Emden-Fowler Equation Research Articles

Highly efficient optimal decomposition approach and its mathematical analysis for solving fourth-order Lane–Emden–Fowler equations

In this paper, an optimal decomposition algorithm is introduced to solve a kind of nonlinear fourth-order Emden–Fowler equations (EFEs) that appear in many applied fields. Transforming the Emden–Fowler equation into a Volterra integral equivalent equation allows us to deal with the singularity at the endpoint x=0. This conversion also helps to reduce the computational cost of solving the problem. The existence and uniqueness of the solution of each integral equation obtained are established in the corresponding theorems. The convergence analysis further supports the theoretical findings. The accuracy and efficiency of the new method are tested against the existing method (Wazwaz et al., 2014) using numerous cases, and the results show that the presented scheme is a reliable method for computing approximate series solutions and even exact solutions. In addition, the new technique overcomes the drawback of the existing method, that provides only an approximation within a limited interval.

Numerical solution of generalized fractional Lane–Emden-Fowler equation using Haar wavelet method

This paper presents a numerical technique to solve the generalized fractional Lane–Emden–Fowler equation. We use the Haar wavelet technique to approximate the solution of the generalized fractional Lane–Emden–Fowler equation. In this method, a differential equation transforms into a system of nonlinear equations, and this system is further solved to obtain Haar coefficients. We also showed the convergence of this method by establishing an upper bound. The rate of convergence is also discussed in our algorithm. Many numerical examples have been taken, and the results of those numerical experiments have been written in tabular form. We also documented the experiments graphically to show the efficiency and accuracy of this method.

FORECASTING THE BEHAVIOR OF FRACTIONAL MODEL OF EMDEN–FOWLER EQUATION WITH CAPUTO–KATUGAMPOLA MEMORY

The main aim of this paper is to analyze the behavior of time-fractional Emden–Fowler (EF) equation associated with Caputo–Katugampola fractional derivative occurring in mathematical physics and astrophysics. A powerful analytical approach, which is an amalgamation of q-homotopy analysis approach and generalized Laplace transform with homotopy polynomials, is implemented to obtain approximate analytical solution of the time-fractional EF equation. Main advantage of this research work is that the implemented technique contains an auxiliary parameter to control the convergence region of obtained series solution. Some examples are considered to illustrate the accuracy and efficiency of the applied technique. Numerical results are demonstrated in the form of tabular and graphical representations.

A Hybrid Numerical Approach to Solve Multi-singular and Nonlinear Emden–Fowler Equations of Fourth Order: HQLMT

A Hybrid Numerical Approach to Solve Multi-singular and Nonlinear Emden–Fowler Equations of Fourth Order: HQLMT

Solve third order Lane–Emden–Fowler equation by Adomian decomposition method and quartic trigonometric B-spline method

In this work, we study the Lane–Emden–Fowler equation which has a wide range of applications in astrophysics, classical and quantum mechanics. Firstly, we convert this nonlinear differential equation into an equivalent integral equation. We employ the powerful Adomian decomposition method to derive an analytical solution that will also lead to numerical solutions. Moreover, we exhibit two theorems to ensure the uniqueness of the solution of the problem and to confirm the convergence of the proposed scheme. In addition, we use quartic trigonometric B-spline as a powerful numerical method to achieve numerical solutions. Finally we present a comparison between the proposed methods to show their accuracy performance. Proper graphs are also provided to illustrate the obtained solutions.

Solving the Lane-Emden and Emden-Fowler equations on Cantor Sets by the Local Fractional Homotopy Analysis Method

Within the definition of the local fractional derivative, we present a reliable and effective method for solving the Lane- Emden and Emden-Fowler models in the current article. Numerous fields of research and engineering can benefit from the use of these models. The proposed technique is named the local fractional homotopy analysis method. The method is applied for both models of k order and tested for several examples. This method proves to be effective in handling such models when compared to other similar models. The results indicate the robustness of the technique and the possibility of its application to other models in the future.

An innovative Fibonacci wavelet collocation method for the numerical approximation of Emden-Fowler equations

This article presents a novel approach using the Fibonacci wavelet collocation method (FWCM) for the numerical solution of Emden-Fowler-type equations. The Emden-Fowler equations are a class of nonlinear differential equations that arise in various fields of science and engineering, particularly in astrophysics and fluid dynamics. Due to their nonlinear and singular nature, the conventional approaches to solving these equations encounter difficulties. This method is particularly effective in handling problems with singularities, as it adapts naturally to the local behaviour of the solution. Here, we introduced the Fibonacci wavelet collocation method as a powerful numerical technique for tackling Emden-Fowler-type equations. The Fibonacci wavelet basis functions possess remarkable properties, including compact support, making them well-suited for approximating solutions to differential equations. The main advantage of this approach lies in its ability to reduce the computational complexity associated with solving Emden-Fowler equations, resulting in accurate and efficient solutions. Comparative analyses with other established numerical methods reveal its superior accuracy and convergence rate performance. Further, several examples demonstrate the method's flexibility when dealing with different singularity levels. This paper contributes to numerical analysis by introducing the Fibonacci wavelet method as a robust tool for solving Emden-Fowler-type equations.

Numerical solution of coupled system of Emden-Fowler equations using artificial neural network technique

In this paper, a deep artificial neural network technique is proposed to solve the coupled system of Emden-Fowler equations. A vectorized form of algorithm is developed. Implementation and simulation of this technique is performed using Python code. This technique is implemented in various numerical examples, and simulations are conducted. We have shown graphically how accurately this method works. We have shown the comparison of numerical solution and exact solution using error tables. We have also conducted a comparative analysis of our solution with alternative methods, including the Bernstein collocation method and the Homotopy analysis method. The comparative results are presented in error tables. The efficiency and accuracy of this method are demonstrated by these graphs and tables.

Solution of Lane Emden–Fowler equations by Taylor series method

In this paper, a new application of the Taylor series method (TSM) and its properties is used to solve partial differential equation problems with a singular limit value problem. We are particularly interested in applying this method to a certain class of physical problems such as Lane Emden–Fowler partial differential equations. The results obtained show that this analytical technique is very simple and effective compared to other methods.

Asymptotic profiles for a nonlinear Kirchhoff equation with combined powers nonlinearity

We study asymptotic behavior of positive ground state solutions of the nonlinear Kirchhoff equation −(a+b∫RN|∇u|2)Δu+λu=uq−1+up−1inRN,(Pλ)as λ→0 and λ→+∞, where N=3 or N=4, 2<q≤p≤2∗, 2∗=2NN−2 is the Sobolev critical exponent, a>0, b≥0 are constants and λ>0 is a parameter. In particular, we prove that in the case 2<q<p=2∗, as λ→0, after a suitable rescaling the ground state solutions of (Pλ) converge to the unique positive solution of the equation −Δu+u=uq−1 and as λ→+∞, after another rescaling the ground state solutions of (Pλ) converge to a particular solution of the critical Emden–Fowler equation −Δu=u2∗−1. We establish a sharp asymptotic characterization of such rescalings, which depends in a non-trivial way on the space dimension N=3 and N=4. We also discuss a connection of our results with a mass constrained problem associated to (Pλ) with normalization constraint ∫RN|u|2=c2. As a consequence of the main results, we obtain the existence, non-existence and asymptotic behavior of positive normalized solutions of such a problem. In particular, we obtain the exact number and their precise asymptotic expressions of normalized solutions if c>0 is sufficiently large or sufficiently small. Our results also show that in the space dimension N=3, there is a striking difference between the cases b=0 and b≠0. More precisely, if b≠0, then both p0≔10/3 and pb≔14/3 play a role in the existence, non-existence, the exact number and asymptotic behavior of the normalized solutions of the mass constrained problem, which is completely different from those for the corresponding nonlinear Schrödinger equation and which reveals the special influence of the nonlocal term.

A Comparative Study using Scale-2 and Scale-3 Haar Wavelet for the Solution of Higher Order Differential Equation

A comparative study of scale-2 and scale-3 Haar wavelet has been presented to illustrate the level of accuracy attained by both the wavelets by applying on higher order differential equations known as Emden fowler equation, which has great importance in the field of astrophysics. Approximation of space variable is done by scale-2 and scale-3 Haar wavelet method by choosing different scales. The method is tested upon several test problems. The results are computed and compared in the form of absolute errors. The numerical tests confirm the accuracy, applicability and efficiency of the proposed method with different levels using both the wavelets. By the help of MATLAB algorithm simplification of the computational process is done.

On the implementation of fractional homotopy perturbation transform method to the Emden–Fowler equations

On the implementation of fractional homotopy perturbation transform method to the Emden–Fowler equations

Numerical solution of general Emden–Fowler equation using Haar wavelet collocation method

This paper deals with the numerical solution of the general Emden–Fowler equation using the Haar wavelet collocation method. This method transforms the differential equation into a system of nonlinear equations. These equations are further solved by Newton's method to obtain the Haar coefficients, and finally the solution to the problem is acquired using these coefficients. We have taken many examples of fifth- and sixth-order equations and implemented our method on those examples. The graphs show the efficiency of the solution for resolution L = 3 and the maximum absolute error of our approach. The error tables give a good picture of the accuracy of this approach.

Approximate Solution of Emden-Fowler Equation Using the Galerkin Method

The Emden-Fowler equation (E-F.Eq.) used in mathematical with other science like physics, chemical physics and astrophysics, also this equation can be reduces to the Lane–Emden equation with specified function and used it in different sciences with mathematics. Many Authors study analytic and numerical methods to find the solution for this kind of the equations in the case linear or nonlinear one of these methods the homotopy-perturbation method.In this work the approximate solution for generalized (E-F.Eq.) in the second order ordinary differential equations was found by Galerkin method which is one of the weighted residual methods and do not need long time also use operator (linear or nonlinear) or differential operator in the any kind of the intervals and compared this solution with the exact solution by discuss the results from applying this method for homogeneous and nonhomogeneous equations and drown the solutions in the same figure to illustrate the results.

Enhanced criteria for detecting oscillations in neutral delay Emden-Fowler differential equations

In this article, we present new oscillation criteria for a specific class of second-order differential equations known as neutral delay Emden-Fowler equations. By utilizing advanced mathematical techniques, we are able to derive conditions that are more accurate and efficient than previous methods. Furthermore, our results also simplify the process of identifying oscillations in these types of equations. Our findings have important implications for various fields such as engineering, physics and mathematics, where these equations are widely used to model dynamic systems. Additionally, We have also highlighted some illustrative examples to further demonstrate the applicability and potential of our results.

Asymptotic profiles for a nonlinear Schrödinger equation with critical combined powers nonlinearity

We study asymptotic behaviour of positive ground state solutions of the nonlinear Schrödinger equation -Δu+u=u2∗-1+λuq-1inRN,(Pλ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} -\\Delta u+u=u^{2^*-1}+\\lambda u^{q-1} \\quad \ extrm{in}\\, {\\mathbb {R}}^N,\\qquad \\qquad \\qquad \\qquad \\qquad {(P_\\lambda )} \\end{aligned}$$\\end{document}where Nge 3 is an integer, 2^{*}=frac{2N}{N-2} is the Sobolev critical exponent, 2<q<2^* and lambda >0 is a parameter. It is known that as lambda rightarrow 0, after a rescaling the ground state solutions of (P_lambda ) converge to a particular solution of the critical Emden-Fowler equation -Delta u=u^{2^*-1}. We establish a novel sharp asymptotic characterisation of such a rescaling, which depends in a non-trivial way on the space dimension N=3, N=4 or N ge 5. We also discuss a connection of these results with a mass constrained problem associated to (P_{lambda }). Unlike previous work of this type, our method is based on the Nehari-Pohožaev manifold minimization, which allows to control the L^{2} norm of the groundstates.

Accurate Laguerre collocation solutions to a class of Emden–Fowler type BVP

We solve numerically the nonlinear and double singular boundary value problem formed by the well known Emden–Fowler equation , s > 1 along with the boundary conditions and In order to capture the exponential decrease of its solution we use the Laguerre–Gauss–Radau collocation method and infer its convergence. We show that the value of u ʹ at origin, which plays a fundamental role in these problems, definitely satisfies some rigorous accepted bounds. A particular attention is paid to the Thomas–Fermi case, i.e. We treat the problems as boundary values ones without any involvement of ones with initial values. The method is robust with respect to scaling and order of approximation.

Solving Emden–Fowler Equations Using Improved Extreme Learning Machine Algorithm Based on Block Legendre Basis Neural Network

Solving Emden–Fowler Equations Using Improved Extreme Learning Machine Algorithm Based on Block Legendre Basis Neural Network

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

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.

Lie algebra classification, conservation laws and invariant solutions for modification of the generalization of the Emden–Fowler equation

We obtain the optimal system’s generating operators associated to a modification of the generalization of the Emden–Fowler Equation. Using those operators we characterize all invariant solutions associated to a generalized. Moreover, we present the variational symmetries and the corresponding conservation laws, using Noether’s theorem and Ibragimov’s method. Finally, we classify the Lie algebra associated to the given equation.