal behaviour; and thus fractional derivatives have applications in long-term relation description along with micro-scaled and macro-scaled phenomena. Formulated by fractional operators, the formulation of a dynamical system is used in [...] Read more.

New fractional operators have the aim of attracting nonlocal problems that display fractal behaviour; and thus fractional derivatives have applications in long-term relation description along with micro-scaled and macro-scaled phenomena. Formulated by fractional operators, the formulation of a dynamical system is used in applications for the description of systems with long-range interactions. Vector-borne illnesses are one of the world’s most serious public health issues with a large economic impact on the nations that are impacted. Population increase, urbanization, globalization, and a lack of public health infrastructure have all had a role in the introduction and reemergence of vector-borne illnesses during the last four decades. The control of these infections are important to lessen the economic burden of vector-borne diseases in infected regions. In this research work, we formulate the transmission process of Zika virus with the impact of sexual incidence rate and vaccination in terms of mathematics. We presented the fundamental theory of fractional operators Caputo–Fabrizio (CF) and Atangana–Baleanu (AB) for the analysis of the proposed system. We examine our system of Zika infection and determined the endemic indicator through a next-generation matrix technique. The uniqueness and existence of the solution has been investigated through fixed point theory. Accordingly, a numerical method has been introduced to investigate the dynamical nature of the system and make a comparison of the outcomes of the operators. The impact of different input factors has been conceptualized through dynamical behaviour of the system. We observed that lowering the index of memory, the fractional system provides accurate results about the recommended Zika dynamics and dramatically reduces infected people. It has been proved that high efficacy of a vaccine can lower the level of infection. Moreover, the impact of other parameters on the system of Zika virus infection are highlighted through numerical results. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2023)

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16 pages, 300 KiB

Open AccessArticle

The Stability Analysis of A-Quartic Functional Equation

by Chinnaappu Muthamilarasi, Shyam Sundar Santra, Ganapathy Balasubramanian, Vediyappan Govindan, Rami Ahmad El-Nabulsi and Khaled Mohamed Khedher

Mathematics 2021, 9(22), 2881; https://0-doi-org.brum.beds.ac.uk/10.3390/math9222881 - 12 Nov 2021

Cited by 4 | Viewed by 1364

Abstract

In this paper, we study the general solution of the functional equation, which is derived from additive–quartic mappings. In addition, we establish the generalized Hyers–Ulam stability of the additive–quartic functional equation in Banach spaces by using direct and fixed point methods. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2023)

12 pages, 276 KiB

Open AccessArticle

Noncanonical Neutral DDEs of Second-Order: New Sufficient Conditions for Oscillation

by Awatif A. Hindi, Osama Moaaz, Clemente Cesarano, Wedad R. Alharbi and Mohamed A. Abdou

Mathematics 2021, 9(17), 2026; https://0-doi-org.brum.beds.ac.uk/10.3390/math9172026 - 24 Aug 2021

Cited by 7 | Viewed by 1725

Abstract

In this paper, new oscillation conditions for the 2nd-order noncanonical neutral differential equation

(a_{0} (t) ((u (t)

+

a_{1} (t) u (g_{0} (t)))^{'} {)^{β})}^{'}

+ [...] Read more.

In this paper, new oscillation conditions for the 2nd-order noncanonical neutral differential equation

(a_{0} (t) ((u (t)

+

a_{1} (t) u (g_{0} (t)))^{'} {)^{β})}^{'}

+

a_{2} (t) u^{β} (g_{1} (t))

=

0,

where

t \geq t_{0},

are established. Using Riccati substitution and comparison with an equation of the first-order, we obtain criteria that ensure the oscillation of the studied equation. Furthermore, we complement and improve the previous results in the literature. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2023)

15 pages, 966 KiB

Open AccessArticle

A Comparative Analysis of Fractional-Order Gas Dynamics Equations via Analytical Techniques

by Shuang-Shuang Zhou, Nehad Ali Shah, Ioannis Dassios, S. Saleem and Kamsing Nonlaopon

Mathematics 2021, 9(15), 1735; https://0-doi-org.brum.beds.ac.uk/10.3390/math9151735 - 22 Jul 2021

Cited by 1 | Viewed by 1706

Abstract

This article introduces two well-known computational techniques for solving the time-fractional system of nonlinear equations of unsteady flow of a polytropic gas. The methods suggested are the modified forms of the variational iteration method and the homotopy perturbation method by the Elzaki transformation. [...] Read more.

