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19 pages, 12533 KiB

Open AccessArticle

A B-Polynomial Approach to Approximate Solutions of PDEs with Multiple Initial Conditions

by Muhammad I. Bhatti and Md. Habibur Rahman

Axioms 2024, 13(12), 833; https://doi.org/10.3390/axioms13120833 - 27 Nov 2024

In this article, we present a novel B-Polynomial Approach for approximating solutions to partial differential equations (PDEs), addressing the multiple initial conditions. Our method stands out by utilizing two-dimensional Bernstein polynomials (B-polynomials) in conjunction with their operational matrices to effectively manage the complexity [...] Read more.

In this article, we present a novel B-Polynomial Approach for approximating solutions to partial differential equations (PDEs), addressing the multiple initial conditions. Our method stands out by utilizing two-dimensional Bernstein polynomials (B-polynomials) in conjunction with their operational matrices to effectively manage the complexity associated with PDEs. This approach not only enhances the accuracy of solutions but also provides a structured framework for tackling various boundary conditions. The PDE is transformed into a system of algebraic equations, which are then solved to approximate the PDE solution. The process is divided into two main steps: First, the PDE is integrated to incorporate all initial and boundary conditions. Second, we express the approximate solution using B-polynomials and determine the unknown expansion coefficients via the Galerkin finite element method. The accuracy of the solution is assessed by adjusting the number of B-polynomials used in the expansion. The absolute error is estimated by comparing the exact and semi-numerical solutions. We apply this method to several examples, presenting results in tables and visualizing them with graphs. The approach demonstrates improved accuracy as the number of B-polynomials increases, with CPU time increasing linearly. Additionally, we compare our results with other methods, highlighting that our approach is both simple and effective for solving multidimensional PDEs imposed with multiple initial and boundary conditions. Full article

(This article belongs to the Special Issue Differential Equations and Related Topics, 2nd Edition)

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12 pages, 259 KiB

Open AccessArticle

Some Theorems of Uncertain Multiple-Delay Differential Equations

by Yin Gao and Han Tang

Axioms 2024, 13(11), 797; https://doi.org/10.3390/axioms13110797 - 18 Nov 2024

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Abstract

Uncertain differential equations with a time delay, called uncertain-delay differential equations, have been successfully applied in feedback control systems. In fact, many systems have multiple delays, which can be described by uncertain differential equations with multiple delays. This paper defines uncertain differential equations [...] Read more.

Uncertain differential equations with a time delay, called uncertain-delay differential equations, have been successfully applied in feedback control systems. In fact, many systems have multiple delays, which can be described by uncertain differential equations with multiple delays. This paper defines uncertain differential equations with multiple delays, which are called uncertain multiple-delay differential equations (UMDDEs). Based on the linear growth condition and the Lipschitz condition, the existence and uniqueness theorem of the solutions to the UMDDEs is proven. In order to judge the stability of the solutions to the UMDDEs, the concept of the stability in measure for UMDDEs is presented. Moreover, two theorems sufficient for use as tools to identify the stability in measure for UMDDEs are proved, and some examples are also discussed in this paper. Full article

(This article belongs to the Special Issue Differential Equations and Related Topics, 2nd Edition)

15 pages, 309 KiB

Open AccessArticle

Existence of Solutions for a Perturbed N-Laplacian Boundary Value Problem with Critical Growth

by Sheng Shi and Yang Yang

Axioms 2024, 13(11), 733; https://doi.org/10.3390/axioms13110733 - 23 Oct 2024

Viewed by 435

Abstract

In this paper, we investigate a perturbed elliptic boundary value problem that exhibits critical growth characterized by a Trudinger–Moser-type inequality. Our primary focus is to establish the existence of two nontrivial solutions. This is achieved by employing a combination of the Trudinger–Moser-type inequality [...] Read more.

In this paper, we investigate a perturbed elliptic boundary value problem that exhibits critical growth characterized by a Trudinger–Moser-type inequality. Our primary focus is to establish the existence of two nontrivial solutions. This is achieved by employing a combination of the Trudinger–Moser-type inequality and a linking theorem based on the

Z_{2}

-cohomological index. The main feature and novelty of this paper lies in extending the equation to N-Laplacian boundary value problems utilizing the aforementioned methods. This extension not only broadens the applicability of these techniques but also enriches the research outcomes in the field of nonlinear analysis. Full article

(This article belongs to the Special Issue Differential Equations and Related Topics, 2nd Edition)

17 pages, 1994 KiB

Open AccessArticle

Notes on Modified Planar Kelvin–Stuart Models: Simulations, Applications, Probabilistic Control on the Perturbations

by Nikolay Kyurkchiev, Tsvetelin Zaevski, Anton Iliev, Vesselin Kyurkchiev and Asen Rahnev

Axioms 2024, 13(10), 720; https://doi.org/10.3390/axioms13100720 - 17 Oct 2024

Viewed by 425

Abstract

In this paper, we propose a new modified planar Kelvin–Stuart model. We demonstrate some modules for investigating the dynamics of the proposed model. This will be included as an integral part of a planned, much more general Web-based application for scientific computing. Investigations [...] Read more.

