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Keywords = Emden–Fowler differential equation

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15 pages, 2705 KiB  
Article
Numerical Solution of Emden–Fowler Heat-Type Equations Using Backward Difference Scheme and Haar Wavelet Collocation Method
by Mohammed N. Alshehri, Ashish Kumar, Pranay Goswami, Saad Althobaiti and Abdulrahman F. Aljohani
Mathematics 2024, 12(23), 3692; https://doi.org/10.3390/math12233692 - 25 Nov 2024
Viewed by 929
Abstract
In this study, we introduce an algorithm that utilizes the Haar wavelet collocation method to solve the time-dependent Emden–Fowler equation. This proposed method effectively addresses both linear and nonlinear partial differential equations. It is a numerical technique where the differential equation is discretized [...] Read more.
In this study, we introduce an algorithm that utilizes the Haar wavelet collocation method to solve the time-dependent Emden–Fowler equation. This proposed method effectively addresses both linear and nonlinear partial differential equations. It is a numerical technique where the differential equation is discretized using Haar basis functions. A difference scheme is also applied to approximate the time derivative. By leveraging Haar functions and the difference scheme, we form a system of equations, which is then solved for Haar coefficients using MATLAB software. The effectiveness of this technique is demonstrated through various examples. Numerical simulations are performed, and the results are presented in graphical and tabular formats. We also provide a convergence analysis and an error analysis for this method. Furthermore, approximate solutions are compared with those obtained from other methods to highlight the accuracy, efficiency, and computational convenience of this technique. Full article
(This article belongs to the Special Issue Exact Solutions and Numerical Solutions of Differential Equations)
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16 pages, 315 KiB  
Article
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
Cited by 4 | Viewed by 1296
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)
16 pages, 303 KiB  
Article
Oscillatory Features of Fourth-Order Emden–Fowler Differential Equations with Sublinear Neutral Terms
by Fahd Masood, Wedad Albalawi, Osama Moaaz and Hamdy El-Metwally
Symmetry 2024, 16(7), 933; https://doi.org/10.3390/sym16070933 - 22 Jul 2024
Cited by 4 | Viewed by 1386
Abstract
This article examines the oscillatory characteristics of a fourth-order Emden–Fowler differential equation, specifically when it includes a sublinear neutral term. Our methodology centers on establishing multiple theorems that introduce innovative conditions to guarantee that there are no positive solutions to the examined equation. [...] Read more.
This article examines the oscillatory characteristics of a fourth-order Emden–Fowler differential equation, specifically when it includes a sublinear neutral term. Our methodology centers on establishing multiple theorems that introduce innovative conditions to guarantee that there are no positive solutions to the examined equation. Due to the symmetry between non-oscillatory solutions, we obtain oscillation conditions by excluding only positive solutions. We employ the Riccati technique in various ways to achieve this objective. The criteria presented in this study complement and generalize many findings published in the literature. We support the efficiency of our findings by applying them to an example. Full article
21 pages, 401 KiB  
Article
Neutral Emden–Fowler Differential Equation of Second Order: Oscillation Criteria of Coles Type
by Amany Nabih, Asma Al-Jaser and Osama Moaaz
Symmetry 2024, 16(7), 931; https://doi.org/10.3390/sym16070931 - 21 Jul 2024
Cited by 1 | Viewed by 1056
Abstract
In this work, we study the asymptotic and oscillatory behavior of solutions to the second-order general neutral Emden–Fowler differential equation (avηxvzv) + qvFxgv = 0, where [...] Read more.
In this work, we study the asymptotic and oscillatory behavior of solutions to the second-order general neutral Emden–Fowler differential equation (avηxvzv) + qvFxgv = 0, where vv0 and the corresponding function z = x + pxh. Besides the importance of equations of the neutral type, studying the qualitative behavior of solutions to these equations is rich in analytical points and interesting issues. We begin by finding the monotonic features of positive solutions. The new properties contribute to obtaining new and improved relationships between x and z for use in studying oscillatory behavior. We present new conditions that exclude the existence of positive solutions to the examined equation, and then we establish oscillation criteria through the symmetry property between non-oscillatory solutions. We use the generalized Riccati substitution method, which enables us to apply the results to a larger area than the special cases of the considered equation. The new results essentially improve and extend previous results in the literature. We support this claim by applying the results to an example and comparing them with previous findings. Moreover, the reduction of our results to Euler’s differential equation introduces the well-known sharp oscillation criterion. Full article
(This article belongs to the Section Mathematics)
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15 pages, 306 KiB  
Article
Some Oscillatory Criteria for Second-Order Emden–Fowler Neutral Delay Differential Equations
by Haifeng Tian and Rongrong Guo
Mathematics 2024, 12(10), 1559; https://doi.org/10.3390/math12101559 - 16 May 2024
Cited by 2 | Viewed by 1251
Abstract
In this paper, by using the Riccati transformation and integral inequality technique, we establish several oscillation criteria for second-order Emden–Fowler neutral delay differential equations under the canonical case and non-canonical case, respectively. Compared with some recent results reported in the literature, we extend [...] Read more.
