Skip to main content
SpringerLink
Log in

A Legendre Approximation for Solving a Fuzzy Fractional Drug Transduction Model into the Bloodstream

  • Conference paper
Recent Advances on Soft Computing and Data Mining

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 287))

Abstract

While an increasing number of fractional order integrals and differential equations applications have been reported in the physics, signal processing, engineering and bioengineering literatures, little attention has been paid to this class of models in the pharmacokinetics-pharmacodynamic (PKPD) literature. In this research, we are confined with the application of Legendre operational matrix for solving fuzzy fractional differential equation arising in the drug delivery model into the bloodstream. The results illustrates the effectiveness of the method which can be in high agreement with the exact solution.

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 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
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. Oldham, K.B., Spainer, J.: The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic Press, New York (1974)

    Google Scholar 

  2. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    Google Scholar 

  3. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    Google Scholar 

  4. Hilfer, R.: Application of Fractional Calculus in Physics. World Scientific, Singapore (2000)

    Google Scholar 

  5. Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus Models and Numerical Methods. World Scientific Publishing Company (2012)

    Google Scholar 

  6. Golmankhaneh, A.K., Yengejeh, A.M., Baleanu, D.: On the fractional Hamilton and Lagrange mechanics. International Journal of Theoretical Physics 51, 2909–2916 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. Lim, S.C., Eab, C.H., Mak, K.H., Li, M., Chen, S.Y.: Solving linear coupled fractional differential equations by direct operationalmethod and some applications. Mathematical Problems in Engineering 2012, 1–28 (2012)

    Article  MathSciNet  Google Scholar 

  8. Pedas, A., Tamme, E.: On the convergence of spline collocation methods for solving fractional differential equations. Journal of Computational and Applied Mathematics 235, 3502–3514 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  9. Jiang, W.H.: Solvability for a coupled system of fractional differential equations at resonance. Nonlinear Analysis: Real World Applications 13, 2285–2292 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  10. Jiang, H., Liu, F., Turner, I., Burrage, K.: Analytical solutions for the multi-term time-space Caputo–Riesz fractional advection–diffusion equationson a finite domain. J. Math. Anal. Appl. 389, 1117–1127 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  11. Jiang, H., Liu, F., Turner, I., Burrage, K.: Analytical solutions for the generalized multi-term time-fractional diffusion-wave/diffusion equation in a finite domain. Comput. Math. Appl. 64, 3377–3388 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  12. Momani, S., Shawagfeh, N.T.: Decomposition method for solving fractional Riccati differential equations. Appl. Math. Comput. 182, 1083–1092 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  13. Song, L., Wang, W.: A new improved Adomian decomposition method and its application to fractional differential equations. Appl. Math. Model. 37, 1590–1598 (2013)

    Article  MathSciNet  Google Scholar 

  14. Odibat, Z., Momani, S., Xu, H.: A reliable algorithm of homotopy analysis method for solving nonlinear fractional differential equations. Appl. Math. Model. 34, 593–600 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hashim, I., Abdulaziz, O., Momeni, S.: Homotopy analysis method for fractional IVPs. Commun. Nonlinear. Sci. Numer. Simul. 14, 674–684 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  16. Doha, E.H., Bhrawy, A.H., Ezz-Eldien, S.S.: Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations. Appl. Math. Model. 35, 5662–5672 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  17. Bhrawy, A.H., Alofi, A.S., Ezz-Eldien, S.S.: A quadrature tau method for variable coefficients fractional differential equations. Appl. Math. Lett. 24, 2146–2152 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  18. Saadatmandi, A., Dehghan, M.: A new operational matrix for solving fractional-order differential equations. Comput. Math. Appl. 59, 1326–1336 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  19. Doha, E.H., Bhrawy, A.H., Ezz-Eldien, S.S.: A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order. Comput. Math. Appl. 62, 2364–2373 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  20. Agarwal, R.P., Lakshmikantham, V., Nieto, J.J.: On the concept of solution for fractional differential equations with uncertainty. Nonlinear Anal. 72, 2859–2862 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  21. Allahviranloo, T., Salahshour, S., Abbasbandy, S.: Explicit solutions of fractional differential equations with uncertainty. Soft Comput. 16, 297–302 (2012)

    Article  MATH  Google Scholar 

  22. Allahviranloo, T., Gouyandeh, Z., Armand, A.: Fuzzy fractional differential equations under generalized fuzzy Caputo derivative. J. Intelligent and Fuzzy Systems (2013), doi:10.3233/IFS-130831

    Google Scholar 

  23. Salahshour, S., Allahviranloo, T., Abbasbandy, S., Baleanu, D.: Existence and uniqueness results for fractional differential equations with uncertainty. Advances in Difference Equations 2012, 112 (2012)

    Article  MathSciNet  Google Scholar 

  24. Salahshour, S., Allahviranloo, T., Abbasbandy, S.: Solving fuzzy fractional differential equations by fuzzy Laplace transforms. Commun. Nonlinear. Sci. Numer. Simulat. 17, 1372–1381 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  25. Ahmadian, A., Suleiman, M., Salahshour, S., Baleanu, D.: A Jacobi operational matrix for solving fuzzy linear fractional differential equation. Adv. Difference Equ. 2013, 104 (2013)

    Article  MathSciNet  Google Scholar 

  26. Ahmadian, A., Suleiman, M., Salahshour, S.: An Operational Matrix Based on Legendre Polynomials for Solving Fuzzy Fractional-Order Differential Equations. Abstract and Applied Analysis 2013, Article ID 505903, 29 pages (2013)

    Google Scholar 

  27. Balooch Shahriyar, M.R., Ismail, F., Aghabeigi, S., Ahmadian, A., Salahshour, S.: An Eigenvalue-Eigenvector Method for Solving a System of Fractional Differential Equations with Uncertainty. Mathematical Problems in Engineering 2013, Article ID 579761, 11 pages (2013)

    Google Scholar 

  28. Ghaemi, F., Yunus, R., Ahmadian, A., Salahshour, S., Suleiman, M., Faridah Saleh, S.: Application of Fuzzy Fractional Kinetic Equations to Modelling of the Acid Hydrolysis Reaction. Abstract and Applied Analysis 2013, Article ID 610314, 19 pages (2013)

    Google Scholar 

  29. Ahmadian, A., Senu, N., Larki, F., Salahshour, S., Suleiman, M., Shabiul Islam, M.: Numerical solution of fuzzy fractional pharmacokinetics model arising from drug assimilation into the blood stream. Abstract and Applied Analysis 2013, Article ID 304739 (2013)

    Google Scholar 

  30. Mazandarani, M., Vahidian Kamyad, A.: Modified fractional Euler method for solving Fuzzy Fractional Initial Value Problem. Commun. Nonlinear Sci. Numer. Simulat. 18, 12–21 (2013)

    Article  MATH  Google Scholar 

  31. Mazandarani, M., Najariyan, M.: Type-2 Fuzzy Fractional Derivatives. Commun. Nonlinear Sci. Numer. Simulat. (2013), doi:10.1016/j.cnsns.2013.11.003

    Google Scholar 

  32. Wu, H.C.: The improper fuzzy Riemann integral and its numerical integration. Information Science 111, 109–137 (1999)

    Article  Google Scholar 

  33. Dubios, D., Prade, H.: Towards fuzzy differential calculus-part3. Fuzzy Sets and Systems 8, 225–234 (1982)

    Article  MathSciNet  Google Scholar 

  34. Anastassiou, G.A.: Fuzzy Mathematics: Approximation Theory. STUDFUZZ, vol. 251. Springer, Heidelberg (2010)

    Book  Google Scholar 

  35. Anastassiou, G.A., Gal, S.: On a fuzzy trigonometric approximation theorem of Weierstrass-type. J. Fuzzy Math. 9, 701–708 (2001)

    MATH  MathSciNet  Google Scholar 

  36. Chalco-Cano, Y., Román-Flores, H.: On new solutions of fuzzy differential equations. Chaos, Solitons & Fractals 38, 112–119 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  37. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, New York (1989)

    Google Scholar 

  38. Bonate, P.L.: Pharmacokinetic-pharmacodynamic modeling and simulation. Springer (2011)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Mathematical Research (INSPEM), Universiti Putra Malaysia, UPM, 43400, Serdang, Selangor, Malaysia

    Ali Ahmadian, Norazak Senu & Mohamed Suleiman

  2. Institute of Microengineering and Nanoelectronics (IMEN), Universiti Kebangsan Malaysia (UKM), 43600 UKM, Bangi, Selangor, Malaysia

    Farhad Larki & Md. Shabiul Islam

  3. Young Researchers and Elite Club, Mobarakeh Branch, Islamic Azad University, Mobarakeh, Iran

    Soheil Salahshour

Authors
  1. Ali Ahmadian
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Norazak Senu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Farhad Larki
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Soheil Salahshour
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Mohamed Suleiman
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Md. Shabiul Islam
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ali Ahmadian .

Editor information

Editors and Affiliations

  1. Department of Information System Faculty of Comp. Sci. & Info. Tech., University of Malaya, Kuala Lumpur, Malaysia

    Tutut Herawan

  2. Faculty of Comp. Sci. and Info. Tech, Universiti Tun Hussein Onn Malaysia, Parit Raja, Malaysia

    Rozaida Ghazali

  3. Faculty of Comp. Sci. and Info. Tech., Universiti Tun Hussein Onn Malaysia, Parit Raja, Malaysia

    Mustafa Mat Deris

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Ahmadian, A., Senu, N., Larki, F., Salahshour, S., Suleiman, M., Islam, M.S. (2014). A Legendre Approximation for Solving a Fuzzy Fractional Drug Transduction Model into the Bloodstream. In: Herawan, T., Ghazali, R., Deris, M. (eds) Recent Advances on Soft Computing and Data Mining. Advances in Intelligent Systems and Computing, vol 287. Springer, Cham. https://doi.org/10.1007/978-3-319-07692-8_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-07692-8_3

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-07691-1

  • Online ISBN: 978-3-319-07692-8

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation