Skip to main content

An Intermediate Nonpolynomial Spline Algorithm for Second Order Nonlinear Differential Problems: Applications to Physiology and Thermal Explosion

  • Conference paper
  • First Online:
Proceedings of Fourth International Conference on Soft Computing for Problem Solving

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 336))

  • 1092 Accesses

Abstract

In this paper, a new nonpolynomial spline scheme based on intermediate stencils for the numerical solution of nonlinear two point boundary value problems is considered. The scheme is compact and applicable to both singular and nonsingular equations. The new scheme can achieve fourth-order accuracy and provides rapidly convergent solution. The convergence analysis of the present method is briefly discussed. Numerical results are shown in terms of maximum absolute errors. The computational illustrations demonstrate reliability, simplicity and efficiency of the proposed method.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Kubicek, M., Hlavacek, V.: Numerical Solutions of Nonlinear Boundary Value Problems with Applications. Dover Publications, New York (2008)

    Google Scholar 

  2. Parter, S.V.: Solution of a differential arising in chemical reactor process. SIAM J. Appl. Math. 26, 687–716 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  3. Aris, R.: The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts. Clarendon Press, Oxford (1975)

    Google Scholar 

  4. Frank-Kamenetskii, D.A.: Diffusion and Heat Transfer in Chemical Kinetics. Plenum Press, Berlin (1969)

    Google Scholar 

  5. Duggan, R.C., Goodman, A.M.: Point wise bounds for a nonlinear heat conduction model for human head. Bull. Math. Biol. 48(2), 229–236 (1989)

    Article  Google Scholar 

  6. Bebernes, J.W., Eberly, D.: Mathematical Problems from Combustion Theory. Springer, Berlin (1989)

    Google Scholar 

  7. Flesh, U.: The distribution of heat sources in the human head: a theoretical consideration. J. Theor. Biol. 54, 285–287 (1975)

    Article  Google Scholar 

  8. Ozisik, M.N.: Boundary value problems of heat conduction. Dover Publications, New York (1989)

    Google Scholar 

  9. Chawla, M.M., Shivkumar, P.N.: An efficient finite difference method for two point boundary value problems. Neural Parallel Sci. Comput. 4, 387–396 (1996)

    MATH  MathSciNet  Google Scholar 

  10. Jha, N., Mohanty, R.K.: TAGE iterative algorithm and nonpolynomial spline basis for the solution of nonlinear singular second order ordinary differential equations. Appl. Math. Comput. 218, 3289–3296 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  11. Simos, T.E.: Exponentially fitted and trigonometrically fitted symmetric linear multistep methods for the numerical integration of orbital problems. Phys. Lett. A 315, 437–446 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  12. Bulatov, M.V., Berghe, G.V.: Two step fourth order methods for linear ODEs of the second order. Numer. Algorithms 51(4), 449–460 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hoffman, J.D.: Numerical Methods for Engineers and Scientists. CRC Press, Boca Raton (2001)

    Google Scholar 

  14. Bieniasz, L.K.: Two new compact finite difference schemes for the solution of boundary value problems in second order nonlinear ordinary differential equations, using non uniform grids. J. Comput. Meth. Sci. Eng. 8, 3–18 (2008)

    MATH  MathSciNet  Google Scholar 

  15. Mohanty, R.K.: A class of non-uniform mesh three point arithmetic average discretizations for y″ = f(x, y, y′) and the estimates of y′. Appl. Math. Comput. 183, 477–485 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kanth, A.S.R.V.: Cubic spline polynomial for nonlinear singular two point boundary value problems. Appl. Math. Comput. 189, 2017–2022 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  17. Lin, B., Li, K., Cheng, Z.: B-spline solution of a singularly perturbed boundary value problems arising in biology. Chaos, Solitons Fractals 42(5), 2934–2948 (2009)

    Article  MATH  Google Scholar 

  18. Ding, H., Zhang, Y., Cao, J., Tian, J.: A class of difference scheme for solving telegraph equation by new non polynomial spline methods. Appl. Math. Comput. 218, 4671–4683 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  19. Vigo-Aguiar, J., Simos, T.E.: An exponentially fitted and trigonometrically fitted method for the numerical solution of orbital problems. Astron. J. 122, 1656–1660 (2001)

    Article  Google Scholar 

  20. Ha, S.N.: A nonlinear shooting method for two point boundary value problems. Comput. Math Appl. 42, 1411–1420 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  21. Islam, S., Ikram, A., Tirmizi, I.A.: Nonpolynomial spline approach to the solution of a system of second order boundary value problems. Appl. Math. Comput. 173, 1208–1218 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  22. Rashidinia, J., Mohammadi, R., Jalilian, R.: The numerical solution of non-linear singular boundary value problems arising in physiology. Appl. Math. Comput. 185, 360–367 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  23. Pandey, R.K., Singh, A.K.: On the convergence of a fourth order method for a class of singular boundary value problems. J. Comput. Appl. Math. 224, 734–742 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  24. Wang, Y.M.: On Numerov’s method for a class of strongly nonlinear two point boundary value problems. Appl. Numer. Math. 61, 38–52 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  25. Gautschi, W.: Numerical Analysis. Springer, Berlin (2011)

    Google Scholar 

  26. Verga, R.S.: Matrix Iterative Analysis. Springer, Berlin (2000)

    Google Scholar 

  27. Henrici, P.: Discrete Variable Methods in Ordinary Differential Equations. Wiley, New York (1962)

    MATH  Google Scholar 

  28. Young, D.M.: Iterative Solution of Large Linear Systems. Academic Press, New York (1971)

    MATH  Google Scholar 

  29. Caglar, H., Caglar, N., Ozer, M.: B-spline solution of nonlinear singular boundary value problems arising in physiology. Chaos, Solitons Fractals 39, 1232–1237 (2009)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Navnit Jha .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer India

About this paper

Cite this paper

Jha, N. (2015). An Intermediate Nonpolynomial Spline Algorithm for Second Order Nonlinear Differential Problems: Applications to Physiology and Thermal Explosion. In: Das, K., Deep, K., Pant, M., Bansal, J., Nagar, A. (eds) Proceedings of Fourth International Conference on Soft Computing for Problem Solving. Advances in Intelligent Systems and Computing, vol 336. Springer, New Delhi. https://doi.org/10.1007/978-81-322-2220-0_52

Download citation

  • DOI: https://doi.org/10.1007/978-81-322-2220-0_52

  • Published:

  • Publisher Name: Springer, New Delhi

  • Print ISBN: 978-81-322-2219-4

  • Online ISBN: 978-81-322-2220-0

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics