MDPI - Publisher of Open Access Journals

13 pages, 272 KiB

Open AccessArticle

Stable Solutions of a Class of Degenerate Elliptic Equations

by Yin Lang and Hairong Liu

Axioms 2024, 13(12), 856; https://doi.org/10.3390/axioms13120856 - 5 Dec 2024

Viewed by 746

Abstract

This paper deals with the second-order semi-linear degenerate elliptic equation [...] Read more.

This paper deals with the second-order semi-linear degenerate elliptic equation

y u_{y y} + b u_{y} + Δ_{x} u + {| u |}^{α - 1} u = 0, (x, y) \in R^{n} \times (0, \infty)

, where

n \geq 1, α > 1

. We establish a Liouville theorem of stable solution of the degenerate equation mentioned above by using the energy method. The classification results for stable solutions belonging to

C^{2}

can be thought of as an analogue of the recent results of Farina for the Lane–Emden equation. Full article

9 pages, 224 KiB

Open AccessArticle

On the Classification of Entire Solutions to the Critical Lane–Emden Equation

by Shenghui Sun and Fei Han

Mathematics 2024, 12(12), 1822; https://doi.org/10.3390/math12121822 - 12 Jun 2024

Viewed by 828

Abstract

The positive solutions to the critical Lane–Emden equation in

R^{n}

have been classified completely by the moving plane method. Afterwards, with additional conditions such as limited energy or volume, there is proof that relies entirely on integration by parts and inequalities techniques. [...] Read more.

The positive solutions to the critical Lane–Emden equation in

R^{n}

have been classified completely by the moving plane method. Afterwards, with additional conditions such as limited energy or volume, there is proof that relies entirely on integration by parts and inequalities techniques. In this paper, the authors provide a new approach to obtain the same classification results without further assumptions. Full article

(This article belongs to the Section C1: Difference and Differential Equations)

11 pages, 255 KiB

Open AccessArticle

Qualitative Properties of the Solutions to the Lane–Emden Equation in the Cylindrical Setup

by Arsen Palestini and Simone Recchi

Mathematics 2024, 12(4), 542; https://doi.org/10.3390/math12040542 - 9 Feb 2024

Viewed by 1292

Abstract

We analyze the Lane–Emden equations in the cylindrical framework. Although the explicit forms of the solutions (which are also called polytropes) are not known, we identify some of their qualitative properties. In particular, possible critical points and zeros of the polytropes are investigated [...] Read more.

We analyze the Lane–Emden equations in the cylindrical framework. Although the explicit forms of the solutions (which are also called polytropes) are not known, we identify some of their qualitative properties. In particular, possible critical points and zeros of the polytropes are investigated and discussed, leading to possible improvements in the approximation methods which are currently employed. The cases when the critical parameter is odd and even are separately analyzed. Furthermore, we propose a technique to evaluate the distance between a pair of polytropes in small intervals. Full article

(This article belongs to the Special Issue Mathematical Modeling with Differential Equations in Biology, Chemistry, Economics, Finance and Physics, Volume 2)

12 pages, 282 KiB

Open AccessArticle

Existence and Uniqueness of Positive Solutions for Semipositone Lane-Emden Equations on the Half-Axis

by Imed Bachar

Mathematics 2023, 11(19), 4184; https://doi.org/10.3390/math11194184 - 6 Oct 2023

Viewed by 958

Abstract

Semipositone Lane–Emden type equations are considered on the half-axis. Such equations have been used in modelling several phenomena in astrophysics and mathematical physics and are often difficult to solve analytically. We provide sufficient conditions for the existence of a positive continuous solution and [...] Read more.

Semipositone Lane–Emden type equations are considered on the half-axis. Such equations have been used in modelling several phenomena in astrophysics and mathematical physics and are often difficult to solve analytically. We provide sufficient conditions for the existence of a positive continuous solution and we describe its global behavior. Our approach is based on a perturbed operator technique and fixed point theorems. Some examples are presented to illustrate the main results. Full article

(This article belongs to the Special Issue Qualitative Analysis of Differential Equations: Theory and Applications)

► Show Figures

Figure 1

15 pages, 368 KiB

Open AccessArticle

A Pair of Optimized Nyström Methods with Symmetric Hybrid Points for the Numerical Solution of Second-Order Singular Boundary Value Problems

by Higinio Ramos, Mufutau Ajani Rufai and Bruno Carpentieri

Symmetry 2023, 15(9), 1720; https://doi.org/10.3390/sym15091720 - 7 Sep 2023

Cited by 2 | Viewed by 1621

Abstract

This paper introduces an efficient approach for solving Lane–Emden–Fowler problems. Our method utilizes two Nyström schemes to perform the integration. To overcome the singularity at the left end of the interval, we combine an optimized scheme of Nyström type with a set of [...] Read more.

This paper introduces an efficient approach for solving Lane–Emden–Fowler problems. Our method utilizes two Nyström schemes to perform the integration. To overcome the singularity at the left end of the interval, we combine an optimized scheme of Nyström type with a set of Nyström formulas that are used at the fist subinterval. The optimized technique is obtained after imposing the vanishing of some of the local truncation errors, which results in a set of symmetric hybrid points. By solving an algebraic system of equations, our proposed approach generates simultaneous approximations at all grid points, resulting in a highly effective technique that outperforms several existing numerical methods in the literature. To assess the efficiency and accuracy of our approach, we perform some numerical tests on diverse real-world problems, including singular boundary value problems (SBVPs) from chemical kinetics. Full article

(This article belongs to the Special Issue Differential Equations and Applied Mathematics)

► Show Figures

Figure 1

28 pages, 910 KiB

Open AccessArticle

Solving General Fractional Lane-Emden-Fowler Differential Equations Using Haar Wavelet Collocation Method

by Kholoud Saad Albalawi, Ashish Kumar, Badr Saad Alkahtani and Pranay Goswami

Fractal Fract. 2023, 7(8), 628; https://doi.org/10.3390/fractalfract7080628 - 17 Aug 2023

Cited by 1 | Viewed by 2013

Abstract

This paper aims to solve general fractional Lane-Emden-Fowler differential equations using the Haar wavelet collocation method. This method transforms the fractional differential equation into a nonlinear system of equations, which is further solved for Haar coefficients using Newton’s method. We have constructed the [...] Read more.

This paper aims to solve general fractional Lane-Emden-Fowler differential equations using the Haar wavelet collocation method. This method transforms the fractional differential equation into a nonlinear system of equations, which is further solved for Haar coefficients using Newton’s method. We have constructed the higher-order Lane-Emden-Fowler equations. We have also discussed the convergence rate and stability analysis of our technique. We have explained the applications and numerically simulated the examples graphically and in tabular format to elaborate on the accuracy and efficiency of this approach. Full article

(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order III)

► Show Figures

Figure 1

13 pages, 774 KiB

Open AccessCommunication

A Generalized Double Chaplygin Model for Anisotropic Matter: The Newtonian Case

by Gabriel Abellán, Ángel Rincón and Eduard Sanchez

Universe 2023, 9(8), 352; https://doi.org/10.3390/universe9080352 - 28 Jul 2023

Cited by 6 | Viewed by 1323

Abstract

In this work, we investigate astrophysical systems in a Newtonian regime using anisotropic matter. For this purpose, we considered that both radial and tangential pressures satisfy a generalized Chaplygin-type equation of state. Using this model, we found the Lane–Emden equation for this system [...] Read more.

In this work, we investigate astrophysical systems in a Newtonian regime using anisotropic matter. For this purpose, we considered that both radial and tangential pressures satisfy a generalized Chaplygin-type equation of state. Using this model, we found the Lane–Emden equation for this system and solved it numerically for several sets of parameters. Finally, we explored the mass supported by this physical system and compared it with the Chandrasekhar mass. Full article

(This article belongs to the Section Gravitation)

► Show Figures

Figure 1

8 pages, 486 KiB

Open AccessArticle

Solving SIVPs of Lane–Emden–Fowler Type Using a Pair of Optimized Nyström Methods with a Variable Step Size

by Mufutau Ajani Rufai and Higinio Ramos

Mathematics 2023, 11(6), 1535; https://doi.org/10.3390/math11061535 - 22 Mar 2023

Cited by 9 | Viewed by 1876

Abstract

This research article introduces an efficient method for integrating Lane–Emden–Fowler equations of second-order singular initial value problems (SIVPs) using a pair of hybrid block methods with a variable step-size mode. The method pairs an optimized Nyström technique with a set of formulas applied [...] Read more.

This research article introduces an efficient method for integrating Lane–Emden–Fowler equations of second-order singular initial value problems (SIVPs) using a pair of hybrid block methods with a variable step-size mode. The method pairs an optimized Nyström technique with a set of formulas applied at the initial step to circumvent the singularity at the beginning of the interval. The variable step-size formulation is implemented using an embedded-type approach, resulting in an efficient technique that outperforms its counterpart methods that used fixed step-size implementation. The numerical simulations confirm the better performance of the variable step-size implementation. Full article

(This article belongs to the Special Issue Numerical Methods for Solving Differential Problems-II)

► Show Figures

Figure 1

15 pages, 452 KiB

Open AccessArticle

Spectral Collocation Approach via Normalized Shifted Jacobi Polynomials for the Nonlinear Lane-Emden Equation with Fractal-Fractional Derivative

by Youssri Hassan Youssri and Ahmed Gamal Atta

Fractal Fract. 2023, 7(2), 133; https://doi.org/10.3390/fractalfract7020133 - 31 Jan 2023

Cited by 52 | Viewed by 2903

Abstract

Herein, we adduce, analyze, and come up with spectral collocation procedures to iron out a specific class of nonlinear singular Lane–Emden (LE) equations with generalized Caputo derivatives that appear in the study of astronomical objects. The offered solution is approximated as a truncated [...] Read more.

Herein, we adduce, analyze, and come up with spectral collocation procedures to iron out a specific class of nonlinear singular Lane–Emden (LE) equations with generalized Caputo derivatives that appear in the study of astronomical objects. The offered solution is approximated as a truncated series of the normalized shifted Jacobi polynomials under the assumption that the exact solution is an element in

L_{2}

. The spectral collocation method is used as a solver to obtain the unknown expansion coefficients. The Jacobi roots are used as collocation nodes. Our solutions can easily be a generalization of the solutions of the classical LE equation, by obtaining a numerical solution based on new parameters, by fixing these parameters to the classical case, we obtain the solution of the classical equation. We provide a meticulous convergence analysis and demonstrate rapid convergence of the truncation error concerning the number of retained modes. Numerical examples show the effectiveness and applicability of the method. The primary benefits of the suggested approach are that we significantly reduce the complexity of the underlying differential equation by solving a nonlinear system of algebraic equations that can be done quickly and accurately using Newton’s method and vanishing initial guesses. Full article

(This article belongs to the Special Issue Applications of Iterative Methods in Solving Nonlinear Equations)

► Show Figures

Figure 1

18 pages, 2852 KiB

Open AccessArticle

A Swarming Approach for the Novel Second Order Perturbed Pantograph Lane–Emden Model Arising in Astrophysics

by Muneerah Al Nuwairan and Zulqurnain Sabir

Axioms 2022, 11(9), 449; https://doi.org/10.3390/axioms11090449 - 2 Sep 2022

Cited by 3 | Viewed by 1834

Abstract

The purpose of this study is to provide a mathematical construction based on the novel singular perturbed model of the second kind (NSPM-SK) using the standard form of the Lane–Emden. The singular Lane–Emden types of the models have abundant applications in astrophysics. The [...] Read more.

