We propose two new modified recursive schemes for solving a class of doubly singular two-point boundary value problems. These schemes are based on Adomian decomposition method (ADM) and new proposed integral operators. We use all the boundary conditions to derive an integral equation before establishing the recursive schemes for the solution components. Thus we develop recursive schemes without any undetermined coefficients while computing successive solution components, whereas several previous recursive schemes have done so. This modification also avoids solving a sequence of nonlinear algebraic or transcendental equations for the undetermined coefficients with multiple roots, which is required to complete calculation of the solution by several earlier modified recursion schemes using the ADM. The approximate solution is computed in the form of series with easily calculable components. The effectiveness of the proposed approach is tested by considering four examples and results are compared with previous known results. 1. Introduction Consider the following class of doubly singular two-point boundary value problems: with boundary conditions where , , , , and are any finite constants. The condition characterizes that problem (1.1) is singular and in addition to this if is allowed to be discontinuous at , then problem (1.1) is called doubly singular Bobisud [1]. Consider problem (1.1) with the following conditions on , , and . Type 1. Dirichlet boundary conditions: , , with in and ; in and ; ？(i) ; (ii) ？ exists and continuous, for all and all real . Type 2. Mixed type boundary conditions: ,？？ ？ ,？？ with in ; ？ in (0,1], and ; ？ the same as ( ). In recent years, the study of such singular boundary value problems (1.1) has attracted the attention of several researchers [2–9]. In particular, if ,？？ and problem (1.1) with Type 1 boundary conditions is known as Thomas-Fermi equation. Thomas [10] and Fermi [11] independently derived a boundary value problem for determining the electrical potential in an atom. The analysis leads to the nonlinear SBVP with boundary conditions given by ,？？ . Chan and Hon [12] considered the generalized Thomas-Fermi equation: , , with , where , , , , which is doubly singular problem. Problem (1.1) with where arises in the study of the distribution of heat sources in the human head [13] with , , . There is a huge literature available on numerical methods for problem (1.1) with , but very few numerical methods are available to tackle doubly singular boundary value problems. Reddien [14] studied the linear form of problem (1.1) and
References
[1]
L. E. Bobisud, “Existence of solutions for nonlinear singular boundary value problems,” Applicable Analysis, vol. 35, no. 1–4, pp. 43–57, 1990.
[2]
T. Aziz and M. Kumar, “A fourth-order finite-difference method based on non-uniform mesh for a class of singular two-point boundary value problems,” Journal of Computational and Applied Mathematics, vol. 136, no. 1-2, pp. 337–342, 2001.
[3]
M. M. Chawla and C. P. Katti, “A uniform mesh finite difference method for a class of singular two-point boundary value problems,” SIAM Journal on Numerical Analysis, vol. 22, no. 3, pp. 561–565, 1985.
[4]
A. Ebaid, “A new analytical and numerical treatment for singular two-point boundary value problems via the Adomian decomposition method,” Journal of Computational and Applied Mathematics, vol. 235, no. 8, pp. 1914–1924, 2011.
[5]
M. Kumar, “A new finite difference method for a class of singular two-point boundary value problems,” Applied Mathematics and Computation, vol. 143, no. 2-3, pp. 551–557, 2003.
[6]
M. M. Chawla and C. P. Katti, “Finite difference methods and their convergence for a class of singular two-point boundary value problems,” Numerische Mathematik, vol. 39, no. 3, pp. 341–350, 1982.
[7]
M. M. Chawla and C. P. Katti, “A finite-difference method for a class of singular two-point boundary value problems,” IMA Journal of Numerical Analysis, vol. 4, no. 4, pp. 457–466, 1984.
[8]
M. M. Chawla, S. McKee, and G. Shaw, “Order 2 method for a singular two-point boundary value problem,” BIT Numerical Mathematics, vol. 26, no. 3, pp. 318–326, 1986.
[9]
S. R. K. Iyengar and P. Jain, “Spline finite difference methods for singular two point boundary value problems,” Numerische Mathematik, vol. 50, no. 3, pp. 363–376, 1987.
[10]
L. Thomas, “The calculation of atomic fields,” in Mathematical Proceedings of the Cambridge Philosophical Society, vol. 23, pp. 542–548, Cambridge University Press, 1927.
[11]
E. Fermi, “Un metodo statistico per la determinazione di alcune priorieta dell’atome,” Rendicondi Accademia Nazionale de Lincei, vol. 32, no. 6, pp. 602–607, 1927.
[12]
C. Y. Chan and Y. C. Hon, “A constructive solution for a generalized Thomas-Fermi theory of ionized atoms,” Quarterly of Applied Mathematics, vol. 45, no. 3, pp. 591–599, 1987.
[13]
R. Duggan and A. Goodman, “Pointwise bounds for a nonlinear heat conduction model of the human head,” Bulletin of Mathematical Biology, vol. 48, no. 2, pp. 229–236, 1986.
[14]
G. W. Reddien, “Projection methods and singular two point boundary value problems,” Numerische Mathematik, vol. 21, pp. 193–205, 1974.
[15]
M. M. Chawla and P. N. Shivakumar, “On the existence of solutions of a class of singular nonlinear two-point boundary value problems,” Journal of Computational and Applied Mathematics, vol. 19, no. 3, pp. 379–388, 1987.
[16]
D. R. Dunninger and J. C. Kurtz, “Existence of solutions for some nonlinear singular boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 115, no. 2, pp. 396–405, 1986.
[17]
R. K. Pandey and A. K. Verma, “A note on existence-uniqueness results for a class of doubly singular boundary value problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 7-8, pp. 3477–3487, 2009.
[18]
G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, vol. 60 of Fundamental Theories of Physics, Kluwer Academic, Dordrecht, The Netherlands, 1994.
[19]
G. Adomian and R. Rach, “Inversion of nonlinear stochastic operators,” Journal of Mathematical Analysis and Applications, vol. 91, no. 1, pp. 39–46, 1983.
[20]
G. Adomian and R. Rach, “A new algorithm for matching boundary conditions in decomposition solutions,” Applied Mathematics and Computation, vol. 58, no. 1, pp. 61–68, 1993.
[21]
G. Adomian and R. Rach, “Modified decomposition solution of linear and nonlinear boundary-value problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 23, no. 5, pp. 615–619, 1994.
[22]
A.-M. Wazwaz, “Approximate solutions to boundary value problems of higher order by the modified decomposition method,” Computers & Mathematics with Applications, vol. 40, no. 6-7, pp. 679–691, 2000.
[23]
A. M. Wazwaz, “A reliable algorithm for obtaining positive solutions for nonlinear boundary value problems,” Computers & Mathematics with Applications, vol. 41, no. 10-11, pp. 1237–1244, 2001.
[24]
M. Benabidallah and Y. Cherruault, “Application of the Adomian method for solving a class of boundary problems,” Kybernetes, vol. 33, no. 1, pp. 118–132, 2004.
[25]
M. Inc and D. J. Evans, “The decomposition method for solving of a class of singular two-point boundary value problems,” International Journal of Computer Mathematics, vol. 80, no. 7, pp. 869–882, 2003.
[26]
B. Jang, “Two-point boundary value problems by the extended Adomian decomposition method,” Journal of Computational and Applied Mathematics, vol. 219, no. 1, pp. 253–262, 2008.
[27]
S. A. Khuri and A. Sayfy, “A novel approach for the solution of a class of singular boundary value problems arising in physiology,” Mathematical and Computer Modelling, vol. 52, no. 3-4, pp. 626–636, 2010.
[28]
D. Lesnic, “A computational algebraic investigation of the decomposition method for time-dependent problems,” Applied Mathematics and Computation, vol. 119, no. 2-3, pp. 197–206, 2001.
[29]
J.-S. Duan and R. Rach, “A new modification of the Adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations,” Applied Mathematics and Computation, vol. 218, no. 8, pp. 4090–4118, 2011.
[30]
K. Abbaoui and Y. Cherruault, “Convergence of Adomian's method applied to differential equations,” Computers & Mathematics with Applications, vol. 28, no. 5, pp. 103–109, 1994.
[31]
Y. Cherruault, “Convergence of Adomian's method,” Kybernetes, vol. 18, no. 2, pp. 31–38, 1989.
[32]
Y. Cherruault and G. Adomian, “Decomposition methods: a new proof of convergence,” Mathematical and Computer Modelling, vol. 18, no. 12, pp. 103–106, 1993.
[33]
M. M. Hosseini and H. Nasabzadeh, “On the convergence of Adomian decomposition method,” Applied Mathematics and Computation, vol. 182, no. 1, pp. 536–543, 2006.
[34]
M. M. Chawla, R. Subramanian, and H. L. Sathi, “A fourth order method for a singular two-point boundary value problem,” BIT Numerical Mathematics, vol. 28, no. 1, pp. 88–97, 1988.
