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In this manuscript, we present an innovative approach to address nonlinear Bratu's problem by applying the Discontinuous Galerkin method. Our method accurately computes two-branched numerical solutions of the problem. Depending on the initial guess function, the resulting numerical solution converges either to the upper or to the lower solution of Bratu's problem. We show that our method achieves an optimal convergence rate of order p+1 when p-degree polynomials are used. To confirm the accuracy and efficiency of our proposed method, we present various numerical results. These findings demonstrate the high precision achieved by Discontinuous Galerkin method, enabling the computation of both branches of the solution.

In this manuscript, we present an innovative approach to address nonlinear Bratu's problem by applying the Discontinuous Galerkin method. Our method accurately computes two-branched numerical solutions of the problem. Depending on the initial guess function, the resulting numerical solution converges either to the upper or to the lower solution of Bratu's problem. We show that our method achieves an optimal convergence rate of order p + 1 when p -degree polynomials are used. To confirm the accuracy and efficiency of our proposed method, we present various numerical results. These findings demonstrate the high precision achieved by Discontinuous Galerkin method, enabling the computation of both branches of the solution. [ABSTRACT FROM AUTHOR]

Abstract We compute the numerical solution of the Bratu’s boundary value problem (BVP) on a Banach space setting. To do this, we embed a Green’s function into a new two-step iteration scheme. After this, under some assumptions, we show that this new iterative scheme converges to a sought solution of the one-dimensional non-linear Bratu’s BVP. Furthermore, we show that the suggested new iterative scheme is essentially weak w 2 $w^{2}$ -stable in this setting. We perform some numerical computations and compare our findings with some other iterative schemes of the literature. Numerical results show that our new approach is numerically highly accurate and stable with respect to different set of parameters.

We compute the numerical solution of the Bratu's boundary value problem (BVP) on a Banach space setting. To do this, we embed a Green's function into a new two-step iteration scheme. After this, under some assumptions, we show that this new iterative scheme converges to a sought solution of the one-dimensional non-linear Bratu's BVP. Furthermore, we show that the suggested new iterative scheme is essentially weak w 2 -stable in this setting. We perform some numerical computations and compare our findings with some other iterative schemes of the literature. Numerical results show that our new approach is numerically highly accurate and stable with respect to different set of parameters. [ABSTRACT FROM AUTHOR]

A brief survey of the properties and different treatments of the one-dimensional Bratu's problems is presented, and a numerical method is stated for solving this problem via Genocchi polynomial Gn(x). In this paper, the obtained error was compared between the proposed technique, Chebyshev wavelets, and Legendre wavelets. Numerical examples are exhibited to verify the efficiency and accuracy of the stated technique. [ABSTRACT FROM AUTHOR]

The present work deals with the construction of a high‐order efficient technique for solving one‐dimensional nonlinear Bratu‐type and Lane‐Emden‐type problems describing various physical, chemical and biological phenomena in applied science and engineering. Convergence analysis of the method is studied. Numerical experiments are carried out to demonstrate the efficiency, accuracy and performance of proposed method. The method is shown to have sixth order convergence. The computed results are compared with those obtained by other numerical techniques to justify the advantage of present scheme. It is shown that the proposed method is fast convergent and provides accurate solutions for Bratu‐type and Lane‐Emden‐type problems with low computational cost. [ABSTRACT FROM AUTHOR]

In this paper, discrete cubic spline method based on central differences is developed to solve one-dimensional (1D) Bratu's and Bratu's type highly nonlinear boundary value problems (BVPs). Convergence analysis is briefly discussed. Four examples are given to justify the presented method and comparisons are made to confirm the advantage of the proposed technique. [ABSTRACT FROM AUTHOR]

The ultimate aim of the article is to apply an efficient semi-analytical method for the approximate solution of the fractional differential equations of Bratu–type. A Laplace Decomposition Method (LDM), which is a combination of Laplace transform and a Decomposition Method, is implemented for the nonlinear fractional Bratu problem that is complemented with initial and boundary conditions. The nonlinear term is decomposed and a recursive algorithm is composed for the determination of the proposed infinite series solution. A patching strategy, based on domain decomposition, is suggested to overcome a deficiency of the LDM. Some examples are selected to explicate the effectiveness and simplicity of the proposed technique. The results assure that this scheme is speedily convergent and quite accurate by which it approximates the solution using only few iterates of its iterative scheme. [ABSTRACT FROM AUTHOR]

In this paper, we propose a variational method in order to solve Bratu's problem for two dimensions in an adequate space of biquadratic spline functions. The solution is obtained by resolving a sequence of boundary value problems. We study some characterizations of the functions of such sequence and we express them as some linear combination of biquadratic spline bases functions. We finish by showing some numerical and graphical examples in order to prove the validity and the effectiveness of our method. [ABSTRACT FROM AUTHOR]

In this study, improved Sinc-Galerkin and Sinc-collocation methods are developed based on double exponential transformation to solve a one-dimensional Bratu-type equation. The properties of these methods are used to reduce the solution of the nonlinear problem to the solution of nonlinear algebraic equations. For simplicity in solving the nonlinear system, a matrix vector form of the nonlinear system is found. The upper bound of the error for the Sinc-Galerkin is determined. Also the numer- ical approximations are compared with the best results reported in the literature. The results confirm that both the Sinc-Galerkin and the Sinc- collocation methods have the same accuracy, but they are significantly more accurate than the other existing methods. [ABSTRACT FROM AUTHOR]

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.

