Collocation Method Based on Bernoulli Polynomial and Shifted Chebychev for Solving the Bratu Equation

In this work, Bernoulli-collocation method is proposed for solving nonlinear Bratu's type equations. The operational matrix of derivative of Bernoulli is introduced. The matrix together with the collocation method are then utilized to reduce the problem into a system of nonlinear algebraic equations. Also, a reliable approach for solving this nonlinear system is discussed. Numerical results and comparisons with other existing methods provided in the literature are made. Citation: EL-Gamel M, Adel W, EL-Azab MS (2018) Collocation Method Based on Bernoulli Polynomial and Shifted Chebychev for Solving the Bratu Equation. J Appl Computat Math 7: 407. doi: 10.4172/2168-9679.1000407


Introduction
Nonlinear boundary value problems (BVPs) of ordinary type plays an important role in all branches of science and engineering especially the two point BVPs. These types of equations appears in a wide variety of problems including but not limited to chemical reactions, hear transfer and solution of optimal control problems. Therefore, the need for fast and efficient methods for solving this type of equations is a must.
In this work we will develop a collocation approach based on Bernoulli polynomials for solving the famous Bratu cos 4 where θ satisfies This type of equation has many applications including the modelling of a combustion in a numerical slab, the ignition of fuel of the thermal theory, the thermal reaction process modelling, the expansion of the universe model and open questions regarding this theory, chemical reaction theory and nanotechnology [1].
Bernoulli polynomials have gained increasing importance in numerical analysis since they are straightforward and need less computational errors. Many researchers have been working on proving the efficiency of this method [38][39][40][41][42][43].
The organization of the paper is as follows. We recall the basic concepts of Bernoulli polynomials and their relevant properties needed hereafter. Bernoulli method is presented for solving the general Bratu's type equations. Some numerical examples are presented along with a comparison with other techniques. Finally, the closing stage which provides the conclusions of the study [44][45][46][47].

Fundamental Relations
Bernoulli polynomials play an important role in different areas of mathematics, including number theory and the theory of finite differences. They are also can be found in the integral representation of the differentiable periodic functions, since they are employed for approximating such functions in term of polynomials. The classical Bernoulli polynomials B n (x) is usually defined by means of exponential generating functions [40].
these polynomials have many interesting properties from which the following Figure 1 ( ) ( 1) x B x nx In the next we will introduce Bernoulli matrix of differentiation that will be needed later.

Bernoulli operational matrix of differentiation
Since D is a lower triangular matrix with nonzero diagonal elements and det (D)=1, so D is an invertible matrix. Thus, the Bernoulli vector can be given directly from note that [] t , denotes transpose of the matrix [] and B t (x) and Ω(x) be the (N+1)×1 and D is the (N+1)×(N+1) operational matrix whose elements are Now, the matrix forms of the solution functions as According to the eqn. (5) the following formula is concluded evidently. Also, the relation between Ω(x) and it's ith derivative and the following formula holds as where

Application of the Proposed Method
First, we need to treat the nonlinear term in eqn. (1) by expanding it using Taylor series expansion in the form By substituting the expanded term from eqn. (10) into eqn. (1), the equation becomes Second, we will use the shifted chebychev defined on the interval [0,1] in the form The matrix form of the boundary conditions represented in eqn. (1) will be in the form By replacing two rows of the augmented matrix [Θ: F] with the boundary conditions defined from eqn. (16), we have Now we have a nonlinear system of N+1 equation in N+1 unknown coefficient c. We can obtain these coefficients by solving the above nonlinear system using the following algorithm.

Numerical Examples
To illustrate the ability, reliability and the performance of the proposed method for Bratu's problem, some examples are provided. The results reveal that the method is very effective and simple. All computations were carried out using Matlab 2014a on a personal computer. The absolute error can be calculated according to the following |, k = 0; 1; 2; : : : :

Example 1
First, we consider the initial value problem in the form [31,9,25,21,15,46] Table 2 a comparison between the reported results in [31,9,25,21,15,46] along with our method at N=18. This table indicates that our method provide better results than the other methods. Figure 2 demonstrates the Bernoulli approximate solution versus the exact solution for x 2 [0; 1].

Theorem
The approximation of the function u v (x k ), k=0,1,…., N can be represented according to the following relation By substituting the above theorem into eqn. (12), we reach the following theorem.

Theorem
If the assumed approximate solution of the problem eqn. (12) is eqn. (8), then the discrete Bernoulli system is Proof: If we replace each term of (12) with its corresponding approximation given by eqns.(2), (9) and (13) and substituting x=x k collocation points.
The matrix form for this system is Θc=F (15) Where ( ) And 0 0 0 subject to the boundary conditions with the exact solution ( ) The form of the above equation is very familiar and has a tremendous work for solving it as mentioned in the literature. We applied our method with various ( Figure 2) values for=0:1; 0:5; 1; 2; 3 and 3:51. The computed maximum absolute errors at different values of N and are tabulated in Table 3. A comparison with the other methods reported in the literature are presented in Table 4 shows that our method is computationally effective even for=3:51 which is near the critical value. The graph of the approximate solutions for different values of has been plotted in Figure 3.

Example 3
Now, we turn our attention to the BVP of Bratu's equation in the form [21,31] (1±sin( x)). This solution with the negative sign blows up at x=0:5 so we will use the solution of positive sign which is convergent and bounded in the form Maximum absolute error is tabulated in Table 5 along with the CPU time and a comparison is made in Table 6 between our method along with the other methods in [21,31]. From Table 6, we noticed that our method is more accurate than the other existing methods. Figure 4 demonstrates the Bernoulli approximate solution and the exact which appears to be in good agreement with each other.

Conclusion
In this paper, we showed that Bernoulli-collocation method can be utilized to and an approximate solution of the nonlinear Bratu's type equations. The method reduces the problem into a system of nonlinear algebraic equations and this system is solved using a novel technique. Also, the efficiency of the method with respect to the other method was shown. In comparison to other methods, we illustrated that Bernoullicollocation method has very high accuracy.