This article introduces two well-known computational techniques for solving the time-fractional system of nonlinear equations of unsteady flow of a polytropic gas. The methods suggested are the modified forms of the variational iteration method and the homotopy perturbation method by the Elzaki transformation. Furthermore, an illustrative scheme is introduced to verify the accuracy of the available techniques. A graphical representation of the exact and derived results is presented to show the reliability of the suggested approaches. It is also shown that the findings of the current methodology are in close harmony with the exact solutions. The comparative solution analysis via graphs also represents the higher reliability and accuracy of the current techniques. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2023)

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20 pages, 318 KiB

Open AccessArticle

Sturm–Liouville Differential Equations Involving Kurzweil–Henstock Integrable Functions

by Salvador Sánchez-Perales, Tomás Pérez-Becerra, Virgilio Vázquez-Hipólito and José J. Oliveros-Oliveros

Mathematics 2021, 9(12), 1403; https://0-doi-org.brum.beds.ac.uk/10.3390/math9121403 - 17 Jun 2021

Cited by 2 | Viewed by 2216

Abstract

In this paper, we give sufficient conditions for the existence and uniqueness of the solution of Sturm–Liouville equations subject to Dirichlet boundary value conditions and involving Kurzweil–Henstock integrable functions on unbounded intervals. We also present a finite element method scheme for Kurzweil–Henstock integrable [...] Read more.

In this paper, we give sufficient conditions for the existence and uniqueness of the solution of Sturm–Liouville equations subject to Dirichlet boundary value conditions and involving Kurzweil–Henstock integrable functions on unbounded intervals. We also present a finite element method scheme for Kurzweil–Henstock integrable functions. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2023)

19 pages, 1360 KiB

Open AccessArticle

Orbital Stability of Solitary Waves to Double Dispersion Equations with Combined Power-Type Nonlinearity

by Natalia Kolkovska, Milena Dimova and Nikolai Kutev

Mathematics 2021, 9(12), 1398; https://0-doi-org.brum.beds.ac.uk/10.3390/math9121398 - 16 Jun 2021

Cited by 1 | Viewed by 1537

Abstract

We consider the orbital stability of solitary waves to the double dispersion equation [...] Read more.

We consider the orbital stability of solitary waves to the double dispersion equation

u_{t t} - u_{x x} + h_{1} u_{x x x x} - h_{2} u_{t t x x} + f {(u)}_{x x} = 0, h_{1} > 0, h_{2} > 0

with combined power-type nonlinearity

f (u) = {a | u |}^{p} u + b {| u |}^{2 p} u, p > 0, a \in R, b \in R, b \neq 0 .

The stability of solitary waves with velocity c,

c^{2} < 1

is proved by means of the Grillakis, Shatah, and Strauss abstract theory and the convexity of the function

d (c)

, related to some conservation laws. We derive explicit analytical formulas for the function

d (c)

and its second derivative for quadratic-cubic nonlinearity

f (u) = a u^{2} + b u^{3}

and parameters

b > 0

,

c^{2} \in [0, min (1, \frac{h_{1}}{h_{2}}))

. As a consequence, the orbital stability of solitary waves is analyzed depending on the parameters of the problem. Well-known results are generalized in the case of a single cubic nonlinearity

f (u) = b u^{3}

. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2023)

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32 pages, 461 KiB

Open AccessArticle

Numerical Solution of the Fredholm and Volterra Integral Equations by Using Modified Bernstein–Kantorovich Operators

by Suzan Cival Buranay, Mehmet Ali Özarslan and Sara Safarzadeh Falahhesar

Mathematics 2021, 9(11), 1193; https://0-doi-org.brum.beds.ac.uk/10.3390/math9111193 - 25 May 2021

Cited by 12 | Viewed by 2704

Abstract

The main aim of this paper is to numerically solve the first kind linear Fredholm and Volterra integral equations by using Modified Bernstein–Kantorovich operators. The unknown function in the first kind integral equation is approximated by using the Modified Bernstein–Kantorovich operators. Hence, by [...] Read more.

The main aim of this paper is to numerically solve the first kind linear Fredholm and Volterra integral equations by using Modified Bernstein–Kantorovich operators. The unknown function in the first kind integral equation is approximated by using the Modified Bernstein–Kantorovich operators. Hence, by using discretization, the obtained linear equations are transformed into systems of algebraic linear equations. Due to the sensitivity of the solutions on the input data, significant difficulties may be encountered, leading to instabilities in the results during actualization. Consequently, to improve on the stability of the solutions which imply the accuracy of the desired results, regularization features are built into the proposed numerical approach. More stable approximations to the solutions of the Fredholm and Volterra integral equations are obtained especially when high order approximations are used by the Modified Bernstein–Kantorovich operators. Test problems are constructed to show the computational efficiency, applicability and the accuracy of the method. Furthermore, the method is also applied to second kind Volterra integral equations. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2023)

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10 pages, 241 KiB

Open AccessArticle

More Effective Results for Testing Oscillation of Non-Canonical Neutral Delay Differential Equations

by Higinio Ramos, Osama Moaaz, Ali Muhib and Jan Awrejcewicz

Mathematics 2021, 9(10), 1114; https://0-doi-org.brum.beds.ac.uk/10.3390/math9101114 - 14 May 2021

Cited by 12 | Viewed by 1253

Abstract

In this work, we address an interesting problem in studying the oscillatory behavior of solutions of fourth-order neutral delay differential equations with a non-canonical operator. We obtained new criteria that improve upon previous results in the literature, concerning more than one aspect. Some [...] Read more.

In this work, we address an interesting problem in studying the oscillatory behavior of solutions of fourth-order neutral delay differential equations with a non-canonical operator. We obtained new criteria that improve upon previous results in the literature, concerning more than one aspect. Some examples are presented to illustrate the importance of the new results. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2023)

11 pages, 243 KiB

Open AccessEditor’s ChoiceArticle

New Oscillation Theorems for Second-Order Differential Equations with Canonical and Non-Canonical Operator via Riccati Transformation

by Shyam Sundar Santra, Abhay Kumar Sethi, Osama Moaaz, Khaled Mohamed Khedher and Shao-Wen Yao

Mathematics 2021, 9(10), 1111; https://0-doi-org.brum.beds.ac.uk/10.3390/math9101111 - 14 May 2021

Cited by 14 | Viewed by 1816

Abstract

In this work, we prove some new oscillation theorems for second-order neutral delay differential equations of the form [...] Read more.

In this work, we prove some new oscillation theorems for second-order neutral delay differential equations of the form

{(a (ξ) ({(v (ξ) + b (ξ) v (ϑ (ξ)))}^{'}))}^{'} + c (ξ) G_{1} (v (κ (ξ))) + d (ξ) G_{2} (v (ς (ξ))) = 0

under canonical and non-canonical operators, that is,

\int_{ξ_{0}}^{\infty} \frac{d ξ}{a (ξ)} = \infty

and

\int_{ξ_{0}}^{\infty} \frac{d ξ}{a (ξ)} < \infty

. We use the Riccati transformation to prove our main results. Furthermore, some examples are provided to show the effectiveness and feasibility of the main results. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2023)

18 pages, 11597 KiB

Open AccessArticle

Fractional System of Korteweg-De Vries Equations via Elzaki Transform

by Wenfeng He, Nana Chen, Ioannis Dassios, Nehad Ali Shah and Jae Dong Chung

Mathematics 2021, 9(6), 673; https://0-doi-org.brum.beds.ac.uk/10.3390/math9060673 - 22 Mar 2021

Cited by 25 | Viewed by 2895

Abstract

In this article, a hybrid technique, called the Iteration transform method, has been implemented to solve the fractional-order coupled Korteweg-de Vries (KdV) equation. In this method, the Elzaki transform and New Iteration method are combined. The iteration transform method solutions are obtained in [...] Read more.

In this article, a hybrid technique, called the Iteration transform method, has been implemented to solve the fractional-order coupled Korteweg-de Vries (KdV) equation. In this method, the Elzaki transform and New Iteration method are combined. The iteration transform method solutions are obtained in series form to analyze the analytical results of fractional-order coupled Korteweg-de Vries equations. To understand the analytical procedure of Iteration transform method, some numerical problems are presented for the analytical result of fractional-order coupled Korteweg-de Vries equations. It is also demonstrated that the current technique’s solutions are in good agreement with the exact results. The numerical solutions show that only a few terms are sufficient for obtaining an approximate result, which is efficient, accurate, and reliable. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2023)

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17 pages, 280 KiB

Open AccessArticle

Existence and Stability Results on Hadamard Type Fractional Time-Delay Semilinear Differential Equations

by Nazim Mahmudov and Areen Al-Khateeb

Mathematics 2020, 8(8), 1242; https://0-doi-org.brum.beds.ac.uk/10.3390/math8081242 - 30 Jul 2020

Cited by 6 | Viewed by 1833

Abstract

A delayed perturbation of the Mittag-Leffler type matrix function with logarithm is proposed. This combines the classic Mittag–Leffler type matrix function with a logarithm and delayed Mittag–Leffler type matrix function. With the help of this introduced delayed perturbation of the Mittag–Leffler type matrix [...] Read more.

A delayed perturbation of the Mittag-Leffler type matrix function with logarithm is proposed. This combines the classic Mittag–Leffler type matrix function with a logarithm and delayed Mittag–Leffler type matrix function. With the help of this introduced delayed perturbation of the Mittag–Leffler type matrix function with a logarithm, we provide an explicit form for solutions to non-homogeneous Hadamard-type fractional time-delay linear differential equations. We also examine the existence, uniqueness, and Ulam–Hyers stability of Hadamard-type fractional time-delay nonlinear equations. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2023)

13 pages, 301 KiB

Open AccessArticle

More Effective Criteria for Oscillation of Second-Order Differential Equations with Neutral Arguments

by Osama Moaaz, Mona Anis, Dumitru Baleanu and Ali Muhib

Mathematics 2020, 8(6), 986; https://0-doi-org.brum.beds.ac.uk/10.3390/math8060986 - 16 Jun 2020

Cited by 34 | Viewed by 1929

Abstract

The motivation for this paper is to create new criteria for oscillation of solutions of second-order nonlinear neutral differential equations. In more than one respect, our results improve several related ones in the literature. As proof of the effectiveness of the new criteria, [...] Read more.

The motivation for this paper is to create new criteria for oscillation of solutions of second-order nonlinear neutral differential equations. In more than one respect, our results improve several related ones in the literature. As proof of the effectiveness of the new criteria, we offer more than one practical example. Full article

(This article belongs to the Special Issue Advances in Differential and Difference Equations with Applications 2023)

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