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.     

A study on the systems of the Volterra integral forms of the Lane–Emden equations by the Adomian decomposition method

In this paper, we introduce systems of Volterra integral forms of the Lane–Emden equations. We use the systematic Adomian decomposition method to handle these systems of integral forms. The Volterra integral forms overcome the singular behavior at the origin x = 0. The Adomian decomposition method gives reliable algorithm for analytic approximate solutions of these systems. Our results are supported by investigating several numerical examples. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

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.     

Numerical solution of singular IVPs of Lane–Emden type using a modified Legendre-spectral method

In this paper, a numerical method for singular initial value problems of the Lane–Emden type in the second-order ordinary differential equations is proposed. The method changes solving the equation to solving a Volterra integral equation. We have applied the improved Legendre-spectral method to solve Lane–Emden type equations. The Legendre Gauss points are used as collocation nodes and Lagrange interpolation is employed in Volterra term. The results reveal that the method is effective, simple and accurate.     

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.     

Numerical Solution of the Two-Sided Space–Time Fractional Telegraph Equation Via Chebyshev Tau Approximation

The operational matrices of left Caputo fractional derivative, right Caputo fractional derivative, and Riemann–Liouville fractional integral, for shifted Chebyshev polynomials, are presented and derived. We propose an accurate and efficient spectral algorithm for the numerical solution of the two-sided space–time Caputo fractional-order telegraph equation with three types of non-homogeneous boundary conditions, namely, Dirichlet, Robin, and non-local conditions. The proposed algorithm is based on shifted Chebyshev tau technique combined with the derived shifted Chebyshev operational matrices. We focus primarily on implementing the novel algorithm both in temporal and spatial discretizations. This algorithm reduces the problem to a system of algebraic equations greatly simplifying the problem. This system can be solved by any standard iteration method. For confirming the efficiency and accuracy of the proposed scheme, we introduce some numerical examples with their approximate solutions and compare our results with those achieved using other methods.     

A fractional‐order Jacobi Tau method for a class of time‐fractional PDEs with variable coefficients

This paper presents a shifted fractional‐order Jacobi orthogonal function (SFJF) based on the definition of the classical Jacobi polynomial. A new fractional integral operational matrix of the SFJF is presented and derived. We propose the spectral Tau method, in conjunction with the operational matrices of the Riemann–Liouville fractional integral for SFJF and derivative for Jacobi polynomial, to solve a class of time‐fractional partial differential equations with variable coefficients. In this algorithm, the approximate solution is expanded by means of both SFJFs for temporal discretization and Jacobi polynomials for spatial discretization. The proposed tau scheme, both in temporal and spatial discretizations, successfully reduced such problem into a system of algebraic equations, which is far easier to be solved. Numerical results are provided to demonstrate the high accuracy and superiority of the proposed algorithm over existing ones. Copyright © 2015 John Wiley & Sons, Ltd.     

Four techniques based on the B‐spline expansion and the collocation approach for the numerical solution of the Lane–Emden equation

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

Transcendental Bernstein series for solving reaction–diffusion equations with nonlocal boundary conditions through the optimization technique

In this paper, we apply transcendental Bernstein series (TBS) for solving reaction–diffusion equations with nonlocal boundary conditions which is the novel approximation tool. To carry out the method, we firstly expand the solution of the system in the term of TBS through the operational matrix scheme. To determine the unknown free coefficients and control parameters appeared in TBS expansion, we define an optimization problem which combines the reaction–diffusion equation with its nonlocal boundary conditions. Then we use the Lagrange multipliers technique for converting the problem under study into a system of algebraic equations. High accuracy and simplicity in reducing the integral boundary conditions are some privileges of the proposed scheme. We emphasize that Bernstein polynomials is the particular case of transcendental Bernstein series. Theoretical discussion about convergence confirms the reliability of the proposed method. Some test problems are chosen to investigate the applicability and computational efficiency. The experimental results confirm that the obtained results are in good agreement with the exact solutions with high rate of convergence.     

High‐order numerical solution of second‐order one‐dimensional hyperbolic telegraph equation using a shifted Gegenbauer pseudospectral method

We present a high‐order shifted Gegenbauer pseudospectral method (SGPM) to solve numerically the second‐order one‐dimensional hyperbolic telegraph equation provided with some initial and Dirichlet boundary conditions. The framework of the numerical scheme involves the recast of the problem into its integral formulation followed by its discretization into a system of well‐conditioned linear algebraic equations. The integral operators are numerically approximated using some novel shifted Gegenbauer operational matrices of integration. We derive the error formula of the associated numerical quadratures. We also present a method to optimize the constructed operational matrix of integration by minimizing the associated quadrature error in some optimality sense. We study the error bounds and convergence of the optimal shifted Gegenbauer operational matrix of integration. Moreover, we construct the relation between the operational matrices of integration of the shifted Gegenbauer polynomials and standard Gegenbauer polynomials. We derive the global collocation matrix of the SGPM, and construct an efficient computational algorithm for the solution of the collocation equations. We present a study on the computational cost of the developed computational algorithm, and a rigorous convergence and error analysis of the introduced method. Four numerical test examples have been carried out to verify the effectiveness, the accuracy, and the exponential convergence of the method. The SGPM is a robust technique, which can be extended to solve a wide range of problems arising in numerous applications. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 307–349, 2016     

An elegant exact solutions for the Emden–Fowler equations of the first kind

We study the exact solutions of the Emden–Fowler equations and generalize the n = 1 and n = 5 Lane–Emden equations. We analyze the series approximations and show that explicit formulas can be written for new classes of equations. Copyright © 2015 John Wiley & Sons, Ltd.     