In this paper, by using the Riccati transformation and integral inequality technique, we establish several oscillation criteria for second-order Emden–Fowler neutral delay differential equations under the canonical case and non-canonical case, respectively. Compared with some recent results reported in the literature, we extend the range of the neutral coefficient. Therefore, our results generalize to some of the results presented in the literature. Furthermore, several examples are provided to illustrate our conclusions. Full article
16 pages, 312 KiB  
Article
Collocation Technique Based on Chebyshev Polynomials to Solve Emden–Fowler-Type Singular Boundary Value Problems with Derivative Dependence
by Shabanam Kumari, Arvind Kumar Singh and Utsav Gupta
Mathematics 2024, 12(4), 592; https://doi.org/10.3390/math12040592 - 17 Feb 2024
Cited by 2 | Viewed by 1604
Abstract
In this work, an innovative technique is presented to solve Emden–Fowler-type singular boundary value problems (SBVPs) with derivative dependence. These types of problems have significant applications in applied mathematics and astrophysics. Initially, the differential equation is transformed into a Fredholm integral equation, which [...] Read more.
In this work, an innovative technique is presented to solve Emden–Fowler-type singular boundary value problems (SBVPs) with derivative dependence. These types of problems have significant applications in applied mathematics and astrophysics. Initially, the differential equation is transformed into a Fredholm integral equation, which is then converted into a system of nonlinear equations using the collocation technique based on Chebyshev polynomials. Subsequently, an iterative numerical approach, such as Newton’s method, is employed on the system of nonlinear equations to obtain an approximate solution. Error analysis is included to assess the accuracy of the obtained solutions and provide insights into the reliability of the numerical results. Furthermore, we graphically compare the residual errors of the current method to the previously established method for various examples. Full article
(This article belongs to the Special Issue Computational Mathematics and Numerical Analysis)
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13 pages, 488 KiB  
Article
On Solutions of the Third-Order Ordinary Differential Equations of Emden-Fowler Type
by Felix Sadyrbaev
Dynamics 2023, 3(3), 550-562; https://doi.org/10.3390/dynamics3030028 - 3 Sep 2023
Cited by 1 | Viewed by 1933
Abstract
For a linear ordinary differential equation (ODE in short) of the third order, results are presented that supplement the theory of conjugate points and extremal solutions by W. Leighton, Z. Nehari, M. Hanan. It is especially noted the sensitivity of solutions to the [...] Read more.
For a linear ordinary differential equation (ODE in short) of the third order, results are presented that supplement the theory of conjugate points and extremal solutions by W. Leighton, Z. Nehari, M. Hanan. It is especially noted the sensitivity of solutions to the initial data, which makes their numerical study difficult. Similar results were obtained for the third-order nonlinear equations of the Emden-Fowler type. Full article
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28 pages, 910 KiB  
Article
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
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16 pages, 315 KiB  
Article
Fourth-Order Emden–Fowler Neutral Differential Equations: Investigating Some Qualitative Properties of Solutions
by Mansour Alatwi, Osama Moaaz, Sameh S. Askar, Ahmad M. Alshamrani and Elmetwally M. Elabbasy
Symmetry 2023, 15(7), 1446; https://doi.org/10.3390/sym15071446 - 19 Jul 2023
Cited by 1 | Viewed by 1160
Abstract
In this article, we investigate some of the qualitative properties of a class of fourth-order neutral differential equations. We start by obtaining new inequalities and relations between the solution and its corresponding function, as well as with its derivatives. The new relations allow [...] Read more.
In this article, we investigate some of the qualitative properties of a class of fourth-order neutral differential equations. We start by obtaining new inequalities and relations between the solution and its corresponding function, as well as with its derivatives. The new relations allow us to improve the monotonic and asymptotic properties of the positive solutions of the studied equation. Then, using an improved approach, we establish new criteria that test the oscillation of all solutions. We also rely on the principle of symmetry between positive and negative solutions to obtain the new criteria. The paper provides illustrative examples that highlight the significance of our findings. Full article
10 pages, 291 KiB  
Article
Oscillation of Emden–Fowler-Type Differential Equations with Non-Canonical Operators and Mixed Neutral Terms
by Sathish Kumar Marappan, Alanoud Almutairi, Loredana Florentina Iambor and Omar Bazighifan
Symmetry 2023, 15(2), 553; https://doi.org/10.3390/sym15020553 - 19 Feb 2023
Cited by 8 | Viewed by 1901
Abstract
The study of the symmetric properties of differential equations is essential for identifying effective methods for solving them. In this paper, we examine the oscillatory behavior of solutions of Emden–Fowler-type mixed non-linear neutral differential equations with both canonical and non-canonical operators. By utilizing [...] Read more.
