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.
Highlights
Introduction e general expression of LiouvilleBratu–Gelfand equation [1, 2]: Δχ(r) + α exp(χ(r)) 0, r ∈ Ω1, (1)with conditions at the ends of the domain χ(0) 0, (3) χ(1) 0.e detailed information of problem (2) is given in [4, 5]. e exact solution of (2) is cosh(0.5θ(r − 0.5)) χ(r) − 2 ln cosh(0.25θ) ,√ where θ 2αcosh(0.25θ). (4)e exponential term guarantees nonlinearity and the bifurcation phenomenon that follows up
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
Summary
Ghulam Mustafa ,1 Syeda Tehmina Ejaz ,2 Sabila Kouser ,2 Shafqat Ali ,1 and Muhammad Aslam 3. Subdivision schemes-based algorithms are not frequently used to find numerical solutions of boundary value problems. Ejaz et al [22, 23] constructed subdivision schemes-based algorithm for solutions of boundary value problems of third and fourth order. 4. Numerical Examples and Comparison e numerical technique discussed previously is illustrated 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. 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. (ii) e fact regarding the solution of Bratu’s problem for α 2 is obtained after fifth iteration, as shown in
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