Nonlinear Lane Research Articles

An efficient recursive technique with Padé approximation for a kind of Lane–Emden type equations emerging in various physical phenomena

The study numerically examined a class of nonlinear singular differential problems known as the Lane–Emden differential equation, which emerges in numerous real-world situations. The primary goal of this work is to formulate a computationally efficient iterative technique for solving the nonlinear Lane–Emden initial value problems. The proposed approach is a hybrid of the homotopy perturbation method and the Padé approximation. The nonlinear singular Lane–Emden initial value problem (SLEIVP) is transformed into an equivalent recursive integral employing the Picard’s approach. To resolve the singularity and nonlinearity, the recursive integral equation is transformed into a system of integral equations by using the homotopy notion. Furthermore, to enhance the convergence rate of the technique, Padé approximation is taken into account. The convergence analysis for the proposed approach is also conducted. The present technique is tested on SLEIVPs and numerical findings are compared with the existing techniques, to demonstrate the accuracy, effectiveness and ease of use.

A novel numerical approach and stability analysis for a class of pantograph delay differential equation

The goal of this research is to introduce a numerical technique based on shifted Chebyshev polynomial and collocation method for a second order non-linear Lane–Emden pantograph delay differential equation (PDDE) subject to initial conditions. The proposed formulation is based on converting the initial value problem (IVP) into an equivalent fundamental algebraic equation which reduces computational expense. The convergence analysis of the proposed numerical technique demonstrates the efficiency of the proposed scheme. The uniqueness and existence, regularity and stability analysis of the Lane–Emden PDDE is discussed and investigated thoroughly by providing sufficient theorems. Due to the non-availability of abundant literature, the new approach is implemented and tested against existing approaches on standard and newly constructed nonlinear examples. The comparison demonstrates that the proposed method is more accurate and consumes lesser CPU time to compute the results than the existing methods.

Principal spectral curves for Lane–Emden fully nonlinear type systems and applications

In this paper we exploit the phenomenon of two principal half eigenvalues in the context of fully nonlinear Lane–Emden type systems with possibly unbounded coefficients and weights. We show that this gives rise to the existence of two principal spectral curves on the plane. We develop an anti-maximum principle, which is a novelty even for Lane–Emden systems involving the Laplacian operator. As applications, we derive a maximum principle in small domains for these systems, as well as existence and uniqueness of positive solutions in the sublinear regime. Most of our results are new even in the scalar case, in particular for a class of Isaac’s operators with unbounded coefficients, whose $$W^{2,\varrho }$$ regularity estimates we also prove.

Wavelet solution of a strongly nonlinear Lane–Emden equation

Wavelet solution of a strongly nonlinear Lane–Emden equation

Analytical and Numerical solutions for fourth order Lane–Emden–Fowler equation

This model attracted attention due to its importance as it emerge in the modeling of many physical systems. We present an approximate solution for the fourth-order nonlinear Lane–Emden–Fowler equation using two simple and powerful methods, the first one is the Adomian decomposition method and the second one is the quintic B-spline method (QBSM). We transform the differential equation into an integral equation to overcome the singularity at the origin. Moreover, we discuss the convergence and the uniqueness of the three types of the model using the Adomian decomposition method. Finally, We present tables and graphs to make a comparison between the two methods and the exact solutions to exhibit the accuracy and efficiency of the proposed methods.

Numerical simulation of variable-order fractional differential equation of nonlinear Lane–Emden type appearing in astrophysics

Abstract This paper suggests the Chebyshev pseudo-spectral approach to solve the variable-order fractional Lane–Emden differential equations (VOFLEDE). The variable-order fractional derivative (VOFD) is defined in the Caputo sense. The proposed method transforms the problem into a set of algebraic equations that can be solved for unknowns. Few examples are discussed to exhibit the viability and effectiveness of the approach. The present study indicates the accuracy, efficiency, and powerfulness of the Chebyshev collocation method in solving the VOFD Lane–Emden equation. Error bound and convergence analysis of the method is also discussed. It is worth noticing that using lesser collocation nodes in computation is another advantage of the technique, which eventually reduces the computational cost.

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

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.

A neuro-swarming intelligent heuristic for second-order nonlinear Lane–Emden multi-pantograph delay differential system

The current study is related to present a novel neuro-swarming intelligent heuristic for nonlinear second-order Lane–Emden multi-pantograph delay differential (NSO-LE-MPDD) model by applying the approximation proficiency of artificial neural networks (ANNs) and local/global search capabilities of particle swarm optimization (PSO) together with efficient/quick interior-point (IP) approach, i.e., ANN-PSOIP scheme. In the designed ANN-PSOIP scheme, a merit function is proposed by using the mean square error sense along with continuous mapping of ANNs for the NSO-LE-MPDD model. The training of these nets is capable of using the integrated competence of PSO and IP scheme. The inspiration of the ANN-PSOIP approach instigates to present a reliable, steadfast, and consistent arrangement relates the ANNs strength for the soft computing optimization to handle with such inspiring classifications. Furthermore, the statistical soundings using the different operators certify the convergence, accurateness, and precision of the ANN-PSOIP scheme.

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.

An efficient approximation technique applied to a non-linear Lane–Emden pantograph delay differential model

The primary focus of our work is to propose a computationally effective approximation algorithm to find the numerical solution of the so-called a new design of second-order Lane–Emden pantograph delayed problem with singularity and non-linearity. Our approach based upon the novel Bessel matrix representation together with the collocation points which transforms the newly designed model problem into a non-linear fundamental matrix equation. To testify the validity and applicability of the proposed method, three test examples with non-linearity are given. The computational results are accurate as compared with the exact solutions as well as with those of numerical values reported in the literature.

Solving a novel designed second order nonlinear Lane–Emden delay differential model using the heuristic techniques

The aim of the present study is to present a new model based on the nonlinear singular second order delay differential equation of Lane–Emden type and numerically solved by using the heuristic technique. Four different examples are presented based on the designed model and numerically solved by using artificial neural networks optimized by the global search, local search methods and their hybrid combinations, respectively, named as genetic algorithm (GA), sequential quadratic programming (SQP) and GA-SQP. The numerical results of the designed model are compared for the proposed heuristic technique with the exact/explicit results that demonstrate the performance and correctness. Moreover, statistical investigations/assessments are presented for the accuracy and performance of the designed model implemented with heuristic methodology.

Numerical investigations of a new singular second-order nonlinear coupled functional Lane–Emden model

AbstractThe present study aims to design a second-order nonlinear Lane–Emden coupled functional differential model and numerically investigate by using the famous spectral collocation method. For validation of the newly designed model, three dissimilar variants have been considered and formulated numerically by applying a famous spectral collocation method. Moreover, a comparison of the obtained results with the exact/true results endorses the effectiveness and competency of the newly designed model, as well as the present technique.

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.

Boundary Values of Degenerate Configuration Core of White Dwarfs using Ramanujan’s Method

In this paper we have found out boundary values of degenerate configuration core of White Dwarf using Ramanujan’s method. By converting non-linear Lane Emden-type equation for degenerate core into a series using an iterative method. Considering the first five terms of the series numerical values has been calculated. The applied method is efficient and simplifies the calculation. The obtained result has been compared with exact values. Ramanujan’s method and the new iterative method are used for the first time to solve the Lane Emden equation for white dwarf core.

Sine–cosine wavelets operational matrix method for fractional nonlinear differential equation

In this paper, we present a solution method for fractional nonlinear ordinary differential equations. We propose a method by utilizing the sine–cosine wavelets (SCWs) in conjunction with quasilinearization technique. The fractional nonlinear differential equations are transformed into a system of discrete fractional differential equations by quasilinearization technique. The operational matrices of fractional order integration for SCW are derived and utilized to transform the obtained discrete system into systems of algebraic equations and the solutions of algebraic systems lead to the solution of fractional nonlinear differential equations. Convergence analysis and procedure of implementation for the proposed method are also considered. To illustrate the reliability and accuracy of the method, we tested the method on fractional nonlinear Lane–Emden type equation and temperature distribution equation.

A Modified Homotopy Perturbation Method for Nonlinear Singular Lane–Emden Equations Arising in Various Physical Models

A modified homotopy perturbation method for solving a class of nonlinear Lane–Emden equations with boundary conditions arising in various physical models is proposed. The proposed algorithm is based on the homotopy perturbation method and integral form of the Lane–Emden equation. The integral form of the problem overcomes the singular behavior at the origin. The accuracy and applicability of our algorithm is examined by solving two singular models: (i) the second kind Lane–Emden equation used to model a thermal explosion in an infinite cylinder or a sphere and (ii) the nonlinear singular problem with Neumann boundary conditions.

A new algorithm for solving the nonlinear Lane–Emden equations arising in astrophysics

Hybrid orthonormal Bernstein and block-pulse function wavelet method for many phenomena in mathematical physics and astrophysics is considered. We use a fundamental administrator and convert Lane–Emden conditions into necessary conditions. The upsides of utilizing the proposed strategy are introduced. At that point, an effective error estimation for the proposed technique is additionally presented lastly a few examinations and their numerical arrangements are given; and looking at between the numerical outcomes acquired from alternate strategies, we demonstrate the high exactness and productivity of the proposed strategy. Our method is characterized then the singular equations are transformed to Volterra integro-differential equations. We modify this equations to an algebraic system of equations. The solution to this application is achieved by solving this system and the constructed solutions are on approximation form. The acquired results guarantee the method provides a truncate solution to the Lane–Emden equations.

A Third-degree B-spline Collocation Scheme for Solving a Class of the Nonlinear Lane–-Emden Type Equations

A Third-degree B-spline Collocation Scheme for Solving a Class of the Nonlinear Lane–-Emden Type Equations

Theoretical study on continuous polynomial wavelet bases through wavelet series collocation method for nonlinear Lane–Emden type equations

In this article, a new method is generated to solve nonlinear Lane–Emden type equations using Legendre, Hermite and Laguerre wavelets. We are interested to note that these wavelets will give same solutions with good accuracy. Theorems on convergence analysis are stated and proved on the spaces, which are created by Legendre, Hermite and Laguerre wavelets bases and justified these spaces are equivalent to polynomial linear space generated by general polynomial basis. The main idea for obtaining numerical solutions depends on converting the differential equation with initial and boundary conditions into a system of linear or nonlinear algebraic equations with unknown coefficients. A very high level of accuracy reflects the reliability of this scheme for such problems.