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A sixth order optimal B-spline collocation method for solving Bratu’s problem

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Abstract

In this paper, we describe an optimal B-spline collocation method to solve one-dimensional non-linear Bratu problem. Convergence result of the method is established through Green’s function technique and it is proved that the method has sixth order convergence. The applicability and accuracy of the method are demonstrated through two numerical examples. It is shown that the computational result is consistent with theoretical prediction. Moreover, the numerical results obtained by the present method are compared with those obtained by two other B-spline collocation approaches reported in the literature. The computed results reveal that our method is accurate and easy to implement.

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Acknowledgements

The authors are very grateful to anonymous referees for their valuable suggestions and comments and thankfully acknowledge the financial support provided by the CSIR, India in the form of Project No. \(25(0286)/18/EMR-II\).

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  1. Department of Mathematics, Visvesvaraya National Institute of Technology, Nagpur, 440010, India

    Pradip Roul & V. M. K. Prasad Goura

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  1. Pradip Roul
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Correspondence to Pradip Roul.

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Appendix

Appendix

The elements of the coefficient matrix P are given as follows:

$$\begin{aligned} \begin{aligned} l_{1}&=717+36I_{0},\;\;l_{2}=1448+936I_{0},\;\;l_{3}=-4300+2376I_{0},\\ l_{4}&=1322+936I_{0},\;\;\\ l_{5}&=938+36I_{0},\;\;k_{1}=1475+18J_{1},\;\;k_{2}=30149+4266J_{1},\\ k_{3}&=-31918+30276J_{1},\;\;\\ k_{4}&=-30322+30276J_{1},\;\;k_{5}=27776+4266J_{1},\;\;k_{6}=3680+18J_{1},\\ q_{1}&=725+36I_{1},\;\;\\ q_{2}&=1454+936I_{1},\;\;q_{3}=-4396+2376I_{1},\;\;q_{4}=1574+936I_{1},\\ q_{5}&=602+36I_{1},\;\;\\ p_{i}&=728+36I_{i+1},\;\;s_{i}=1406+936I_{i+1},\;\;t_{i}=-4270+2376I_{i+1},\\ u_{i}&=1406+936I_{i+1},\;\;\\ v_{i}&=728+36I_{i+1},\;\;1\le i \le n-3,\;\;{\tilde{q}}_{1}=602+36I_{n-1},\\ {\tilde{q}}_{2}&=1574+936I_{n-1},\;\;\\ {\tilde{q}}_{3}&=-4396+2376I_{n-1},\;\;{\tilde{q}}_{4}=1454+936I_{n-1},\\ {\tilde{q}}_{5}&=725+36I_{n-1},\;\;{\tilde{k}}_{1}=3680+18J_{n},\;\;\\ {\tilde{k}}_{2}&=27776+4266J_{n},\;\;{\tilde{k}}_{3}=-30322+30276J_{n},\;\; {\tilde{k}}_{4}=-31918+30276J_{n},\;\;\\ {\tilde{k}}_{5}&=30149+4266J_{n},\;\;{\tilde{k}}_{6}=1475+18J_{n},\;\; {\tilde{l}}_{1}=938+36I_{n},\\ {\tilde{l}}_{2}&=1322+936I_{n},\;\;\\ {\tilde{l}}_{3}&=-4300+2376I_{n},\;\;{\tilde{l}}_{4}=1448+936I_{n},\;\; {\tilde{l}}_{5}=717+36I_{n},\;\;I_{i}=h^{2}{G}_{i},\\ 0&\le i \le n, J_{i}=h^{2}{G}(\tau _{i}),i=1,n. \end{aligned} \end{aligned}$$

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Roul, P., Goura, V.M.K.P. A sixth order optimal B-spline collocation method for solving Bratu’s problem. J Math Chem 58, 967–988 (2020). https://doi.org/10.1007/s10910-020-01105-6

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