The study of the symmetric properties of differential equations is essential for identifying effective methods for solving them. In this paper, we examine the oscillatory behavior of solutions of Emden–Fowler-type mixed non-linear neutral differential equations with both canonical and non-canonical operators. By utilizing integral conditions and the integral averaging method, we present new sufficient conditions to ensure that all solutions are oscillatory. Our results enhance and extend previous findings in the literature and are illustrated with suitable examples to demonstrate their effectiveness. Full article
12 pages, 292 KiB  
Article
On the Asymptotic Behavior of Noncanonical Third-Order Emden–Fowler Delay Differential Equations with a Superlinear Neutral Term
by Qingmin Liu, Said R. Grace, Irena Jadlovská, Ercan Tunç and Tongxing Li
Mathematics 2022, 10(16), 2902; https://doi.org/10.3390/math10162902 - 12 Aug 2022
Cited by 6 | Viewed by 1524
Abstract
The present paper is concerned with the asymptotic behavior of solutions to a class of noncanonical third-order Emden–Fowler delay differential equations with a superlinear neutral term. Using a Riccati-type transformation as well as integral criteria, we establish some new sufficient conditions guaranteeing that [...] Read more.
The present paper is concerned with the asymptotic behavior of solutions to a class of noncanonical third-order Emden–Fowler delay differential equations with a superlinear neutral term. Using a Riccati-type transformation as well as integral criteria, we establish some new sufficient conditions guaranteeing that every solution of the equation considered either oscillates or converges to zero asymptotically. The results are illustrated with two examples. Full article
(This article belongs to the Special Issue Mathematical Modeling and Simulation of Oscillatory Phenomena)
12 pages, 280 KiB  
Article
Oscillation of Second Order Nonlinear Neutral Differential Equations
by Yingzhu Wu, Yuanhong Yu and Jinsen Xiao
Mathematics 2022, 10(15), 2739; https://doi.org/10.3390/math10152739 - 2 Aug 2022
Cited by 7 | Viewed by 1785
Abstract
The study of the oscillatory behavior of solutions to second order nonlinear differential equations is motivated by their numerous applications in the natural sciences and engineering. In the presented research, some new oscillation criteria for a class of damped second order neutral differential [...] Read more.
The study of the oscillatory behavior of solutions to second order nonlinear differential equations is motivated by their numerous applications in the natural sciences and engineering. In the presented research, some new oscillation criteria for a class of damped second order neutral differential equations with noncanonical operators are established. The results extend and improve on those reported in the literature. Moreover, some examples are provided to show the significance of the results. Full article
(This article belongs to the Special Issue Mathematical Modeling and Simulation of Oscillatory Phenomena)
16 pages, 326 KiB  
Article
Charged Shear-Free Fluids and Complexity in First Integrals
by Sfundo C. Gumede, Keshlan S. Govinder and Sunil D. Maharaj
Entropy 2022, 24(5), 645; https://doi.org/10.3390/e24050645 - 4 May 2022
Cited by 3 | Viewed by 1633
Abstract
The equation yxx=f(x)y2+g(x)y3 is the charged generalization of the Emden-Fowler equation that is crucial in the study of spherically symmetric shear-free spacetimes. This version arises from the [...] Read more.
The equation yxx=f(x)y2+g(x)y3 is the charged generalization of the Emden-Fowler equation that is crucial in the study of spherically symmetric shear-free spacetimes. This version arises from the Einstein–Maxwell system for a charged shear-free matter distribution. We integrate this equation and find a new first integral. For this solution to exist, two integral equations arise as integrability conditions. The integrability conditions can be transformed to nonlinear differential equations, which give explicit forms for f(x) and g(x) in terms of elementary and special functions. The explicit forms f(x)1x511x11/5 and g(x)1x611x12/5 arise as repeated roots of a fourth order polynomial. This is a new solution to the Einstein-Maxwell equations. Our result complements earlier work in neutral and charged matter showing that the complexity of a charged self-gravitating fluid is connected to the existence of a first integral. Full article
(This article belongs to the Special Issue Complexity of Self-Gravitating Systems)
13 pages, 1368 KiB  
Article
New Monotonic Properties of the Class of Positive Solutions of Even-Order Neutral Differential Equations
by Barakah Almarri, Higinio Ramos and Osama Moaaz
Mathematics 2022, 10(9), 1470; https://doi.org/10.3390/math10091470 - 27 Apr 2022
Cited by 6 | Viewed by 1851
Abstract
In this study, new asymptotic properties of positive solutions of the even-order neutral delay differential equation with the noncanonical operator are established. The new properties are of an iterative nature, which allows it to be applied several times. Using these properties, we obtain [...] Read more.
In this study, new asymptotic properties of positive solutions of the even-order neutral delay differential equation with the noncanonical operator are established. The new properties are of an iterative nature, which allows it to be applied several times. Using these properties, we obtain new criteria to exclude a class from the positive solutions of the studied equation, using the comparison principles. Full article
(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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12 pages, 251 KiB  
Article
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
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