Solving singular two-point boundary value problems using continuous genetic algorithm. (English) Zbl 1261.34018

Summary: A continuous genetic algorithm is applied for the solution of singular two-point boundary value problems, where smooth solution curves are used throughout the evolution of the algorithm to obtain the required nodal values. The proposed technique might be considered as a variation of the finite difference method in the sense that each of the derivatives is replaced by an appropriate difference quotient approximation. This novel approach possesses some advantages: it can be applied without any limitation on the nature of the problem, the type of singularity, and the number of mesh points. Numerical examples are included to demonstrate the accuracy, applicability and generality of the presented technique. The results reveal that the algorithm is very effective, straightforward and simple.


34A45 Theoretical approximation of solutions to ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
Full Text: DOI


[1] H. \cCa\vglar, N. \cCa\vglar, and M. Özer, “B-spline solution of non-linear singular boundary value problems arising in physiology,” Chaos, Solitons and Fractals, vol. 39, no. 3, pp. 1232-1237, 2009. · Zbl 1197.65107
[2] P. Hiltman and P. Lory, “On oxygen diffusion in a spherical cell with Michaelis-Menten oxygen uptake kinetics,” Bulletin of Mathematical Biology, vol. 45, no. 5, pp. 661-664, 1983. · Zbl 0512.92008
[3] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover, New York, NY, USA, 1981. · Zbl 0142.44103
[4] U. M. Ascher, R. M. M. Mattheij, and R. D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, vol. 13 of Classics in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 1995. · Zbl 0843.65054
[5] P. Jamet, “On the convergence of finite-difference approximations to one-dimensional singular boundary-value problems,” Numerische Mathematik, vol. 14, pp. 355-378, 1969/1970. · Zbl 0179.22103
[6] R. K. Mohanty, “A family of variable mesh methods for the estimates of (du/dr) and solution of non-linear two-point boundary-value problems with singularity,” Journal of Computational and Applied Mathematics, vol. 182, no. 1, pp. 173-187, 2005. · Zbl 1071.65113
[7] A. S. Bataineh, M. S. M. Noorani, and I. Hashim, “Approximate solutions of singular two-point BVPs by modified homotopy analysis method,” Physics Letters A, vol. 372, pp. 4062-4066, 2008. · Zbl 1217.35101
[8] A. S. V. Ravi Kanth and Y. N. Reddy, “Higher order finite difference method for a class of singular boundary value problems,” Applied Mathematics and Computation, vol. 155, no. 1, pp. 249-258, 2004. · Zbl 1058.65078
[9] A. S. V. Ravi Kanth and Y. N. Reddy, “Cubic spline for a class of singular two-point boundary value problems,” Applied Mathematics and Computation, vol. 170, no. 2, pp. 733-740, 2005. · Zbl 1103.65086
[10] R. K. Mohanty, P. L. Sachdev, and N. Jha, “An O(h4) accurate cubic spline TAGE method for nonlinear singular two point boundary value problems,” Applied Mathematics and Computation, vol. 158, no. 3, pp. 853-868, 2004. · Zbl 1060.65080
[11] M. Cui and F. Geng, “Solving singular two-point boundary value problem in reproducing kernel space,” Journal of Computational and Applied Mathematics, vol. 205, no. 1, pp. 6-15, 2007. · Zbl 1149.65057
[12] C. Chun, A. Ebaid, M. Y. Lee, and E. Aly, “An approach for solving singular two-point boundary value problems: analytical and numerical treatment,” ANZIAM Journal, vol. 53, pp. E21-E43, 2011. · Zbl 1333.65091
[13] M. Kumar and N. Singh, “A collection of computational techniques for solving singular boundary-value problems,” Advances in Engineering Software, vol. 40, no. 4, pp. 288-297, 2009. · Zbl 1159.65076
[14] Z. S. Abo-Hammour, Advanced continuous genetic algorithms and their applications in the motion planning of robotic manipulators and the numerical solution of boundary value problems [Ph.D. thesis], Quiad-Azam University, 2002.
[15] Z. S. Abo-Hammour, N. M. Mirza, S. M. Mirza, and M. Arif, “Cartesian path generation of robot manipulators using continuous genetic algorithms,” Robotics and Autonomous Systems, vol. 41, no. 4, pp. 179-223, 2002. · Zbl 1010.68962
[16] Z. S. Abo-Hammour, “A novel continuous genetic algorithms for the solution of the cartesian path generation problem of robot manipulators,” in Robot Manipulators: New Research, J. X. Lui, Ed., pp. 133-190, Nova Science, New York, NY, USA, 2005.
[17] Z. S. Abo-Hammour, M. Yusuf, N. M. Mirza, S. M. Mirza, M. Arif, and J. Khurshid, “Numerical solution of second-order, two-point boundary value problems using continuous genetic algorithms,” International Journal for Numerical Methods in Engineering, vol. 61, no. 8, pp. 1219-1242, 2004. · Zbl 1073.65543
[18] Z. S. Abo-Hammour, A. G. Asasfeh, A. M. Al-Smadi, and O. M. K. Alsmadi, “A novel continuous genetic algorithm for the solution of optimal control problems,” Optimal Control Applications and Methods, vol. 32, no. 4, pp. 414-432, 2011. · Zbl 1260.49054
[19] Z. S. Abo-Hammour, O. Alsmadi, S. I. Bataineh, M. A. Al-Omari, and N. Affach, “Continuous genetic algorithms for collision-free cartesian path planning of robot manipulators,” International Journal of Advanced Robotic Systems, vol. 8, pp. 14-36, 2011.
[20] Z. S. Abo-Hammour, R. B. Albadarneh, and M. S. Saraireh, “Solution of Laplace equation using continuous genetic algorithms,” Kuwait Journal of Science & Engineering, vol. 37, no. 2, pp. 1-15, 2010. · Zbl 1243.65145
[21] O. Abu Arqub, Z. Abo-Hammour, and S. Momani, “Application of continuous genetic algorithm for nonlinear system of second-order boundary value problems,” Applied Mathematics and Information Sciences. In press. · Zbl 1261.34018
[22] O. Abu Arqub, Numerical solution of fuzzy differential equation using continuous genetic algorithms [Ph. D. thesis], University of Jordan, 2008.
[23] J. Li, “General explicit difference formulas for numerical differentiation,” Journal of Computational and Applied Mathematics, vol. 183, no. 1, pp. 29-52, 2005. · Zbl 1077.65021
[24] D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, Addison-Wesley, Reading, Mass, USA, 1989. · Zbl 0721.68056
[25] U. Hörchner and J. H. Kalivas, “Further investigation on a comparative study of simulated annealing and genetic algorithm for wavelength selection,” Analytica Chimica Acta, vol. 311, no. 1, pp. 1-13, 1995.
[26] J. H. Jiang, J. H. Wang, X. Chu, and R. Q. Yu, “Clustering data using a modified integer genetic algorithm (IGA),” Analytica Chimica Acta, vol. 354, no. 1-3, pp. 263-274, 1997.
[27] E. M. Rudnick, J. H. Patel, G. S. Greenstein, and T. M. Niermann, “A genetic algorithm framework for test generation,” IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, vol. 16, no. 9, pp. 1034-1044, 1997.
[28] J. R. Vanzandt, “A genetic algorithm for search route planning,” ESD TR-92-262, United States Air Force, 1992.
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