Abstract In this paper, an efficient analytical iterative method is introduced for solving the initial value and boundary value Bratu problems. A quasilinearization method with an optimal parameter is derived and utilized to reduce the highly nonlinear Bratu equations to a sequence of linear problems. An optimal Picard iteration method is used to obtain approximate analytical solution of linearized versions of Bratu's equations. The numerical results presented the effectiveness and high performance of the proposed method and comparisons are made with many other existing numerical solution techniques. The comparisons reveal the efficiency and applicability of the present work. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we propose an iterative finite difference (IFD) scheme to simultaneously approximate both branches of a two-branched solution to the one-dimensional Bratu's problem. We first introduce a transformation to convert Bratu's problem into a simpler one. The transformed nonlinear ordinary differential equation is discretized using the Newton–Raphson–Kantorovich approximation in function space. The convergence of the sequence of approximations is proved to be quadratic. Then, we apply the classical finite difference method to approximate the sequence of approximations. The proposed new scheme has two main advantages. First, it produces accurate numerical solutions with low computational cost. Second, it is able to compute the two branches of the solution of Bratu's problem, even for small values of the transition parameter λ , where the numerical computation of the upper branch of the solution becomes challenging. Numerical examples are provided to show the efficiency and accuracy of the proposed scheme. Highlights • A new transformation of Bratu's problem conserving its bifurcated behavior. • Computation of the two different branches of the solution for Bratu's problem. • An iterative numerical method based on Newton–Raphson–Kantorovich approximation. • A linear iterative scheme yielding accurate solutions of Bratu's problem. • Convergence of the method is proved to be quadratic, and its accuracy is confirmed numerically. [ABSTRACT FROM AUTHOR]

Recently, Caglar et al. [B-spline method for solving Bratu's problem, Int. J. Comput. Math. 87(8) (2010), pp. 1885-1891] proposed a numerical technique based on cubic B-spline for solving a Bratu-type problem. This method provides a second-order convergent approximation to the solution of the problem. In this paper, we develop a high-order numerical method based on quartic B-spline collocation approach for the Bratu-type and Lane-Emden problems. The error analysis of the quartic B-spline interpolation is carried out. Some numerical examples are provided to demonstrate the efficiency and applicability of the method and to verify its rate of convergence. The numerical results are compared with exact solutions and a numerical method based on cubic B-spline approach. Comparison reveals that our method produces more accurate results than the method proposed by Caglar et al. [B-spline method for solving Bratu's problem, Int. J. Comput. Math. 87(8) (2010), pp. 1885-1891]. [ABSTRACT FROM AUTHOR]

The aim of this manuscript is to investigate an accurate discretization method to solve the one-, two-, and three-dimensional highly nonlinear Bratu-type problems. By discretization of the nonlinear equation via a fourth-order nonstandard compact finite difference formula, the considered problem is reduced to the solution of a highly nonlinear algebraic system. To solve the derived nonlinear system, a modified nonlinear solver is used. The new scheme is accurate, fast, straightforward and very effective to find the lower and upper branches of the Bratu’s problem. Numerical simulations and comparative results for the one-, two-, and three-dimensional cases verify that the new technique is easy to implement and more accurate than the other existing methods in the literature. [ABSTRACT FROM AUTHOR]

This paper presents a semi-analytical method for solving fractional differential equations with strong terms like (exp, sin, cos,. . .). An auxiliary parameter is introduced into the well-known Picard's method and so called controlled Picard's method. The proposed approach is based on a combination of controlled Picard's method with Simpson rule. This approach can cover a wider range of integer and fractional orders differential equations due to the extra auxiliary parameter which enhances the convergence and is suitable for higher order differential equations. The proposed approach can be effectively applied to Bratu's problem in fractional order domain to predict and calculate all branches of problem solutions simultaneously. Also, it is tested on other fractional differential equations like nonlinear fractional order Sine-Gordon equation. The results demonstrate reliability, simplicity and efficiency of the approach developed. [ABSTRACT FROM AUTHOR]

In this paper, a three-step iterative method with sixth-order local convergence for approximating the solution of a nonlinear system is presented. From Ostrowski’s scheme adding one step of Newton with ’frozen’ derivative and by using a divided difference operator we construct an iterative scheme of order six for solving nonlinear systems. The computational efficiency of the new method is compared with some known ones, obtaining good conclusions. Numerical comparisons are made with other existing methods, on standard nonlinear systems and the classical 1D-Bratu problem by transforming it in a nonlinear system by using finite differences. From this numerical examples, we confirm the theoretical results and show the performance of the presented scheme. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to propose an efficient numerical method for solving the differential equations of Bratu-type with fractional order in reproducing kernel Hilbert space. The exact solution is calculated in the form of a convergent series with easily computable components. Finally, some examples are given to illustrate the efficiency and applicability of the method. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, a new algorithm for the numerical solution of the Bratu-type and the Lane-Emden problems with boundary conditions is presented. The proposed algorithm is based on the variational iteration method (VIM), where all the boundary conditions are used before designing the recursive scheme for the approximate solutions of considered boundary value problems. Unlike the VIM, the proposed algorithm avoids solving a sequence of nonlinear algebraic or (transcendental) equations for the undetermined coefficients. Convergence and error analysis of the proposed method is also given. Illustrative examples, of two different models, are examined to demonstrate the accuracy, applicability, and generality of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper we propose a new iterative finite difference (IFD) scheme based on the Newton–Raphson–Kantorovich approximation method in function space to solve the classical one-dimensional Bratu’s problem. This new numerical method produces accurate solutions with low computational cost. The effectiveness and accuracy of the IFD method are confirmed through several numerical examples and compared to some existing numerical solvers. [ABSTRACT FROM AUTHOR]

We transform the Bratu's problem directly into an unconstrained optimization problem, and then solve it by a simple algorithm, which combining the particle swarm optimization algorithm with Runge-kutta method together. We compare the numerical performance of the proposed optimization algorithm with the existing methods, and the numerical results show that the proposed new method is quite promising. [ABSTRACT FROM AUTHOR]

In this paper, we use the reproducing kernel Hilbert space method for solving a boundary value problem for the second order Bratu's differential equation. Convergence analysis of presented method is discussed. The numerical approximations to the exact solution are computed and compared with other existing methods. Our presented method produces more accurate results in comparison with those obtained by Adomian decomposition, Laplace decomposition, B-spline, non-polynomial spline and Lie-group shooting methods. Our yardstick is absolute error. The comparison of the results with exact ones is made to confirm the validity and efficiency. [ABSTRACT FROM AUTHOR]

One of the newest analytical methods to solve nonlinear equations is Variational Iteration Method-II which is an accurate and a rapid convergence method in finding the approximate solution for nonlinear equations. In these papers, VIM-II is implemented to solve the initial and boundary value problems of Bratu-type which are widely applicable in fuel ignition of the combustion theory and heat transfer. Several examples are given to confirm the efficiency and the accuracy of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Abstract: A brief survey of the properties and different treatments of the one-dimensional (1D) and (2D) Bratu problems is presented. Different iterative treatments of the resulting nonlinear system of equations are discussed. The finite-difference treatment of the problem is considered. Nonstandard finite-difference methods with a simple sinusoidal starting function having an appropriate amplitude are recommended. Bounds on the amplitude for yielding both lower and upper solutions are given. [Copyright &y& Elsevier]

Abstract: A brief survey of the properties and different treatments of the one-dimensional (1D) planar Bratu problem is presented. The treatment using Green’s function is considered. Different iterative treatments of the resulting nonlinear system of equations are discussed. Starting the solution with simple sinusoidal function having appropriate amplitude is recommended. Then the resulting system is solved using a nonlinear conjugate-gradient (NL-CG) method. The importance of the proper choice of the starting function as well as the stopping criterion of the iterative solution are stressed. [Copyright &y& Elsevier]

Study of the performance of the Galerkin method using sinc basis functions for solving Bratu's problem is presented. Error analysis of the presented method is given. The method is applied to two test examples. By considering the maximum absolute errors in the solutions at the sinc grid points are tabulated in tables for different choices of step size. We conclude that the Sinc-Galerkin method converges to the exact solution rapidly, with order, $O(\exp{(-c \sqrt{n}}))$ accuracy, where c is independent of n. [ABSTRACT FROM AUTHOR]

In this paper, we propose a B-spline method for solving the one-dimensional Bratu's problem. The numerical approximations to the exact solution are computed and then compared with other existing methods. The effectiveness and accuracy of the B-spline method is verified for different values of the parameter, below its critical value, where two solutions occur. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present a framework to determine exact solutions of Bratu-type equations. The algorithm rests mainly on the Adomian decomposition method. The proposed scheme is illustrated by studying two boundary value problems and an initial value problem of Bratu-type. The first type gives a solution that blows up at the middle of the domain, whereas the other equations give bounded solutions. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]

The main purpose of this article is to present a special case of Hermite interpolation, namely the two-point Taylor formula, to approximate dual solutions of the strongly nonlinear one-dimensional Bratu problem. The proposed method is able to effectively predict and approximate both branches of the solution of the problem for different values of Bratu's parameter λ in an acceptable computational time. The other advantage of this method is its capability to accurately estimate the unique solution of Bratu's problem for the critical value of the parameter λ The convergence analysis of the presented scheme is discussed. Some numerical results are reported to validate the efficiency and precision of the proposed method. [ABSTRACT FROM AUTHOR]