Skip to main content
SpringerLink
Log in

Application of Geometric Explicit Runge–Kutta Methods to Pharmacokinetic Models

  • Conference paper
Modeling and Simulation in Engineering, Economics and Management (MS 2012)

Part of the book series: Lecture Notes in Business Information Processing ((LNBIP,volume 115))

Abstract

In an earlier work, the authors proposed a class of geometric explicit Runge– Kutta methods for solving one-dimensional first order Initial Value Problems (IVPs). In this work, some members of this class of schemes which were found to be more accurate are applied to systems of first order ordinary differential equations (ODEs). We present the development of these selected schemes and also study their basic properties vis-a-vis systems of ODEs. We then apply this approach to solve some mathematical models arising in Pharmacokinetics.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 54.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Akanbi, M.A.: On 3–stage Geometric Explicit Runge–Kutta Method for Singular Autonomous Initial Value Problems in Ordinary Differential Equations. Computing 92, 243–263 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Allen, L.J.S.: An Introduction To Mathematical Biology. Pearson Education, Inc., Upper Saddle River (2007)

    Google Scholar 

  3. Evans, D.J., Sanugi, B.B.: A new fourth order Runge-Kutta method for inilia/value problems. In: Fatunla, S.O. (ed.) Computat. Math. II. Boole Press (1986)

    Google Scholar 

  4. Evans, D.J., Sanugi, B.B.: A new 4th order Runge-Kutta formula for y = Ay with stepwise control. Camp. Math. Applic. 15, 991–995 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  5. Evans, D.J.: New Runge-Kutta methods for initial value problems. Appl. Math. Letters 2, 25–28 (1989)

    Article  MATH  Google Scholar 

  6. Evans, D.J., Yaacob, N.B.: A New Fourth Order Runge-Kutta Formula Based On Harmonic Mean. Department of Computer Studies. Loughborough University of Technology, Loughborough (1993)

    Google Scholar 

  7. Evans, D.J., Yaacob, N.B.: A Fourth order Runge-Kutta Method Based on the Heronian Mean Formula. Intern. J. Comput. Math. 58, 103–115 (1995)

    Article  MATH  Google Scholar 

  8. Evans, D.J., Yaacob, N.B.: A new Runge Kutta RK (4,4) method. Intern. J. Comput. Math. 58, 169–187 (1995)

    Article  MATH  Google Scholar 

  9. Evans, D.J., Yaacob, N.B.: A Fourth Order Runge-Kurla Method Based On The Heronian Mean. Intern. J. Computer Math. 59, 12 (1995)

    Article  Google Scholar 

  10. Griffiths, D.F., Higham, D.J.: Numerical Methods for Ordinary Differential Equations. Springer Undergraduate Mathematics Series. Springer, London (2010), doi:10.1007/978-0-85729-148-6

    Book  MATH  Google Scholar 

  11. Heun, K.: Neue Methoden zur approximativen Integration der Differential-gleichungen einer unabhangigen Veranderlichen. Z. Math. Phys. 45, 23–38 (1900)

    Google Scholar 

  12. Hǔta, A.: Une amelioration de la methode de Runge-Kutta-Nyström pour la resolution numerique des equations differentielles du premier ordre. Ada Fac. Nat. Univ. Comenian. Math. 1, 201–224 (1956)

    Google Scholar 

  13. Kutta, W.: Beitrag zur Naherungs-weissen Integration tolaken Differential-gleichungen. Z. Maths Phys. 46, 435–453 (1901)

    Google Scholar 

  14. Murugesan, K., Paul Dhayabaran, D.P., Amirtharaj, E.C.H., Evans, D.J.: A fourth order Embedded Runge-Kutta RKCeM(4,4) Method based on Arithmetic and Centroidal Means with Error Control. International J. Comput. Math. 79(2), 247–269 (2002)

    Article  MATH  Google Scholar 

  15. Nestorov, I.: Whole body Pharmacokinetic Models. Clinical Pharmacokinetic 42(10) (2003)

    Google Scholar 

  16. Nyström, E.J.: Uber die numerische Integration von Differentialgleichungen. Ada Soc. Sci. Fennicae 50(13), 55 (1925)

    Google Scholar 

  17. Pang, K.S., Weiss, M., Macheras, P.: Advanced Pharmacokinetic Models Based on Organ Clearance. Circulatory, and Fractal Concepts The AAPS Journal 9(2), Article 30, E268–E283 (2007)

    Google Scholar 

  18. Ponalagusamy, R., Senthilkumar, S.: A comparison of RK-fourth orders of variety of means on multilayer raster CNN simulation. Trends in Applied Science and Research 3(3), 242–252 (2008)

    Article  Google Scholar 

  19. Ponalagusamy, R., Senthilkumar, S.: A New Fourth Order Embedded RKAHeM(4,4) Method with Error Control on Multilayer Raster Cellular Neural Network. Signal Image and Video Processing (2008) (accepted in press)

    Google Scholar 

  20. Razali, N., Ahmad, R.: New Fifth-Order Runge-Kutta Methods for Solving Ordinary Differential Equation. Proceeding of Seminar on Engineering Mathematics 2, 155–162 (2008)

    Google Scholar 

  21. Runge, C.: Uber die numerische Auflosung von differntialglechungen. Math. Ann. 46, 167–178 (1895)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sanugi, B.B., New Numerical Strategies for Initial Value Type Ordinary Differential Equations, Ph.D. Thesis, Loughborough University of Technology, U.K. (1986)

    Google Scholar 

  23. Sanugi, B.B.: Ph.D. Thesis Loughborough University of Technology (1986)

    Google Scholar 

  24. Sanugi, B.B., Evans, D.J.: A New Fourth Order Runge-Kutta method based on Harmonic Mean. Comput. Stud. Rep., LUT (June 1993)

    Google Scholar 

  25. Sanugi, B.B., Evans, D.J.: A new fourth order Runge-Kutta formulae based on harmonic mean. Intern. J. Comput. Math. 50, 113–118 (1994)

    Article  MATH  Google Scholar 

  26. Sanugi, B.B., Yaacob, N.B.: A new fifth order Runge-Kurra method for Initial Value type problems in ODEs. Intern. J. Comput. Math. 59, 187–207 (1995)

    Article  Google Scholar 

  27. Shonkwiler, R.W., Herod, J.: Mathematical Biology: An Introduction with Maple and Matlab, 2nd edn. Undergraduate Texts in Mathematics. Springer, Heidelberg (2009)

    MATH  Google Scholar 

  28. Wazwaz, A.M.: A modified third order Runge-Kutta method. Appl. Math. Letter 3, 123–125 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wazwaz, A.M.: A Comparison of Modified Runge-Kutta formulas based on Variety of means. Intern. J. Comput. Math. 50, 105–112 (1994)

    Article  MATH  Google Scholar 

  30. Yaacob, N.B., Evans, D.J.: A new fourth order Runge-Kutta method based on the Root Mean Square formula, Computer Studies, Report, 862, Louborough University of Technology, U.K (December 1993)

    Google Scholar 

  31. Yaacob, N.B., Sanugi, B.B.: A New Fifth-Order Five-Stage Explicit HaM-RK5(5) Method For Solving Initial Value Problems in ODEs. Laporan Teknik. Jab. Mat., UTM (1995)

    Google Scholar 

  32. Yaacob, N.B., Sanugi, B.B.: A New Fourth-Order Embedded Method Based on the Harmonic mean. Mathematika, Jilid, hml. 1–6 (1998)

    Google Scholar 

  33. Yaacob, N.B., Evans, D.J.: New Runge-Kutta Starters for Multi-step Methods. Intern. J. Comput. Math. 71, 99–104 (1999)

    Article  MathSciNet  Google Scholar 

  34. Yaacob, N.B., Evans, D.J.: A fourth order Runge-Kutta RK(4,4) method with Error Control. Intern. J. Comput. Math. 71, 383–411 (1999)

    Article  MathSciNet  Google Scholar 

  35. Yeargers, E.K., Shonkwiler, R.W., Herod, J.V.: An Introduction to the Mathematics of Biology, Birkhau

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Lagos State University, P.M.B. 0001 LASU Post Office, Lagos, Nigeria

    Moses A. Akanbi

  2. Department of Mathematics & Applied Mathematics, University of the Western Cape, Private Bag X17, Bellville, 7535, Cape Town, South Africa

    Kailash C. Patidar

Authors
  1. Moses A. Akanbi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kailash C. Patidar
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Hagan School of Business, Iona College, New Rochelle, NY, USA

    Kurt J. Engemann

  2. Department of Business Administration, University of Barcelona, Barcelona, Spain

    Anna M. Gil-Lafuente  & José M. Merigó  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Akanbi, M.A., Patidar, K.C. (2012). Application of Geometric Explicit Runge–Kutta Methods to Pharmacokinetic Models. In: Engemann, K.J., Gil-Lafuente, A.M., Merigó, J.M. (eds) Modeling and Simulation in Engineering, Economics and Management. MS 2012. Lecture Notes in Business Information Processing, vol 115. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30433-0_26

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-30433-0_26

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-30432-3

  • Online ISBN: 978-3-642-30433-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation