Abstract
The purpose of this article is to employ the subdivision collocation method to resolve Bratu’s boundary value problem by using approximating subdivision scheme. The main purpose of this researcher is to explore the application of subdivision schemes in the field of physical sciences. Our approach converts the problem into a set of algebraic equations. Numerical approximations of the solution of the problem and absolute errors are compared with existing methods. The comparison shows that the proposed method gives a more accurate solution than the existing methods.
1. Introduction
The general expression of Liouville–Bratu–Gelfand equation [1, 2]:where parameter and is a bounded domain. We consider the Bratu’s boundary value problem in one-dimensional planar coordinates [2–4] of the formwith conditions at the ends of the domain
The detailed information of problem (2) is given in [4, 5]. The exact solution of (2) is
The exponential term guarantees nonlinearity and the bifurcation phenomenon that follows up. In particular, one can verify the following for different values of , i.e., problem (2) has no solution for , unique solution for , and two bifurcated solutions have been obtained for , where is the critical value given as . It is the solution of .
In science and engineering, Bratu’s problem is often used to characterize complex physical and chemical models. For example, Bratu’s problem is used in a wide range of applications, including the thermal combustion theory’s fuel ignition model, the model of the thermal reaction mechanism, the Chandrasekhar model of the universe’s expansion, chemical reaction theory, radiative heat transfer, and nanotechnology.
Many researchers have developed analytical and numerical methods to solve the Bratu’s problem, including the B-spline method [6], Adomian decomposition method [7, 8], Chebyshev polynomial approximation method [2], homotopy analysis method [9], homotopy perturbation method [10, 11], differential transform method [12], Laplace transformed decomposition method [13], method of weighted residuals [14], and variational iteration method [15, 16]. Moreover, the solutions of the problem have been reported by Jalilian [17] using the nonpolynomial spline method, by Boyd [18] with the one-point pseudospectral collocation method, and by Abbasbandy et al. [19] with the Lie-group shooting method.
Our goal is to make use of subdivision schemes for solving Bratu’s problem. Subdivision schemes-based algorithms are not frequently used to find numerical solutions of boundary value problems. The approximate solutions of boundary value problems have been found by subdivision-based algorithms. Initially, these algorithms were constructed by Qu and Agarwal [20, 21]. Their constructed algorithms were based on an interpolatory subdivision algorithm and formulated only for the second-order two-point boundary value problems. After that, Ejaz et al. [22, 23] constructed subdivision schemes-based algorithm for solutions of boundary value problems of third and fourth order. We present a subdivision collocation algorithm for solving Bratu’s problem in this paper.
We organize our paper in the following way. In Section 2, we present some important properties of 6-point binary approximating subdivision scheme. In Section 3, subdivision collocation algorithm is formulated for the solution of (2). The convergence and error estimation of the proposed algorithm are also discussed in this section. Numerical results based on the proposed algorithm, comparison with other existing methods, and conclusion based on the obtained results are given in Section 4.
2. Subdivision Scheme and Derivatives of Its Two-Scale Relation
In this section, we define 6-point binary approximating subdivision scheme (6PBASS) [24] aswith , , , , , , where and are tension parameters. The 6PBASS scheme possesses some of the following properties:(i)The scheme (5) is -continuous for , .(ii)It has support width .(iii)Its approximation order is fourth.(iv)Its fundamental solution is and it satisfies the two-scale relation where is the mask of the scheme (5). Since 6PBASS is -continuous by [24], so its 2-scale relations are also -continuous.(v)For the computation of the first- and second-order derivatives of (7), we adopt similar approach of [22, 23].
The first two derivatives of (7) are given in the following equations:
3. Subdivision Collection Algorithm for Bratu’s Problem
In this section, we have constructed a subdivision collocation algorithm for the solution of (2), which is based upon the fundamental solution of the subdivision scheme and its derivatives. Convergence and error estimations results are also presented in this section.
3.1. Formulation of Subdivision Collection Algorithm for Bratu’s Problem
The detail of the proposed algorithm is given as follows:
Letbe an approximate solution of (2) and must be greater than or equal to four, is the step size and is defined as , , where to , and are the unknowns to be determined. From (10), we get
By using (10) and (11) in (2), we getwhere , and the conditions given at the ends of the domain (3) become
The matrix representation of equation (12) iswhere
Since system (14) is underdetermined because it has fewer equations than unknowns, so it requires eight more equations to get a unique solution. Two conditions are given in (13) at the ends of the domain of (2) and the detail of the remaining six conditions is given in the next section.
3.2. Forced Conditions
As we require six more conditions to get a unique solution of (14), so we will construct three conditions at the left and three conditions at the right end of the domain. Since 6PBASS reproduces third degree polynomial with order of approximation four, so the order of new conditions is four and these conditions are known as forced conditions. Let and represent the left end points and right end points. These left and right end points can be computed by using polynomial of degree three which interpolates the data , for , i.e., left end conditions are obtained fromwhere
Since by (10), for and substituting by in (19), we have
So, the following conditions can be used at the left end and
Similarly, at the right end, we have the following conditions:
Finally, we get system of nonlinear equationsorwherewhere is defined in (15) and and are constrained as follows: the matrix is obtained from all the conditions defined at the left end of the domain, i.e., first three rows obtained from (21) and fourth row of obtained from (13) at . Similarly, the matrix is obtained from all the conditions defined at the right end of the domain, i.e., first row comes from (13) at and remaining rows come from (22). Hence,
The column vector is defined in (16) and is defined aswhere is given in (17).
3.3. Iterative Algorithm
To find the numerical solutions of nonlinear system of equation (24), we define an iterative algorithm. The iterative algorithm includes the following steps:(i)Formulation of initial solution: the initial approximate solution is selected to find the following system: where where . The column vector is the linear approximation of the column vector (27).(ii)Iterative scheme: the following iterative scheme is used to find the approximate solution ,(iii)Terminating criteria: the following condition is used to stop the iteration at level, for any ; let ,
3.4. Convergence and Error Estimation
In this section, we present results of convergence and error estimation of the proposed iterative algorithm. The convergence of the iterative algorithm is guaranteed by the following proposition.
Proposition 1. The approximate solution founded by (28) and (30) linearly converges to the approximate solution of (24) with the supposition that step size and the Lipschitz constants are small, i.e.,
The proof is similar to [23].
The main result of error estimation is given by the following proposition.
Theorem 1. Let exact solution and be obtained by solving (24) with the fourth order boundary treatment at the end points. Then, we have
4. Numerical Examples and Comparison
The numerical technique discussed previously is illustrated in this section by applying subdivision collection algorithm to the planar one-dimensional Bratu’s problem (2) for three distinct values of , which guarantee the existence of two locally unique solutions. We have created comparison tables using and 3.51 to show the consistency of our approach in comparison to the exact solution as well as the solutions of other methods. All calculations have been performed using MATLAB .(i)The fact regarding the solution of Bratu’s problem for is obtained after third iteration, as shown in Table 1. Comparison between the numerical results and absolute errors obtained by our subdivision collection algorithm and decomposition method [25] are presented in Tables 2 and 3, respectively. From the tabulated results, we observed that the numerical results obtained by our subdivision collection algorithm are better than the decomposition method [25].(ii)The fact regarding the solution of Bratu’s problem for is obtained after fifth iteration, as shown in Table 4. Comparison between the numerical results and absolute errors obtained by our subdivision collection algorithm, decomposition [25], and Laplace method [5] are presented in Tables 5 and 6, respectively. From the tabulated results, we observed that the numerical results obtained by our subdivision collection algorithm are better than [5, 25].(iii)The fact regarding the solution of Bratu’s problem for is obtained after forty-two iterations, as shown in Table 7. Comparisons between the absolute errors obtained by our subdivision collection algorithm and B-spline [6] method are presented in Table 8. From the tabulated results, we observed that the numerical results obtained by our subdivision collection algorithm give better approximation than [6].
5. Concluding Remarks
In this paper, we have established a subdivision collocation algorithm for the solution of one-dimensional nonlinear Bratu’s problem. The numerical results obtained by subdivision collection algorithm showed that the algorithm is suitable for the approximate solution of (2). We have concluded that the numerical results converge to the exact solution for the small step size. We have also presented a comparison of absolute errors of the solution obtained from subdivision collection algorithm with decomposition method [25], Laplace method, [5] and B-spline method [6] for different values of . We conclude that our algorithm gives smaller absolute errors as compared with the other existing methods [5, 6, 25].
Data Availability
The data used to support the findings of the study are available within this paper.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.