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Hanta-virüs Modelinden Elde Edilen Lojistik Diferansiyel Denklem

Year 2016, Volume: 11 Issue: 1, 82 - 91, 24.05.2016

Abstract

Kesirli mertebeden Hanta-virüs modeli olarak alınan lineer olmayan diferansiyel denklem sistemi
〖(_c^)D〗_(0,t)^α X(t)=(b-c)X(t)+bY(t)-(X^2 (t))/K-((1+aK)/K)X(t)Y(t)
〖(_c^)D〗_(0,t)^α Y(t)=-cY(t)-(Y^2 (t))/K-((1-aK)/K)X(t)Y(t)

(1)
şeklinde tanımlanmıştır. Burada 〖(_c^)D〗_(0,t)^α kesirli türev (Caputo) operatörünü göstermektedir. (1) sistemini ayrıklaştırmak için Grünwald-Letnikov türev operatörü ve Standart Olmayan Sonlu Farklar (SOSF) Yöntemi uygulanacaktır. (1) sistemindeki bazı düzenlemeler ile kesirli mertebeden Lojistik denklem elde edilip, bulgular bazı grafikler ve tablolar yardımı ile desteklenecektir.
Anahtar kelimeler: Kesirli Diferansiyel Denklem, Hanta-virüs, Lojistik Diferansiyel Denklem.

Logistic Differential Equations Obtained from Hanta-virus Model

Abstract: Fractional-order Hanta-virüs Model as received nonlinear differential equation system is
〖(_c^)D〗_(0,t)^α X(t)=(b-c)X(t)+bY(t)-(X^2 (t))/K-((1+aK)/K)X(t)Y(t)
〖(_c^)D〗_(0,t)^α Y(t)=-cY(t)-(Y^2 (t))/K-((1-aK)/K)X(t)Y(t)

(1)
Here, 〖(_c^)D〗_(0,t)^α denotes the fractional derivative (Caputo) operator. The Grünwald-Letnikov operator and Nonstandart Finite Diference (SOSF) schemes will be applied to discretize the fractional-order nonlinear system (1). Fractional order logistic equation optioned with some adjustments in (1) system. The findings will be supported with the help of some of the graphs and tables.
Key words: Fractional diferantial equation, Hanta-virus, Logistic Differential Equation.

References

  • [1] Abramson G., Kenkre V.M., 2002. Spatio-temporal patterns in the Hantavirus infected, Physcial Rewiew E, 66, 011912.
  • [2] Mickens,R.E., 2007. Calculation of denominator functions for nonstandart finite difference schemes for differential equations satisfying a positivity condition, Numerical Methods for Partial Differential Equations, 23 (3) : 672-691.
  • [3] Chen M., Clemence D.P.,2006. Stability properties of a nonstandard finite difference scheme for a hantavirus epidemic model, Journal of Difference Equations and Applications, 12 (12), 1243- 1256.
  • [4] Chen M., Clemence D.P., 2007. Analysis of and numerical schemes for a mouse population model in Hantavirus epidemics, Journal of Difference Equations and Applications, 12 (9), 887-899.
  • [5] Abdullah F.A.,2011, Simulations of the spread of the Hantavirus using fractional differential equations, Matematika, 27, 149-158
  • [6] Matignon D.,1996, Stability results for fractional differential equations with applications to control processing. Computational Engineering in Systems Applications, 2: 817-823.
  • [7] Lubich CH.,1986. Discretized Fractional Calculus, SIAM Journal on Mathematical Analysis (SIMA) 17 (3): 704-719.
  • [8] Podlubny I., 1999. Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications. Academic Press, San Diego-Boston-New York-London-Tokyo-Toronto, p.368.
Year 2016, Volume: 11 Issue: 1, 82 - 91, 24.05.2016

Abstract

References

  • [1] Abramson G., Kenkre V.M., 2002. Spatio-temporal patterns in the Hantavirus infected, Physcial Rewiew E, 66, 011912.
  • [2] Mickens,R.E., 2007. Calculation of denominator functions for nonstandart finite difference schemes for differential equations satisfying a positivity condition, Numerical Methods for Partial Differential Equations, 23 (3) : 672-691.
  • [3] Chen M., Clemence D.P.,2006. Stability properties of a nonstandard finite difference scheme for a hantavirus epidemic model, Journal of Difference Equations and Applications, 12 (12), 1243- 1256.
  • [4] Chen M., Clemence D.P., 2007. Analysis of and numerical schemes for a mouse population model in Hantavirus epidemics, Journal of Difference Equations and Applications, 12 (9), 887-899.
  • [5] Abdullah F.A.,2011, Simulations of the spread of the Hantavirus using fractional differential equations, Matematika, 27, 149-158
  • [6] Matignon D.,1996, Stability results for fractional differential equations with applications to control processing. Computational Engineering in Systems Applications, 2: 817-823.
  • [7] Lubich CH.,1986. Discretized Fractional Calculus, SIAM Journal on Mathematical Analysis (SIMA) 17 (3): 704-719.
  • [8] Podlubny I., 1999. Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications. Academic Press, San Diego-Boston-New York-London-Tokyo-Toronto, p.368.
There are 8 citations in total.

Details

Primary Language Turkish
Subjects Mathematical Sciences
Journal Section Makaleler
Authors

Zarife Gökçen Karadem This is me

Mevlüde Yakıt Ongun

Publication Date May 24, 2016
Published in Issue Year 2016 Volume: 11 Issue: 1

Cite

IEEE Z. G. Karadem and M. Yakıt Ongun, “Hanta-virüs Modelinden Elde Edilen Lojistik Diferansiyel Denklem”, Süleyman Demirel University Faculty of Arts and Science Journal of Science, vol. 11, no. 1, pp. 82–91, 2016.