Geodesic
Efficient Jacobi–Gauss collocation method for solving initial value problems of Bratu-type
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 9

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In this paper, we propose the shifted Jacobi–Gauss collocation spectral method for solving initial value problems of Bratu type, which is widely applicable in fuel ignition of the combustion theory and heat transfer. The spatial approximation is based on shifted Jacobi polynomials $J_n^{(\alpha,\beta)}(x)$ with $\alpha, \beta \in(-1,\infty)$, $x\in[0,1]$ and $n$ the polynomial degree. The shifted Jacobi–Gauss points are used as collocation nodes. Illustrative examples have been discussed to demonstrate the validity and applicability of the proposed technique. Comparing the numerical results of the proposed method with some well-known results show that the method is efficient and gives excellent numerical results. 
@article{ZVMMF_2013_53_9_a4,
     author = {E. H. Doha and A. H. Bhrawy and D. Baleanu and R. H. Hafez},
     title = {Efficient {Jacobi{\textendash}Gauss} collocation method for solving initial value problems of {Bratu-type}},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
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     publisher = {mathdoc},
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     number = {9},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_9_a4/}
}
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E. H. Doha; A. H. Bhrawy; D. Baleanu; R. H. Hafez. Efficient Jacobi–Gauss collocation method for solving initial value problems of Bratu-type. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 9. http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_9_a4/