The purpose of this study is to provide a mathematical construction based on the novel singular perturbed model of the second kind (NSPM-SK) using the standard form of the Lane–Emden. The singular Lane–Emden types of the models have abundant applications in astrophysics. The inclusive features of this model are provided using the perturbed, pantograph, singular point together and the shape factor based on the NSPM-SK. These models become more complicated by using these factors through the artificial neural networks (ANNs) together with the optimization procedures of the swarming particle swarm optimization (PSO) paradigms and the local sequential quadratic programming (SQP). An objective function is provided based on the differential form of the NSPM-SK and then optimization is performed through the hybridization of the PSOSQP. The exactness of the model is attained to solve three different variations of the mathematical NSPM-SK by using the overlapping of the obtained and true results. The stability, robustness, and convergence of the designed numerical approach are perceived by using different statistical performances of the ANNs together with the optimization of the PSOSQP for 30 independent executions. Full article

► Show Figures

Figure 1

12 pages, 251 KiB

Open AccessArticle

A Method for the Solution of Coupled System of Emden–Fowler–Type Equations

by Aishah A. Alsulami, Mariam AL-Mazmumy, Huda O. Bakodah and Nawal Alzaid

Symmetry 2022, 14(5), 843; https://doi.org/10.3390/sym14050843 - 19 Apr 2022

Cited by 5 | Viewed by 1708

Abstract

A dependable semi-analytical method via the application of a modified Adomian Decomposition Method (ADM) to tackle the coupled system of Emden–Fowler-type equations has been proposed. More precisely, an effective differential operator together with its corresponding inverse is successfully constructed. Moreover, this operator is [...] Read more.

A dependable semi-analytical method via the application of a modified Adomian Decomposition Method (ADM) to tackle the coupled system of Emden–Fowler-type equations has been proposed. More precisely, an effective differential operator together with its corresponding inverse is successfully constructed. Moreover, this operator is able to navigate to the closed-form solution easily without resorting to converting the coupled system to a system of Volterra integral equations; as in the case of a well-known reference in the literature. Lastly, the effectiveness of the method is demonstrated on some coupled systems of the governing model, and a speedier convergence rate was noted. Full article

(This article belongs to the Special Issue Ordinary and Partial Differential Equations: Theory and Applications II)

14 pages, 1011 KiB

Open AccessArticle

An Effective Approximation Algorithm for Second-Order Singular Functional Differential Equations

by Mohammad Izadi, Hari M. Srivastava and Waleed Adel

Axioms 2022, 11(3), 133; https://doi.org/10.3390/axioms11030133 - 14 Mar 2022

Cited by 11 | Viewed by 2717

Abstract

In this research study, a novel computational algorithm for solving a second-order singular functional differential equation as a generalization of the well-known Lane–Emden and differential-difference equations is presented by using the Bessel bases. This technique depends on transforming the problem into a system [...] Read more.

In this research study, a novel computational algorithm for solving a second-order singular functional differential equation as a generalization of the well-known Lane–Emden and differential-difference equations is presented by using the Bessel bases. This technique depends on transforming the problem into a system of algebraic equations and by solving this system the unknown Bessel coefficients are determined and the solution will be known. The method is tested on several test examples and proves to provide accurate results as compared to other existing methods from the literature. The simplicity and robustness of the proposed technique drive us to investigate more of their applications to several similar problems in the future. Full article

(This article belongs to the Special Issue Mathematical Analysis and Applications III)

► Show Figures

Figure 1

14 pages, 1417 KiB

Open AccessArticle

A Neuro-Evolution Heuristic Using Active-Set Techniques to Solve a Novel Nonlinear Singular Prediction Differential Model

by Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Thongchai Botmart and Wajaree Weera

Fractal Fract. 2022, 6(1), 29; https://doi.org/10.3390/fractalfract6010029 - 4 Jan 2022

Cited by 20 | Viewed by 1915

Abstract

In this study, a novel design of a second kind of nonlinear Lane–Emden prediction differential singular model (NLE-PDSM) is presented. The numerical solutions of this model were investigated via a neuro-evolution computing intelligent solver using artificial neural networks (ANNs) optimized by global and [...] Read more.

In this study, a novel design of a second kind of nonlinear Lane–Emden prediction differential singular model (NLE-PDSM) is presented. The numerical solutions of this model were investigated via a neuro-evolution computing intelligent solver using artificial neural networks (ANNs) optimized by global and local search genetic algorithms (GAs) and the active-set method (ASM), i.e., ANN-GAASM. The novel NLE-PDSM was derived from the standard LE and the PDSM along with the details of singular points, prediction terms and shape factors. The modeling strength of ANN was implemented to create a merit function based on the second kind of NLE-PDSM using the mean squared error, and optimization was performed through the GAASM. The corroboration, validation and excellence of the ANN-GAASM for three distinct problems were established through relative studies from exact solutions on the basis of stability, convergence and robustness. Furthermore, explanations through statistical investigations confirmed the worth of the proposed scheme. Full article

(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order II)

► Show Figures

Figure 1

17 pages, 1952 KiB

Open AccessArticle

Swarm Intelligence Procedures Using Meyer Wavelets as a Neural Network for the Novel Fractional Order Pantograph Singular System

by Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Juan L. G. Guirao and Tareq Saeed

Fractal Fract. 2021, 5(4), 277; https://doi.org/10.3390/fractalfract5040277 - 17 Dec 2021

Cited by 8 | Viewed by 2488

Abstract

The purpose of the current investigation is to find the numerical solutions of the novel fractional order pantograph singular system (FOPSS) using the applications of Meyer wavelets as a neural network. The FOPSS is presented using the standard form of the Lane–Emden equation [...] Read more.

The purpose of the current investigation is to find the numerical solutions of the novel fractional order pantograph singular system (FOPSS) using the applications of Meyer wavelets as a neural network. The FOPSS is presented using the standard form of the Lane–Emden equation and the detailed discussions of the singularity, shape factor terms along with the fractional order forms. The numerical discussions of the FOPSS are described based on the fractional Meyer wavelets (FMWs) as a neural network (NN) with the optimization procedures of global/local search procedures of particle swarm optimization (PSO) and interior-point algorithm (IPA), i.e., FMWs-NN-PSOIPA. The FMWs-NN strength is pragmatic and forms a merit function based on the differential system and the initial conditions of the FOPSS. The merit function is optimized, using the integrated capability of PSOIPA. The perfection, verification and substantiation of the FOPSS using the FMWs is pragmatic for three cases through relative investigations from the true results in terms of stability and convergence. Additionally, the statics’ descriptions further authorize the presentation of the FMWs-NN-PSOIPA in terms of reliability and accuracy. Full article

(This article belongs to the Special Issue Numerical Methods and Simulations in Fractal and Fractional Problems)

► Show Figures

Figure 1

10 pages, 458 KiB

Open AccessArticle

Stellar Structure in a Newtonian Theory with Variable G

by Júlio C. Fabris, Túlio Ottoni, Júnior D. Toniato and Hermano Velten

Physics 2021, 3(4), 1123-1132; https://doi.org/10.3390/physics3040071 - 25 Nov 2021

Cited by 4 | Viewed by 2610

Abstract

A Newtonian-like theory inspired by the Brans–Dicke gravitational Lagrangian has been recently proposed by us. For static configurations, the gravitational coupling acquires an intrinsic spatial dependence within the matter distribution. Therefore, the interior of astrophysical configurations may provide a testable environment for this [...] Read more.

A Newtonian-like theory inspired by the Brans–Dicke gravitational Lagrangian has been recently proposed by us. For static configurations, the gravitational coupling acquires an intrinsic spatial dependence within the matter distribution. Therefore, the interior of astrophysical configurations may provide a testable environment for this approach as long as no screening mechanism is evoked. In this work, we focus on the stellar hydrostatic equilibrium structure in such a varying Newtonian gravitational coupling G scenario. A modified Lane–Emden equation is presented and its solutions for various values of the polytropic index are discussed. The role played by the theory parameter

ω

, the analogue of the Brans–Dicke parameter, in the physical properties of stars is discussed. Full article

(This article belongs to the Special Issue Light on Dark Worlds—A Themed Issue in Honor of Professor Maxim Yu. Khlopov on the Occasion of His 70th Birthday)

► Show Figures

Figure 1

Search Results (37)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (37)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI