Subdivision Collocation Method for One-Dimensional Bratu’s Problem

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.

(4) e exponential term guarantees nonlinearity and the bifurcation phenomenon that follows up. In particular, one can verify the following for different values of α, i.e., problem (2) has no solution for α > α c , unique solution for α � α c , and two bifurcated solutions have been obtained for 0 < α < α c , where α c is the critical value given as α c � 3.51380719. It is the solution of 1 � 0. 25 ��� 2α c sinh(0.25θ). 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.
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. e approximate solutions of boundary value problems have been found by subdivisionbased algorithms. Initially, these algorithms were constructed by Qu and Agarwal [20,21]. eir constructed algorithms were based on an interpolatory subdivision algorithm and formulated only for the second-order twopoint 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). e 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.

Subdivision Scheme and Derivatives of Its Two-Scale Relation
In this section, we define 6-point binary approximating subdivision scheme (6PBASS) [24] as , where α and β are tension parameters. e 6PBASS scheme possesses some of the following properties: (i) e scheme (5) is C 2 -continuous for α � (1/5), β � (13/1000). (ii) It has support width (5, 5). (iii) Its approximation order is fourth. (iv) Its fundamental solution is and it satisfies the two-scale relation where a k is the mask of the scheme (5). Since 6PBASS is C 2 -continuous by [24], so its 2-scale relations ](x) are also C 2 -continuous. (v) For the computation of the first-and second-order derivatives of (7), we adopt similar approach of [22,23].

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.

Formulation of Subdivision Collection Algorithm for
Bratu's Problem. e detail of the proposed algorithm is given as follows: be an approximate solution of (2) and N must be greater than or equal to four, h is the step size and is defined as , and w i are the unknowns to be determined. From (10), we get By using (10) and (11) in (2), we get where j � 0, 1, . . . , N, and the conditions given at the ends of the domain (3) become e matrix representation of equation (12) is where 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.

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 w − 3 , w − 2 , w − 1 and w N+1 , w N+2 , w N+3 represent the left end points and right end points. ese left and right end points can be computed by using polynomial of degree three which interpolates the data (y i , w i ), for 0 ≤ i ≤ 3, i.e., left end conditions are obtained from where Since by (10), W(y i ) � w i for i � 1, 2, 3 and substituting 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 (N + 9) × (N + 9) nonlinear equations Journal of Mathematics or where where A is defined in (15) and J c 0 and J c N are constrained as follows: the matrix J c 0 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 J c 0 obtained from (13) at W(0) � w 0 � 0. Similarly, the matrix J c N is obtained from all the conditions defined at the right end of the domain, i.e., first row comes from (13) at W(1) � w N � 0 and remaining rows come from (22). Hence, e column vector W is defined in (16) and D is defined as where D 1 is given in (17).

Iterative Algorithm.
To find the numerical solutions of nonlinear system of equation (24), we define an iterative algorithm. e iterative algorithm includes the following steps: (i) Formulation of initial solution: the initial approximate solution W 0 is selected to find the following system: where where i ∈ 0, 1, . . . , N { }. e column vector R 0 is the linear approximation of the column vector (27).
(ii) Iterative scheme: the following iterative scheme is used to find the approximate solution W * , (iii) Terminating criteria: the following condition is used to stop the iteration at k − th level, for any ε; let ε � 10 − 4 ,

Convergence and Error Estimation.
In this section, we present results of convergence and error estimation of the proposed iterative algorithm. e convergence of the iterative algorithm is guaranteed by the following proposition.

Proposition 1.
e approximate solution W k founded by (28) and (30) linearly converges to the approximate solution W * of (24) with the supposition that step size and the Lipschitz constants r 0 , r 1 are small, i.e., e proof is similar to [23]. e main result of error estimation is given by the following proposition. Theorem 1. Let exact solution g(y) ∈ C 4 [0, 1] and w i be obtained by solving (24) with the fourth order boundary treatment at the end points. en, we have

Numerical Examples and Comparison
e 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 α, which guarantee the existence of two locally unique solutions. We have created comparison tables using α � 1, 2 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) e fact regarding the solution of Bratu's problem for α � 1 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) e fact regarding the solution of Bratu's problem for α � 2 is obtained after fifth iteration, as shown in Table 1: Numerical results of (2) by subdivision collection algorithm for case α � 1.   Table 3: Comparison between the absolute errors of (2) for case α � 1.
x By subdivision collection algorithm By [25] By [5] 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) e fact regarding the solution of Bratu's problem for α � 3.15 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].

Concluding Remarks
In this paper, we have established a subdivision collocation algorithm for the solution of one-dimensional nonlinear Bratu's problem. e 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 α. We conclude that our algorithm gives smaller absolute errors as compared with the other existing methods [5,6,25].

Data Availability
e data used to support the findings of the study are available within this paper.