Collocation method for the numerical solutions of Lane–Emden type equations using cubic Hermite spline functions

Three numerical techniques based on cubic Hermite spline functions are presented for the solution of Lane–Emden equation. Some properties of Hermite splines are presented and are utilized to reduce the solution of Lane–Emden equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of these techniques. Copyright © 2013 John Wiley & Sons, Ltd.     

A solution to the Lane–Emden equation in the theory of stellar structure utilizing the Tau method

In this paper, we propose a Tau method for solving the singular Lane–Emden equation—a nonlinear ordinary differential equation on a semi‐infinite interval. We applied collocation, Galerkin, and Tau methods for solving this problem, and according to the results, the solution of Tau method is the most accurate. The operational derivative and product matrices of the modified generalized Laguerre functions are presented. These matrices, in conjunction with the Tau method, are then utilized to reduce the solution of the Lane–Emden equation to that of a system of algebraic equations. We also present a comparison of this work with some well‐known results and show that the present solution is highly accurate. Copyright © 2012 John Wiley & Sons, Ltd.     

Solution of Lane–Emden type equations using rational Bernoulli functions

In this paper, a numerical method for solving Lane‐Emden type equations, which are nonlinear ordinary differential equations on the semi‐infinite domain, is presented. The method is based upon the modified rational Bernoulli functions; these functions are first introduced. Operational matrices of derivative and product of modified rational Bernoulli functions are then given and are utilized to reduce the solution of the Lane‐Emden type equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. Copyright © 2015 John Wiley & Sons, Ltd.     

An optimal sixth‐order quartic B‐spline collocation method for solving Bratu‐type and Lane‐Emden–type problems

This paper is concerned with the numerical solutions of Bratu‐type and Lane‐Emden–type boundary value problems, which describe various physical phenomena in applied science and technology. We present an optimal collocation method based on quartic B‐spine basis functions to solve such problems. This method is constructed by perturbing the original problem and on a uniform mesh. The method has been tested by four nonlinear examples. In order to show the advantage of the new method, numerical results are compared with those obtained by some of the existing methods, such as normal quartic B‐spline collocation method and the finite difference method (FDM). It has been observed that the order of convergence of the proposed method is six, which is two orders of magnitude larger than the normal quartic B‐spline collocation method. Moreover, our method gives highly accurate results than the FDM.     

An operational matrix method for solving Lane–Emden equations arising in astrophysics

This paper deals with the numerical solution of Lane–Emden equations in arising in astrophysics by using truncated shifted Chebyshev series together with the operational matrix. The main characteristic behind the approach using this technique is that it reduces such problems to those of solving a system of algebraic equations those greatly simplifying the problem. Some examples are included to demonstrate the validity and applicability of the this technique. Copyright © 2013 John Wiley & Sons, Ltd.     

An extension of the Tau numerical algorithm for the solution of linear and nonlinear Lane–Emden equations

The purpose of this paper is to present a numerical algorithm for solving the Lane–Emden equations as singular initial value problems. The proposed algorithm is based on an operational Tau method (OTM). The main idea behind the OTM is to convert the desired problem to some operational matrices. Firstly, we use a special integral operator and convert the Lane–Emden equations to integral equations. Then, we use OTM to linearize the integral equations to some operational matrices and convert the problem to an algebraic system. The concepts, properties, and advantages of OTM and its application for solving Lane–Emden equations are presented. Some orthogonal polynomials are also used to reduce the volume of computations. Finally, several experiments of Lane–Emden equations including linear and nonlinear terms are given to illustrate the validity and efficiency of the proposed algorithm. Copyright © 2012 John Wiley & Sons, Ltd.     

WEB‐spline based mesh‐free finite element analysis for the heat equation and the time‐dependent Navier–Stokes equation: A survey

In this article, we give some numerical techniques and error estimates using web‐spline based mesh‐free finite element method for the heat equation and the time‐dependent Navier–Stokes equations on bounded domains. The web‐spline method uses weighted extended B‐splines on a regular grid as basis functions and does not require any grid generation. We demonstrate the method by providing numerical results for the Poisson's and stationary Stokes equation. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Numerical solution of Fokker‐Planck equation using the cubic B‐spline scaling functions

In this article a numerical technique is presented for the solution of Fokker‐Planck equation. This method uses the cubic B‐spline scaling functions. The method consists of expanding the required approximate solution as the elements of cubic B‐spline scaling function. Using the operational matrix of derivative, the problem will be reduced to a set of algebraic equations. Some numerical examples are included to demonstrate the validity and applicability of the technique. The method is easy to implement and produces very accurate results. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

B‐spline collocation algorithm for numerical solution of the generalized Burger's‐Huxley equation

The cubic B‐spline collocation scheme is implemented to find numerical solution of the generalized Burger's–Huxley equation. The scheme is based on the finite‐difference formulation for time integration and cubic B‐spline functions for space integration. Convergence of the scheme is discussed through standard convergence analysis. The proposed scheme is of second‐order convergent. The accuracy of the proposed method is demonstrated by four test problems. The numerical results are found to be in good agreement with the exact solutions. Results are compared with other results given in literature. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A numerical approach for solving a class of the nonlinear Lane–Emden type equations arising in astrophysics

In this paper, a collocation method based on the Bessel polynomials is presented for the approximate solution of a class of the nonlinear Lane–Emden type equations, which have many applications in mathematical physics. The exact solution can be obtained if the exact solution is polynomial. In other cases, such as an increasing number of nodes, a good approximation can be obtained with applicable errors. In addition, the method is presented with error and stability analysis. The numerical results show the effectiveness of the method for this type of equations. Comparing the methodology with some known techniques shows that the present approach is relatively easy and highly accurate. Copyright © 2011 John Wiley & Sons, Ltd.     

