This paper deals with the two singular boundary values problems of second order. Two singular points are both boundary values points of the differential equation. The numerical solutions are developed by modified differential transform method (DTM) for expanded point. Linear and nonlinear models are solved by this method to get more reliable and efficient numerical results. It can also solve ordinary differential equations where the traditional one fails. Besides, we give the convergence of this new method. 1. Introduction In the present paper, we consider the following two singular boundary value problems (BVPs) of second order: where , and are known and continuous functions, , and is a nonlinear function of . The equation is singular at these two boundary values . Scientists and engineers have been interested in the singular equation because of its importance in many applications such as physical and mathematical models. There are many research directions on these equations. Some studied their qualitative properties [1, 2]. For example, Bartolucci and Montefusco [1] studied the concentration-compactness problem and the mass quantization properties. Some others used theorems to establish the existence and uniqueness of solution [3, 4]. For example, Guo et al. [3] got the existence and uniqueness of solution using a fixed point theorem. Recently great attention had been paid to numerical solutions [5–13]. For example, Duan and Rach [5] solved boundary value problems using a new modified Adomian decomposition method. Chowdhury and Hashim [7] used homotopy asymptotic method for finding the approximate solutions. Wazwaz [6, 8, 9] used Adomian decomposition method to get the numerical solutions. Puhov in 1976 [14] proposed the concept of DTM. DTM is the extension of Taylor series method and had been applied to solve analytic solutions of ordinary [15, 16], partial [17–19], differential-algebraic equations [20, 21], differential-difference equations [22, 23], and integrodifferential equations [24, 25]. Numerical solutions are also obtained [26]. Furthermore, Alquran and Al-Khaled [27] applied DTM to solve some eigenvalue problems. In the present paper, there are two singular points and these two singular points are just boundary values of the equation. Traditional DTM only can solve one-point singular BVP or two-point BVP (but not both the two points are just boundary values). We add some operation properties of DTM; then DTM can be used to calculate this type of problem. Furthermore, we have a convergent analysis of this method. 2. Modified Technique Now,
References
[1]
D. Bartolucci and E. Montefusco, “Blow-up analysis, existence and qualitative properties of solutions for the two-dimensional Emden-Fowler equation with singular potential,” Mathematical Methods in the Applied Sciences, vol. 30, no. 18, pp. 2309–2327, 2007.
[2]
M. Lefranc and J. Mawhin, “ètude qualitative des solutions de l'équation différentielle d'Emden-Fowler,” Académie Royale de Belgique Bulletin de la Classe des Sciences, vol. 55, no. 5, p. 763–770, 1969 (French).
[3]
C. Guo, C. Zhai, and R. Song, “An existence and uniqueness result for the singular Lane-Emden-Fowler equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1275–1279, 2010.
[4]
A. E. Ebaid, “Exact solutions for a class of nonlinear singular two-point boundary value problems: the decomposition method,” Zeitschrift fur Naturforschung, vol. 65, no. 3, pp. 145–150, 2010.
[5]
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.
[6]
A. M. Wazwaz, “A new method for solving singular initial value problems in the second-order ordinary differential equations,” Applied Mathematics and Computation, vol. 128, no. 1, pp. 45–57, 2002.
[7]
M. S. H. Chowdhury and I. Hashim, “Solutions of Emden-Fowler equations by homotopy-perturbation method,” Nonlinear Analysis: Real World Applications, vol. 10, no. 1, pp. 104–115, 2009.
[8]
A. M. Wazwaz, “Adomian decomposition method for a reliable treatment of the Emden-Fowler equation,” Applied Mathematics and Computation, vol. 161, no. 2, pp. 543–560, 2005.
[9]
A. M. Wazwaz, “A new algorithm for solving differential equations of Lane-Emden type,” Applied Mathematics and Computation, vol. 118, no. 2-3, pp. 287–310, 2001.
[10]
P. M. Lima and A. M. Oliveira, “Numerical solution of a singular boundary value problem for a generalized Emden-Fowler equation,” Applied Numerical Mathematics, vol. 45, no. 4, pp. 389–409, 2003.
[11]
S. A. Khuri, “A new approach to Bratu's problem,” Applied Mathematics and Computation, vol. 147, no. 1, pp. 131–136, 2004.
[12]
P. Y. Tsai and C. K. Chen, “An approximate analytic solution of the nonlinear Riccati differential equation,” Journal of the Franklin Institute, vol. 347, no. 10, pp. 1850–1862, 2010.
[13]
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.
[14]
G. è. Puhov, “The application of Taylor transforms to the solution of differential equations,” Elektronikai Modelirovanie Vyp, vol. 11, no. 128, pp. 18–23, 1976 (Russian).
[15]
A. Kurnaz and G. Oturan？, “The differential transform approximation for the system of ordinary differential equations,” International Journal of Computer Mathematics, vol. 82, no. 6, pp. 709–719, 2005.
[16]
F. Mirzaee, “Differential transform method for solving linear and nonlinear systems of ordinary differential equations,” Applied Mathematical Sciences, vol. 5, no. 70, pp. 3465–3472, 2011.
[17]
C. K. Chen and S. H. Ho, “Solving partial differential equations by two-dimensional differential transform method,” Applied Mathematics and Computation, vol. 106, no. 2-3, pp. 171–179, 1999.
[18]
M. J. Jang, C. L. Chen, and Y. C. Liu, “Two-dimensional differential transform for partial differential equations,” Applied Mathematics and Computation, vol. 121, no. 2-3, pp. 261–270, 2001.
[19]
A. ？etinkaya, O. K？ymaz, and J. ？aml？, “Solutions of nonlinear PDE's of fractional order with generalized differential transform method,” International Mathematical Forum, vol. 6, no. 1, pp. 39–47, 2011.
[20]
H. Liu and Y. Song, “Differential transform method applied to high index differential-algebraic equations,” Applied Mathematics and Computation, vol. 184, no. 2, pp. 748–753, 2007.
[21]
F. Ayaz, “Applications of differential transform method to differential-algebraic equations,” Applied Mathematics and Computation, vol. 152, no. 3, pp. 649–657, 2004.
[22]
L. Zou, Z. Wang, and Z. Zong, “Generalized differential transform method to differential-difference equation,” Physics Letters A, vol. 373, no. 45, pp. 4142–4151, 2009.
[23]
A. Arikoglu and I. Ozkol, “Solution of differential-difference equations by using differential transform method,” Applied Mathematics and Computation, vol. 181, no. 1, pp. 153–162, 2006.
[24]
A. Tari and S. Shahmorad, “Differential transform method for the system of two-dimensional nonlinear Volterra integro-differential equations,” Computers and Mathematics with Applications, vol. 61, no. 9, pp. 2621–2629, 2011.
[25]
D. Nazari and S. Shahmorad, “Application of the fractional differential transform method to fractional-order integro-differential equations with nonlocal boundary conditions,” Journal of Computational and Applied Mathematics, vol. 234, no. 3, pp. 883–891, 2010.
[26]
J. Liu and G. Hou, “Numerical solutions of the space- and time-fractional coupled Burgers equations by generalized differential transform method,” Applied Mathematics and Computation, vol. 217, no. 16, pp. 7001–7008, 2011.
[27]
M. T. Alquran and K. Al-Khaled, “Approximations of Sturm-Liouville eigenvalues using sinc-Galerkin and differential transform methods,” Applications and Applied Mathematics, vol. 5, no. 1, pp. 128–147, 2010.
[28]
F. Geng and M. Cui, “Solving singular nonlinear boundary value problems by combining the homotopy perturbation method and reproducing kernel Hilbert space method,” International Journal of Computer Mathematics, vol. 87, no. 9, pp. 2024–2031, 2010.
[29]
Y. W. Lin, T. T. Lu, and C. K. Chen, “Adomian decomposition method using integrating factor,” Communications in Theoretical Physics, vol. 60, no. 2, pp. 159–164, 2013.