A nonlocal multi-point singular fractional integro-differential problem of Lane–Emden type

In this paper, using Riemann–Liouville integral and Caputo derivative, we study a nonlinear singular integro-differential equation of Lane–Emden type with nonlocal multi-point integral conditions. We prove the existence and uniqueness of solutions by application of Banach contraction principle. Also, we prove an existence result using Schaefer fixed point theorem. Then, we present some examples to show the applicability of the main results.     

Reconstruction of a time-dependent coefficient in nonlinear Klein–Gordon equation using Bernstein spectral method

In this work, we present a spectral method for recovering an unknown time-dependent lower-order coefficient and unknown wave displacement in a nonlinear Klein–Gordon equation with overdetermination at a boundary condition. We apply the initial and boundary conditions to construct the satisfier function and use this function in a transformation to convert the main problem to a nonclassical hyperbolic equation with homogeneous initial and boundary conditions. Then, we utilize the orthonormal Bernstein basis functions to approximate the solution of the reformulated problem and use a direct technique based on the operational matrices of integration and differentiation of these basis functions together with the collocation technique to reduce the problem to a system of nonlinear algebraic equations. Regarding the perturbed measurements, the method takes advantage of the mollification method in order to derive stable numerical derivatives. Numerical simulations for solving several test examples are presented to show the applicability of the proposed method for obtaining accurate and stable results.     

A new Jacobi operational matrix: An application for solving fractional differential equations

In this paper, we derived the shifted Jacobi operational matrix (JOM) of fractional derivatives which is applied together with spectral tau method for numerical solution of general linear multi-term fractional differential equations (FDEs). A new approach implementing shifted Jacobi operational matrix in combination with the shifted Jacobi collocation technique is introduced for the numerical solution of nonlinear multi-term FDEs. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem. The proposed methods are applied for solving linear and nonlinear multi-term FDEs subject to initial or boundary conditions, and the exact solutions are obtained for some tested problems. Special attention is given to the comparison of the numerical results obtained by the new algorithm with those found by other known methods.     

Newton's method based on bifurcation for solving multiple solutions of nonlinear elliptic equations

On the basis of bifurcation theory, we use Newton's method to compute and visualize the multiple solutions to a series of typical semilinear elliptic boundary value problems with a homogeneous Dirichlet boundary condition in . We present three algorithms on the basis of the bifurcation method to solving these multiple solutions. We will compute and visualize the profiles of such multiple solutions, thereby exhibiting the geometrical effects of the domains on the multiplicity. The domains include the square, disk, symmetric or nonsymmetric annuli and dumbbell. The nonlinear partial differential equations include the Lane–Emden equation, concave–convex nonlinearities, Henon equation, and generalized Lane–Emden system. Copyright © 2013 John Wiley & Sons, Ltd.     

Numerical solution of fractional differential,integral and integro-differential equations by using piecewise constant orthogonal functions

We use operational matrices of piecewise constant orthogonal functions on the interval [0, 1] to solve fractional differential, integral and integro-differential equations. We first use operational matrices and then convert the problem to a system of linear equations. Numerical examples show that the approximate solutions have a good degree of accuracy. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical solution of Volterra integral and integro-differential equations of convolution type by using operational matrices of piecewise constant orthogonal functions

In this paper, we use operational matrices of piecewise constant orthogonal functions on the interval [0,1)$[0,1)$ to solve Volterra integral and integro-differential equations of convolution type without solving any system. We first obtain Laplace transform of the problem and then we find numerical inversion of Laplace transform by operational matrices. Numerical examples show that the approximate solutions have a good degree of accuracy.     

Numerical technique based on generalized Laguerre and shifted Chebyshev polynomials

In this study, we present a numerical scheme for solving a class of fractional partial differential equations. First, we introduce psi -Laguerre polynomials like psi-shifted Chebyshev polynomials and employ these newly introduced polynomials for the solution of space-time fractional differential equations. In our approach, we project these polynomials to develop operational matrices of fractional integration. The use of these orthogonal polynomials converts the problem under consideration into a system of algebraic equations. The solution of this system provide us the desired results. The convergence of the proposed method is analyzed. Finally, some illustrative examples are included to observe the validity and applicability of the proposed method.     

On the numerical solution of linear and nonlinear volterra integral and integro-differential equations

Sinc bases are developed to approximate the solutions of linear and nonlinear Volterra integral and integro-differential equations. Properties of these sinc bases and some operational matrices are first presented. These properties are then used to reduce the integral and integro-differential equations to systems of linear or nonlinear algebraic equations. Numerical examples illustrate the pertinent features of the method and its applicability to a large variety of problems. The examples include convolution type, singular as well as singularly-perturbed problems.