Lane-Emden Type Equations Research Articles

Boubaker Wavelets Functions: Properties and Applications

تم في هذا البحث تقديم شرح تفصيلي لدوال متعددة حدود بوبكر المتعامدة مع بعض الخواص ذات الاهمية، كذلك استنتاج تعريف متعددات حدود بوبكر الموجية في الفترة (1, 0] وذلك بالاستفادة من بعض الخواص المهمة لمتعددة حدود بوبكر. تمتلك هذه الدوال الاساسية خاصية العيارية المتعامدة بالإضافة الى ضرورة تواجد المنطلق المرصوص. لهذه الدوال الموجية العديد من المزايا وقد استخدمت في المجال النظري بالإضافة الى المجال العملي وتم استخدامها مع متعددات الحدود المتعامدة لغرض طرح طريقة جديدة للتعامل مع العديد من المسائل في العلوم والهندسة ولذلك تعتبر طريقة استخدام الموجبات ذات اهمية كبيرة عند الاستفادة منها في المجالات ذات العلاقة. بالإضافة الى الاستفادة من موجبات بوبكر للحصول على خاصية جديدة وهي مشتقات دالة بوبكر الموجية. استخدمت موجية بوبكر مع طريقة الترصيف للحصول على حل عددي تقريبي لمعادلات لان ايمدن من النوع الخطي المنفرد. تصف معادلات لان ايمدن العديد من الظواهر المهمة في علم الرياضيات والفيزياء السماوي مثل الانفجارات الحرارية الكونية وتكوين النجوم. وتعتبر احدى حالات مسائل القيمة الابتدائية المنفردة للمعادلات التفاضلية اللاخطية من الرتبة الثانية. تقوم هذه الطريقة المقترحة بتحويل معادلة لان ايمدن الى نظام من المعادلات التفاضلية الخطية والتي يمكن حلها بسهولة باستخدام الحاسبة. بناءً على هذا فقد ظهر تطابق الحل العددي مع الحل التحليلي بالرغم من استخدام عدد قليل من متعددات حدود بوبكر الموجية لغرض ايجاد هذا الحل. كذلك، تم في هذا البحث البرهنة على قيمه قيد الخطأ المستخرج من هذه الطريقة. وتضمن هذا البحث على ثلاث امثلة عددية من نوع معادلات لان ايمدن لتوضيح قابلية استخدام الطريقة المقترحة. تم توضيح النتائج الحقيقة مع النتائج التقريبية في شكل جداول ورسوم هندسية لغرض المقارنة.

Mathematical Analysis of Non-Isothermal Reaction–Diffusion Models Arising in Spherical Catalyst and Spherical Biocatalyst

The theory of dynamical systems and their widespread applications involve, for example, the Lane–Emden-type equations which are known to arise in initial- and boundary-value problems with singularity at the time t=0. The main objective of this paper is to make use of some mathematical analytic tools and techniques in order to numerically solve some reaction–diffusion equations, which arise in spherical catalysts and spherical biocatalysts, by applying the Chebyshev spectral collocation method. The proposed scheme has good accuracy. The results are demonstrated by means of illustrative graphs and numerical tables. The accuracy of the proposed method is verified by a comparison with the results which are derived by using analytical methods.

Generalized Bessel Quasilinearization Technique Applied to Bratu and Lane–Emden-Type Equations of Arbitrary Order

The ultimate goal of this study is to develop a numerically effective approximation technique to acquire numerical solutions of the integer and fractional-order Bratu and the singular Lane–Emden-type problems especially with exponential nonlinearity. Both the initial and boundary conditions were considered and the fractional derivative being considered in the Liouville–Caputo sense. In the direct approach, the generalized Bessel matrix method based on collocation points was utilized to convert the model problems into a nonlinear fundamental matrix equation. Then, the technique of quasilinearization was employed to tackle the nonlinearity that arose in our considered model problems. Consequently, the quasilinearization method was utilized to transform the original nonlinear problems into a sequence of linear equations, while the generalized Bessel collocation scheme was employed to solve the resulting linear equations iteratively. In particular, to convert the Neumann initial or boundary condition into a matrix form, a fast algorithm for computing the derivative of the basis functions is presented. The error analysis of the quasilinear approach is also discussed. The effectiveness of the present linearized approach is illustrated through several simulations with some test examples. Comparisons with existing well-known schemes revealed that the presented technique is an easy-to-implement method while being very effective and convenient for the nonlinear Bratu and Lane–Emden equations.

