International E-publication: Publish Projects, Dissertation, Theses, Books, Souvenir, Conference Proceeding with ISBN.  International E-Bulletin: Information/News regarding: Academics and Research

Memetic Heuristic Computation for Solving Nonlinear Singular Boundary Value Problems Arising in Physiology

    , , ,

Author Affiliations

  • 11Department of Electronic Engineering, Faculty of Engineering and Technology, International Islamic University, Islamabad, PAKISTAN
  • 2Department of Electrical Engineering, Air University, Islamabad, PAKISTAN
  • 3Institute of Signals, Systems and Soft computing, Islamabad, PAKISTAN

Res. J. Recent Sci., Volume 2, Issue (9), Pages 47-55, September,2 (2013)

Abstract

We present a stochastic numerical method based on memetic heuristic computing for the approximate numerical solution of a class of nonlinear singular boundary value problems arising in physiology. The solution of the nonlinear problem is represented by the linear combination of some log sigmoid basis functions. A fitness function representing the mean square error consisting of unknown adaptable parameters (chromosome) is formulated. The minimization of the fitness function is carried out by employing Genetic algorithm (GA), Interior Point algorithm (IPA), and hybrid scheme combining GA and IPA (GA-IPA) for the optimal values of the chromosome. The efficiency of the presented method is testified by solving two examples from physiology. The results prove that the proposed heuristic method provides the approximate numerical solution comparable to some of the existing conventional numerical solutions.

References

  1. Kumar M. and Singh N., A collection of computational techniques for solving singular boundary-value problems, Adv. in Engg. Soft.,40, 288–297 (2009)
  2. Caglar H., Caglar N. and Özer M., B-spline solution of non-linear singular boundary value problems arising in physiology, Chaos, Solitons and Fractals39, 1232–1237 (2009)
  3. Ravi Kanth A.S.V., Bhattacharya V., Cubic spline for a class of non-linear singular boundary value problems arising in physiology, Appl. Math. Comput.,174 , 768–774 (2006)
  4. Alipanah A., Nonclassical pseudospectral method for a class of singular boundary value problems arising in physiology, Appl. Math., 2(2), 1-4 (2012)
  5. Khuri S.A., Sayfy A., A novel approach for the solution of a class of singular boundary value problems arising in physiology, Math. Comput. Model., 52, 626-636 (2010)
  6. Adam J. A., A simplified mathematical model of tumor growth, Math Biosci., 81(2) , 229-244 (1986)
  7. Greenspan H. P., Models for the growth of a solid tumor by diffusion, Stud. Appl. Math.,, 317–340 (1972)
  8. Lin H. S., Oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics, J. Theor. Biol., 60 449-457 (1976)
  9. McElwain D. L.S., A re-examination of oxygen diffusion in a spherical cell with Michaelis-Menten oxygen uptake kinetics, J. Theor. Biol., 71, 255-263 (1978)
  10. Anderson N. and Arthurs A. M., Complementary variational principles for diffusion problems with Michaelis-Menten Kinetics, Bull. math. Biol. 42, 131-135 (1980)
  11. Hiltmann P and Lory P, On oxygen diffusion in a spherical cell with Michaelis–Menten oxygen uptake kinetics, Bull. Math. Biol., 45, 661–664 (1983)
  12. Duggan R.C. and Goodman A.M., Point wise bounds for a nonlinear heat conduction model of the human head, Bull. Math. Biol., 48 229–236 (1986)
  13. Flesch U., The distribution of heat sources in the human head, a theoretical consideration, J. Theor. Biol.,54 285–287 (1975)
  14. Garner J.B. and Shivaji R., Diffusion problems with mixed nonlinear boundary condition, J. Math. Anal. Appl.,148 422–430 (1990)
  15. Rashidinia J., Mohammadi R. and Jalilian R., The numerical solution of non-linear singular boundary value problems arising in physiology, Appl. Math. Comput., 185, 360–367 (2007)
  16. Motsa S.S. and Sibanda P., A linearisation method for non-linear singular boundary value problems, Comput. Math. Appl.,63, 1197–1203 (2012)
  17. Arqub O.A., Abo-Hammour Z., Momani S. and Shawagfeh N., Solving singular two-point boundary value problems using continuous genetic algorithm, Abstract and Applied Analysis,(2012)
  18. Khan J.A., Raja M.A.Z. and Qureshi I.M., An application of evolutionary computational technique to non-linear singular system arising in polytrophic and isothermal sphere, Global J.Res. Engg., 12(1), 9-15 (2012)
  19. Malik S.A., Qureshi I.M., Zubair M. and Haq I., Solutionto force-free and forced Duffing-van der pol oscillator using memetic computing, J. Basic. Appl. Sci. Res., 2(11),, 11136-11148 (2012)
  20. Behrang, M.A., Ghalambaz, M., Assareh, E., and Noghrehabadi A.R., A new solution for natural convection of darcian fluid about a vertical full cone embedded in porous media prescribed wall temperature by using a hybrid neural network-particle swarm optimization method, World Acad. Sci. Engg. Tech., 49, 1098-1103 (2011)
  21. Grosan C. and Abraham A., Hybrid Evolutionary Algorithms: methodologies, architectures, and reviews, Stud. Comput. Intel.,75, 1–17 (2007)
  22. Fam D.F., Koh S.P., Tiong S.K. and Chong K.H., Qualitative analysis of stochastic operations in dual axis solar tracking environment, Res.J.Recent Sci., 1(9), 74-78 (2012)
  23. Mitchell M., Genetic algorithms: An overview, Complexity, , 31–39 (1995)
  24. Lesaja G., Introducing interior-point methods for introductory operations research courses and/or linear programming courses, The Open Operational Res. J., , 1-12 (2009)