In this paper, we propose a new modified planar Kelvin–Stuart model. We demonstrate some modules for investigating the dynamics of the proposed model. This will be included as an integral part of a planned, much more general Web-based application for scientific computing. Investigations in light of Melnikov’s approach are considered. Some simulations and applications are also presented. The proposed new modifications of planar Kelvin–Stuart models contain many free parameters (the coefficients

g_{i}, i = 1, 2, \dots, N

), which makes them attractive for use in engineering applications such as the antenna feeder technique (a possible generating and simulating of antenna factors) and the theory of approximations (a possible good approximation of a given electrical stage). The probabilistic control of the perturbations is discussed. Full article

(This article belongs to the Special Issue Differential Equations and Related Topics, 2nd Edition)

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15 pages, 275 KiB

Open AccessArticle

Fredholm Determinant and Wronskian Representations of the Solutions to the Schrödinger Equation with a KdV-Potential

by Pierre Gaillard

Axioms 2024, 13(10), 712; https://doi.org/10.3390/axioms13100712 - 15 Oct 2024

Viewed by 428

Abstract

From the finite gap solutions of the KdV equation expressed in terms of abelian functions we construct solutions to the Schrödinger equation with a KdV potential in terms of fourfold Fredholm determinants. For this we establish a connection between Riemann theta functions and [...] Read more.

From the finite gap solutions of the KdV equation expressed in terms of abelian functions we construct solutions to the Schrödinger equation with a KdV potential in terms of fourfold Fredholm determinants. For this we establish a connection between Riemann theta functions and Fredholm determinants and we obtain multi-parametric solutions to this equation. As a consequence, a double Wronskian representation of the solutions to this equation is constructed. We also give quasi-rational solutions to this Schrödinger equation with rational KdV potentials. Full article

(This article belongs to the Special Issue Differential Equations and Related Topics, 2nd Edition)

16 pages, 315 KiB

Open AccessArticle

Second-Order Neutral Differential Equations with a Sublinear Neutral Term: Examining the Oscillatory Behavior

by Ahmed Alemam, Asma Al-Jaser, Osama Moaaz, Fahd Masood and Hamdy El-Metwally

Axioms 2024, 13(10), 681; https://doi.org/10.3390/axioms13100681 - 1 Oct 2024

Viewed by 702

Abstract

This article highlights the oscillatory properties of second-order Emden–Fowler delay differential equations featuring sublinear neutral terms and multiple delays, encompassing both canonical and noncanonical cases. Through the proofs of several theorems, we investigate criteria for the oscillation of all solutions to the equations [...] Read more.

This article highlights the oscillatory properties of second-order Emden–Fowler delay differential equations featuring sublinear neutral terms and multiple delays, encompassing both canonical and noncanonical cases. Through the proofs of several theorems, we investigate criteria for the oscillation of all solutions to the equations under study. By employing the Riccati technique in various ways, we derive results that expand the scope of previous research and enhance the cognitive understanding of this mathematical domain. Additionally, we provide three illustrative examples to demonstrate the validity and applicability of our findings. Full article

(This article belongs to the Special Issue Differential Equations and Related Topics, 2nd Edition)

21 pages, 608 KiB

Open AccessArticle

On Extending the Applicability of Iterative Methods for Solving Systems of Nonlinear Equations

by Indra Bate, Muniyasamy Murugan, Santhosh George, Kedarnath Senapati, Ioannis K. Argyros and Samundra Regmi

Axioms 2024, 13(9), 601; https://doi.org/10.3390/axioms13090601 - 4 Sep 2024

Viewed by 474

Abstract

In this paper, we present a technique that improves the applicability of the result obtained by Cordero et al. in 2024 for solving nonlinear equations. Cordero et al. assumed the involved operator to be differentiable at least five times to extend a two-step [...] Read more.

In this paper, we present a technique that improves the applicability of the result obtained by Cordero et al. in 2024 for solving nonlinear equations. Cordero et al. assumed the involved operator to be differentiable at least five times to extend a two-step p-order method to order

p + 3

. We obtained the convergence order of Cordero et al.’s method by assuming only up to the third-order derivative of the operator. Our analysis is in a more general commutative Banach algebra setting and provides a radius of the convergence ball. Finally, we validate our theoretical findings with several numerical examples. Also, the concept of basin of attraction is discussed with examples. Full article

(This article belongs to the Special Issue Differential Equations and Related Topics, 2nd Edition)

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23 pages, 338 KiB

Open AccessArticle

Uniform Stabilization and Asymptotic Behavior with a Lower Bound of the Maximal Existence Time of a Coupled System’s Semi-Linear Pseudo-Parabolic Equations

by Nian Liu

Axioms 2024, 13(9), 575; https://doi.org/10.3390/axioms13090575 - 23 Aug 2024

Viewed by 441

Abstract

This article discusses the initial boundary value problem for a class of coupled systems of semi-linear pseudo-parabolic equations on a bounded smooth domain. Global solutions with exponential decay and asymptotic behavior are obtained when the maximal existence time has a lower bound for [...] Read more.

This article discusses the initial boundary value problem for a class of coupled systems of semi-linear pseudo-parabolic equations on a bounded smooth domain. Global solutions with exponential decay and asymptotic behavior are obtained when the maximal existence time has a lower bound for both low and overcritical energy cases. A sharp condition linking these phenomena is derived, and it is demonstrated that global existence also applies to the case of the potential well family. Full article

(This article belongs to the Special Issue Differential Equations and Related Topics, 2nd Edition)

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