Laguerre collocation method for solving Lane-Emden type equations

In this paper, a Laguerre collocation method is presented in order to obtain numerical solutions for linear and nonlinear Lane-Emden type equations and their initial conditions. The basis of the present method is operational matrices with respect to modified generalized Laguerre polynomials(MGLPs) that transforms the solution of main equation and its initial conditions to the solution of a matrix equation corresponding to the system of algebraic equations with the unknown Laguerre coefficients. By solving this system, coefficients of approximate solution of the main problem will be determined. Implementation of the method is easy and has more accurate results in comparison with results of other methods.

Approximate solution of Lane-Emden problem via modified Hermite operation matrix method


 
 
 
 Lane-Emden equations are singular initial value problems and they are important in mathematical physics and astrophysics. The aim of this present paper is presenting a new numerical method for finding approximate solution to Lane-Emden type equations arising in astrophysics based on modified Hermite operational matrix of integration. The proposed technique is based on taking the truncated modified Hermite series of the solution function in the Lane-Emden equation and then transferred into a matrix equation together with the given conditions. The obtained result is system of linear algebraic equation using collection points. The suggested algorithm is applied on some relevant physical problems as Lane-Emden type equations. 
 
 
 

An efficient hybrid computational technique for the time dependent Lane-Emden equation of arbitrary order

The study of dynamic behaviour of nonlinear models that arise in ocean engineering play a vital role in our daily life. There are many examples of ocean water waves which are nonlinear in nature. In shallow water, the linearization of the equations imposes severe conditions on wave amplitude than it does in deep water, and the strong nonlinear effects are observed. In this paper, q-homotopy analysis Laplace transform scheme is used to inspect time dependent nonlinear Lane-Emden type equation of arbitrary order. It offers the solution in a fast converging series. The uniqueness and convergence analysis of the considered model is presented. The given examples confirm the competency as well as accuracy of the presented scheme. The behavior of obtained solution for distinct orders of fractional derivative is discussed through graphs. The auxiliary parameter ħ offers a suitable mode of handling the region of convergence. The outcomes reveal that the q-HATM is attractive, reliable, efficient and very effective.

Improved Numerical Computation to Solve Lane-Emden type Equations

Lane-Emden equation is also of fundamental importance in mathematical physics, celestial mechanics,and computer science. It can be used to describe stellar structures, equilibrium density distribution in polytrophicisothermal gas, thermal behavior in mutual attraction of its molecules. An improved numerical method is developed for solving Lane-Emden type differential equations. The method is based on power series solutions of differential equations and Maclaurin series expansion. A python program is written to carry out numerical calculations. Five examples are solved and shown in this paper, the solutions obtained by the program are compared with the exact solutions of differential equation, an excellent agreement is found between them. The present method improves runtime.

Solvability for a class of nonlocal singular fractional differential equations of Lane Emden type

In this paper, we study a class of nonlocal singular differential equations of fractional orders. The considered class involves Caputo derivatives and Riemann-Liouville integral in its linearities. It is a more general problem for which the question of existence and uniqueness of solutions is investigated. For this class, the question of existence of at least one solution is also discussed. At the end, an illustrative example is presented.

Stability of solutions for two classes of fractional differential equations of Lane-Emden type

By introducing some appropriate definitions for Ulam type stabilities, in this paper, we discuss the Ulam and Hyers and the generalized Ulam and Hyers stabilities for two different classes of nonlinear singular fractional integro-differential problems of Lane and Emden type. The existence and uniqueness results for our two classes have already been published respectively in Math. Method. Appl. Sci., 2020, and Moroc. J. Pure Appl. Anal., 2020.

Singular curve and critical curve for doubly nonlinear Lane–Emden type equations

This paper is concerned with singular curve and critical curve for the periodic doubly nonlinear Lane–Emden type equation . In 2010, under a convex assumption on the domain Ω, Wang et al. (2010) considered a partial case of . While, there are no results about the cases of and before the present work. In this paper, we fill the gap for these two cases and give a total classification for the exponents. Furthermore, by constructing a special blow‐up sequence, we remove the convex assumption on the domain Ω, not only for the partial case considered in Wang et al. (2010) but also for other remaining cases.

A novel operational matrix for the numerical solution of nonlinear Lane–Emden system of fractional order

In this work, we introduce a numerical method for solving nonlinear fractional system of Lane–Emden type equations. The proposed technique is based on Dickson operational matrix of a fractional derivative. First, we deduce the Dickson operational matrix of the fractional derivative using Dickson polynomial, and then, the obtained matrix is unitized to convert the fractional Lane–Emden system with its initial conditions into a system of nonlinear algebraic equations. This system of algebraic equations can be solved numerically via Newton’s iteration method. An error estimate of the proposed method is derived. Numerical examples are provided to demonstrate the validity, applicability, and accuracy of the new technique.

Orthonormal Bernoulli wavelets neural network method and its application in astrophysics

The purpose of this paper is to present two efficient computational schemes for solving Lane–Emden type equations based on an artificial neural network. The mentioned neural network consists of three layers (input layer, hidden layer, output layer). We use orthonormal Bernoulli wavelets and arctanh(x) functions as activation functions of the hidden layer and output layer, respectively. Finally, the classical optimization and collocation methods are applied to train this neural network in first and second methods, respectively. In addition, to show the accuracy and efficiency of the current techniques, we test those on various types of Lane–Emden equations and our schemes are compared with some other numerical methods.

A discontinuous finite element approximation to singular Lane-Emden type equations

In this article, we develop the local discontinuous Galerkin finite element method for the numerical approximations of a class of singular second-order ordinary differential equations known as the Lane-Emden type equations equipped with initial or boundary conditions. These equations have been considered via different models that naturally appear for example in several phenomena in astrophysical science. By converting the governing equations into a first-order systems of differential equations, the approximate solution is sought in a piecewise discontinuous polynomial space while the natural upwind fluxes are used at element interfaces. The existence-uniqueness of the weak formulation is provided and the numerical stability of the method in the L∞ norm is established. Five illustrative test problems are given to demonstrate the applicability and validity of the scheme. Comparisons between the numerical results of the proposed method with existing results are carried out in order to show that the new approximation algorithm provides accurate solutions even near the singular point.

A novel Chebyshev neural network approach for solving singular arbitrary order Lane-Emden equation arising in astrophysics

ABSTRACT The motivation of this investigation is to develop a single-layer Chebyshev Neural Network (ChNN) model to handle singular fractional (arbitrary)-order Lane-Emden type equations. These equations are well-known application problems of astrophysics and quantum mechanics. Fractional Lane-Emden equations are singular so it is very difficult to solve analytically. Thus, an efficient method is required to handle the above equations. Here, our main aim is to use a single-layer ChNN model for solving fractional Lane-Emden equations. ChNN model is one kind of Functional Link Neural Network (FLNN) in which the hidden layer is replaced by a functional expansion block of the input pattern using orthogonalshifted Chebyshev polynomials (SChP). Thus, the network parameters of ChNN are less than the Multi-Layer Artificial Neural Network (MLANN). We have considered factional-order singular nonlinear problems of astrophysics to show the computational effort of the proposed method. Back Propagation algorithm of the unsupervised version has been considered for minimizing the error function and updating the weights of the ChNN model. Computed results are displayed in terms of tables and graphs.

Perturbed Collocation Method For Solving Singular Multi-order Fractional Differential Equations of Lane-Emden Type

In this work, a general class of multi-order fractional differential equations of Lane-Emden type is considered. Here, an assumed approximate solution is substituted into a slightly perturbed form of the general class and the resulting equation is collocated at equally spaced interior points to give a system of linear algebraic equations which are then solved by suitable computer software; Maple 18.

Applications of modified Mickens-type NSFD schemes to Lane–Emden equations

If there is a jump discontinuity present in the forcing term of a boundary value problem (BVP), the nonstandard finite difference (NSFD) and finite difference (FD) methods do not approximate the solutions very well. Here we use fuzzy transforms (FTs) and derive fuzzy transformed NSFD schemes that are referred to as non-standard fuzzy transform methods (NFTMs). The convergence of the derived NFTMs is established. Numerical solutions of Lane–Emden type equations are obtained using NFTMs. We show that NFTMs provide better results than NSFD and FD methods when the forcing term has a jump discontinuity. Even for large jumps, the NFTMs provide more accurate results than the other methods.

Modified Hermite Operational Matrix Method for Nonlinear Lane-Emden Problem

Nonlinear Lane –Emden equations are significant in astrophysics and mathematical physics. The aim of this paper is submitted a new approximate method for finding solution to nonlinear Lane-Emden type equations appearing in astrophysics based on modified Hermite integration operational matrix. The suggest technique is based on taking the truncated modified Hermite series of the solution function in the nonlinear Lane-Emden equation and then changed into a matrix equation with the given conditions. Nonlinear system of algebraic equation using collection points is obtained. The present method is applied on some relevant physical problems as nonlinear Lane-Emden type equations.

Numerical solution of Lane-Emden type equations using Adomian decomposition method with unequal step-size partitions

PurposeThe purpose of this paper is to obtain the highly accurate numerical solution of Lane–Emden-type equations using modified Adomian decomposition method (MADM) for unequal step-size partitions.Design/methodology/approachFirst, the authors describe the standard Adomian decomposition scheme and the Adomian polynomials for solving nonlinear differential equations. After that, for the fast calculation of the Adomian polynomials, an algorithm is presented based on Duan’s corollary and Rach’s rule. Then, MADM is discussed for the unequal step-size partitions of the domain, to obtain the numerical solution of Lane–Emden-type equations. Moreover, convergence analysis and an error bound for the approximate solution are discussed.FindingsThe proposed method removes the singular behaviour of the problems and provides the high precision numerical solution in the large effective region of convergence in comparison to the other existing methods, as shown in the tested examples.Originality/valueUnlike the other methods, the proposed method does not require linearization or perturbation to obtain an analytical and numerical solution of singular differential equations, and the obtained results are more physically realistic.

Analytical solution to the conformable fractional Lane-Emden type equations arising in astrophysics

Lane-Emden equations have many applications in physical science and so make great attention to researchers. In most cases there is are no exact solutions to these equations and only approximate solutions could be performed. This paper introduces analytical solutions to Lane-Emden type equations using the fractional Adomian decomposition method. We solve six examples related to astrophysical and physical problems. The approximate solutions are derived in the form of a divergent series. In the case of the fractional parameter equal one, the integer order, we obtained the same results as obtained by the method of Adomian decomposition.

A Deep-Learning-Based Method for Solving Nonlinear Singular Lane-Emden Type Equation

The nonlinear Lane-Emden type equation can be used to describe many physical phenomena. To solve this type of equation, a method based on deep neural network is proposed. The output layer of this network has two layers, the last one of which scaling the outputs of their neighbors with an aim at coping with issues where the values of function to be approximated are much less or larger than the order of 1. The Lane-Emden equation and its initial conditions are employed to construct the loss function, and the problem of solving Lane-Emden equation is transformed into an optimization problem. A hybrid method combined Adam and L-BFGS-B methods is used to solve the optimization problem and consequently the Lane-Emden type equation is solved. To increase the accuracy, an adaptive strategy is incorporated into the training data sampling method. Especially, a strategy in coping with the issue about solving white-dwarf problem is proposed. Numerical experiments are conducted in which reference solutions including analytical solutions and numerical solutions given by Runge-Kutta method are used to verify the effectiveness of our proposed method. More importantly, the solutions over large domains are calculated by the use of the proposed method. The results show that the results given by our method are in good agreement with the reference solutions, and in cases where existent neural-network-based method fails, our method is able to produce convincible results.