[1] Ahlberg, J.H., Nilson, E.N. and Walsh, J.L. The Theory of Splines and Their Applications, Academic Press, New York. (1967),
[2] Akram, G. and Siddiqi, S.S. Nonic spline solutions of eighth-order boundary value problems, Appl. Math. Comput. 182 (2006), 829–845.
[3] Akram, G. and Siddiqi, S.S. Solution of sixth order boundary value problems using non-polynomial spline technique, Appl. Math. Comput. 181 (2006), 708–720.
[4] Aziz, T. and Khan, A. Quintic spline approach to the solution of a singularly-perturbed boundary-value problem, J. Optim. Theory Appl. 112 (2002), 517–527.
[5] Buckmire, R. Application of a Mickens finite-difference scheme to the cylindrical Bratu-Gelfand problem, Numer. Meth. Part. Differ. Equat. 20 (2004), 327–337.
[6] Caglar, H., Caglar, N., Ozer, M., Valaristos, A. and Anagnostopoulos, A.N. B-spline method for solving Bratu’s problem, Int. J. Comput. Math. 87 (2010), 1885–1891.
[7] Farajeyan, K., Rashidinia, J. and Jalilian, R. Classes of high-order nu-merical methods for solution of certain problem in calculus of variations, Cogent. Math. Stat. 4 (2017), 1–15.
[8] Farajeyan, K., Rashidinia, J., Jalilian, R. and Maleki, N.R. Application of spline to approximate the solution of singularly perturbed boundary-value problems, Comput. Methods Differ. Equ. 8 (2020), 373–388.
[9] Greville, T.N.E. Introduction to spline functions, in: Theory and Ap-plication of Spline Functions, Academic Press, New York. (1969),
[10] Henrici, P. Discrete variable methods in ordinary differential equations, New York, Wiley. (1961),
[11] Islam, S. U., Tirmizi, I.A. , Haq, F. and Khan, M.A. Non-polynomial splines approach to the solution of sixth-order boundary-value problems, Appl. Math. Comput. 195 (2008), 270–284.
[12] Jacobsen, J. and Schmitt, K. The Liouville-Bratu-Gelfand problem for radial operators, J. Differ. Equat. 184 (2002), 283-298.
[13] Jalilian, R. Non-polynomial spline method for solving Bratu’s problem, Comput. Phys. Commun. 181 (2010), 1868–1872.
[14] Jalilian, R., Rashidinia, J., Farajyan, K. and Jalilian, H. Non-Polynomial Spline for the Numerical Solution of Problems in Calculus of Variations, Int. J. Math. Comput. 5 (2015), 1–14.
[15] Khan, A. Parametric cubic spline solution of two point boundary value problems, Appl. Math. Comput. 154 (2004), 175–182.
[16] Khan, A., Khan, I. and Aziz, T. A survey on parametric spline function approximation, Appl. Math. Comput. 171 (2005), 983–1003.
[17] Khan, A., Khan, I. and Aziz, T. Sextic spline solution of a singularly perturbed boundary-value problems, Appl. Math. Comput. 181 (2006), 432–439.
[18] Rashidinia, J. and Golbabaee, A. Convergence of numerical solution of a fourth-order boundary value problem, Appl. Math. Comput. 171 (2005), 1296–1305.
[19] Rashidinia, J., Jalilian, R. and Farajeyan, K. Spline approximate solu-tion of eighth-order boundary-value problems, Int. J. Comput. Math. 86 (2009), 1319–1333.
[20] Rashidinia, J., Jalilian, R. and Farajeyan, K. Non polynomial spline solutions for special linear tenth-order boundary value problems, World J. Model. Simul. 7 (2011), 40–51.
[21] Rashidinia, J., Jalilian, R. and Mohammadi, R. Convergence analysis of spline solution of certain two-point boundary value problems, Computer Science and Engineering and Electrical Engineering. 16 (2009), 128–136.
[22] Rashidinia, J. and Mahmoodi, Z. Non-polynomial spline solution of a singularly perturbed boundary-value problems, Int. J. Contemp. Math. Sciences. 2 (2007), 1581–1586.
[23] Razzaghi, M. and Yousefi, S. Legendre wavelets direct method for vari-ational problems, Math. Comput. Simulat. 53 (2000), 185–192.
[24] Siddiqi, S.S. and Akram, G. Solution of tenth-order boundary value problems using eleventh degree spline, Appl. Math. Comput. 185 (2007), 115–127.
[25] Siddiqi, S.S. and Akram, G. Solutions of 12th order boundary value problems using non-polynomial spline technique, Appl. Math. Comput. 199 (2008), 559–571.
[26] Siddiqi, S.S. and Akram, G. Septic spline solutions of sixth-order bound-ary value problems, J. Comput. Appl. Math. 215 (2008), 288–301.
[27] Surla, K. and Vukoslavcevi´c, V. A spline difference scheme for boundary value problems with a small parameter, Zb. Rad. Prirod.-Mat. Fak. Ser. Mat. 25 (1995), 159–168.
[28] Zarebnia, M. and Aliniya, N. Sinc-Galerkin method for the solution of problems in calculus of variations, Int. J. Nat. Eng. Sci. 5 (2011), 140–145.
[29] Zarebnia, M. and Sarvari, Z. Numerical solution of the boundary value problems in calculus of variations using parametric cubic spline method, J. Inform. Comput. Sci. 8 (2013), 275–282.
[30] Zarebnia, M. and Sarvari, Z. Numerical solution of Variational Problems via parametric quintic spline method, J. Hyperstruct. 3 (2014), 40–52.
[31] Zarebnia, M. and Sarvari, Z. Parametric spline method for solving Bratu’s problem, Int. J. Nonlinear Sci. 14 (2012), 3–10.
Send comment about this article