The use of Chebyshev cardinal functions for solution of the second‐order one‐dimensional telegraph equation

A numerical technique is presented for the solution of the second order one‐dimensional linear hyperbolic equation. This method uses the Chebyshev cardinal functions. The method consists of expanding the required approximate solution as the elements of Chebyshev cardinal functions. Using the operational matrix of derivative, the problem is reduced to a set of algebraic equations. Some numerical examples are included to demonstrate the validity and applicability of the technique. The method is easy to implement and produces very accurate results. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

The use of cubic B‐spline scaling functions for solving the one‐dimensional hyperbolic equation with a nonlocal conservation condition

Problems for parabolic partial differential equations with nonlocal boundary conditions have been studied in many articles, but boundary value problems for hyperbolic partial differential equations have so far remained nearly uninvestigated. In this article a numerical technique is presented for the solution of a nonclassical problem for the one‐dimensional wave equation. This method uses the cubic B‐spline scaling functions. Some numerical results are reported to support our study. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Effective computation of exact and analytic approximate solutions to singular nonlinear equations of Lane–Emden–Fowler type

The particular motivation of this work is to develop a computational method to calculate exact and analytic approximate solutions to singular strongly nonlinear initial or boundary value problems of Lane–Emden–Fowler type which model many phenomena in mathematical physics and astrophysics. A powerful algorithm is proposed based on the series representation of the solution via suitable base functions. The utilization of such functions converts the solution of a given nonlinear differential equation to the solution of algebraic equations. Error analysis and convergence of the method is presented. Comparisons with the other methods reveal validity, applicability and great potential of the method. Several physical problems are treated to illustrative the good performance and high accuracy of the technique.     

Numerical solution of the Rosenau–KdV–RLW equation by operator splitting techniques based on B‐spline collocation method

In the present study, the operator splitting techniques based on the quintic B‐spline collocation finite element method are presented for calculating the numerical solutions of the Rosenau–KdV–RLW equation. Two test problems having exact solutions have been considered. To demonstrate the efficiency and accuracy of the present methods, the error norms L₂ and L_∞ with the discrete mass Q and energy E conservative properties have been calculated. The results obtained by the method have been compared with the exact solution of each problem and other numerical results in the literature, and also found to be in good agreement with each other. A Fourier stability analysis of each presented method is also investigated.     

Taylor‐Galerkin B‐spline finite element method for the one‐dimensional advection‐diffusion equation

The advection‐diffusion equation has a long history as a benchmark for numerical methods. Taylor‐Galerkin methods are used together with the type of splines known as B‐splines to construct the approximation functions over the finite elements for the solution of time‐dependent advection‐diffusion problems. If advection dominates over diffusion, the numerical solution is difficult especially if boundary layers are to be resolved. Known test problems have been studied to demonstrate the accuracy of the method. Numerical results show the behavior of the method with emphasis on treatment of boundary conditions. Taylor‐Galerkin methods have been constructed by using both linear and quadratic B‐spline shape functions. Results shown by the method are found to be in good agreement with the exact solution. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Analysis and computational method based on quadratic B‐spline FEM for the Rosenau–Burgers equation

L_∞‐error estimates for B‐spline Galerkin finite element solution of the Rosenau–Burgers equation are considered. The semidiscrete B‐spline Galerkin scheme is studied using appropriate projections. For fully discrete B‐spline Galerkin scheme, we consider the Crank–Nicolson method and analyze the corresponding error estimates in time. Numerical experiments are given to demonstrate validity and order of accuracy of the proposed method. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 877–895, 2016     

Quartic B‐spline collocation algorithms for numerical solution of the RLW equation

The collocation method based on quartic B‐spline interpolation is studied for numerical solution of the regularized long wave (RLW) equation. The time‐split RLW equation is also solved with the quartic B‐spline collocation method. Numerical accuracy is tested by obtaining the single solitary wave solution. Then, interaction, undulation and evolution of solitary waves are studied. Solutions are compared with available results. Conservation quantities are computed for all test experiments. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

Numerical solution of volterra functional integral equation by using cubic B‐spline scaling functions

In this article, we consider a class of nonlinear functional integral equations which has rather general form and contains a lot of particular cases such as functional equations and nonlinear integral equations of Volterra type. We use a combination of a fixed point method and cubic semiorthogonal B‐spline scaling functions to solve the integral equation numerically. We provide an error analysis for the method which shows that the approximate solution converges to the exact solution. Some numerical results for several test problems are given to confirm the accuracy and the ease of implementation of the method. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 699–722, 2014     

Numerical solution of a coupled Korteweg–de Vries equations by collocation method

A numerical method for solving the coupled Korteweg‐de Vries (CKdV) equation based on the collocation method with quintic B‐spline finite elements is set up to simulate the solution of CKdV equation. Invariants and error norms are studied wherever possible to determine the conservation properties of the algorithm. Simulation of single soliton, interaction of two solitons, and birth of solitons are presented. A linear stability analysis shows the scheme to be unconditionally stable. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009