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Diferansiyel Dönüşüm Yöntemi İle Rastgele Kalp Atış Modelinin Analizi

Year 2019, Volume: 40 Issue: 2, 285 - 298, 30.06.2019

Pınar Oral Mehmet Merdan , Zafer Bekiryazıcı

https://doi.org/10.17776/csj.460984
https://izlik.org/JA48DB56AM

Abstract

Bu çalışmada, rastgele Zeeman Kalpatış Modelinin incelenmesi için
diferansiyel dönüşüm yöntemi kullanılmıştır. Modelin bazı parametreleri ve
başlangıç koşulları sırasıyla Beta ve Normal dağılımlara sahip rastgele
değişkenler olarak alınmıştır. Rastgele Zeeman Modelinin yaklaşık analitik
çözümü elde edilmiş ve model bileşenlerinin beklenen değer ve varyansı elde
edilmiştir. Beta ve normal olarak dağılmış rastgele etkiler altında, rastgele
modellerin sonuçları karşılaştırılmış ve bu durumlar için elde edilen yaklaşık
sayısal karakteristikler karşılaştırılmıştır. Elde edilen yaklaşık formüllere,
yaklaşımların yakınsama aralığını artırmak için Laplace-Padé Metodu uygulanarak
iyileştirilmiş çözümler bulunmuştur.

Keywords

Zeeman Kalpatış Modeli Rastgele diferansiyel denklem Beklenen değer Varyans Padé yaklaşımı

References

  • [1] Merdan M. and Khaniyev T., On the Behavior of Solutions Under the Influence of Stochastic Effect of Avian-Human Influenza Epidemic Model, International Journal of Biotechnology and Biochemistry, 4-1 (2008) 75-100.
  • [2] Merdan M., Bekiryazici Z., Kesemen T. and Khaniyev T., Comparison of stochastic and random models for bacterial resistance. Advances in Difference Equations, 2017-1 (2017) 133.
  • [3] Zeeman E.C., Differential Equations for the Heartbeat and Nerve Impulse, Mathematics Institute, University of Warwick, Coventry, UK, 1972.
  • [4] Zeeman E.C., Catastrophe Theory, Selected Papers 1972–1977, Addison-Wesley, Reading, MA, 1977.
  • [5] Zeeman E.C., Differential equations for the heartbeat and nerve impulse, In: Waddington, C.H. (Ed.), Towards a Theoretical Biology 4: Essays, 8–67, Edinburgh University Press, 1972.
  • [6] Bekiryazici Z., Merdan M., Kesemen T. and Khaniyev T., Mathematical Modeling of Biochemical Reactions under Random Effects, Turkish Journal of Mathematics and Computer Science, 5 (2016) 8-18.
  • [7] Zhou J.K., Differential Transformation and its Applications for Electrical Circuits, Huarjung University Press, Wuuhahn, China, 1986.
  • [8] Ev Pukhov G.G., Differential Transforms and Circuit theory, Int. J. Circuit Theory Appl., 10-3 (1982) 265–276.
  • [9] Forbes C., Evans M., Hastings N. and Peacock B., Statistical Distributions, 4th Edition, John Wiley & Sons, New Jersey, 2011.
  • [10] Villafuerte L. and Chen-Charpentier B.M., A random differential transform method: Theory and applications, Applied Mathematics Letters, 25-10 (2012) 1490-1494.
  • [11] Khudair A.R., Haddad S.A.M. and Khalaf S.L., Mean square solutions of second-order random differential equations by using the differential transformation method, Open Journal of Applied Sciences, 6-4 (2016) 287-297.
  • [12] Villafuerte L. and Cortés J.-C., Solving Random Differential Equations by Means of Differential Transform Methods, Advances in Dynamical Systems and Applications, 8-2 (2013) 413-425.
  • [13] Calbo G., Cortés J.-C., Jódar L. and Villafuerte L., Solving the random Legendre differential equation: mean square power series solution and its statistical functions, Comput. Math. Appl., 61-9 (2011) 2782–2792.
  • [14] Villafuerte L., Braumann C.A., Cortés J.-C. and Jódar L., Random differential operational calculus: Theory and applications, Computers and Mathematics with Applications, 59-1 (2010) 115-125.
  • [15] Gökdoğan A., Merdan M. and Yildirim A., The Modified Algorithm for the Differential Transform Method to Solution of Genesio Systems, Communications in Nonlinear Science and Numerical Simulation, 17-1 (2012) 45-51.
  • [16] Benhammouda B. and Vazquez-Leal H., Analytical solutions for systems of partial differential–algebraic equations, SpringerPlus, 3-1 (2014) 137.
  • [17] Sungu, I.C. and Demir, H., A Computational Method for the Time-Fractional Navier-Stokes Equation, Cumhuriyet Science Journal, 39-4 (2018) 900-911.
  • [18] Merdan, M., Bekiryazici, Z., Kesemen, T. and Khaniyev, T., Analyzing the dynamics of Ebola transmission with random effects. Communications in Mathematical Biology and Neuroscience, 2018 (2018) Article-ID 22.
  • [19] Mehdi Rashidi, M. andErfani, E., The modified differential transform method for investigating nano boundary-layers over stretching surfaces. International Journal of Numerical Methods for Heat & Fluid Flow, 21-7 (2011) 864-883.
  • [20] Onwubuoya, C., Nwanze, D. E., Erejuwa, J. S. and Akinyemi, S. T., An Approximate Solution of a Computer Virus Model with Antivirus using Modifed Differential Transform Method, International Journal of Engineering Research & Technology, 7-4 (2018) 154-161.
  • [21] Bekiryazici, Z., Bazi Kompartmanli Modellerin Rastgele Ektiler Altinda Incelenmesi, PhD Thesis, Karadeniz Technical University, Trabzon, 2017.

Analysis of a Random Zeeman Heartbeat Model with Differential Transformation Method

Year 2019, Volume: 40 Issue: 2, 285 - 298, 30.06.2019

Pınar Oral Mehmet Merdan , Zafer Bekiryazıcı

https://doi.org/10.17776/csj.460984
https://izlik.org/JA48DB56AM

Abstract

In this paper, the differential transformation method is used
to examine the random Zeeman Heartbeat Model. Some of the parameters and the
initial conditions of the model are taken as random variables with Beta and
Normal distributions, respectively. The approximate analytical solution of the
random Zeeman Model is obtained and used to find the expectation and variance
of the model components. The results from the random models including Beta and
normal distributed random effects are compared and the approximate numerical
characteristics are obtained for these cases. The approximate formulas are also
modified by using Laplace-Padé Method to increase the convergence interval of
the approximations.

Keywords

Zeeman Heartbeat Model Variance Random differential equation Expected value Padé approximation

References

  • [1] Merdan M. and Khaniyev T., On the Behavior of Solutions Under the Influence of Stochastic Effect of Avian-Human Influenza Epidemic Model, International Journal of Biotechnology and Biochemistry, 4-1 (2008) 75-100.
  • [2] Merdan M., Bekiryazici Z., Kesemen T. and Khaniyev T., Comparison of stochastic and random models for bacterial resistance. Advances in Difference Equations, 2017-1 (2017) 133.
  • [3] Zeeman E.C., Differential Equations for the Heartbeat and Nerve Impulse, Mathematics Institute, University of Warwick, Coventry, UK, 1972.
  • [4] Zeeman E.C., Catastrophe Theory, Selected Papers 1972–1977, Addison-Wesley, Reading, MA, 1977.
  • [5] Zeeman E.C., Differential equations for the heartbeat and nerve impulse, In: Waddington, C.H. (Ed.), Towards a Theoretical Biology 4: Essays, 8–67, Edinburgh University Press, 1972.
  • [6] Bekiryazici Z., Merdan M., Kesemen T. and Khaniyev T., Mathematical Modeling of Biochemical Reactions under Random Effects, Turkish Journal of Mathematics and Computer Science, 5 (2016) 8-18.
  • [7] Zhou J.K., Differential Transformation and its Applications for Electrical Circuits, Huarjung University Press, Wuuhahn, China, 1986.
  • [8] Ev Pukhov G.G., Differential Transforms and Circuit theory, Int. J. Circuit Theory Appl., 10-3 (1982) 265–276.
  • [9] Forbes C., Evans M., Hastings N. and Peacock B., Statistical Distributions, 4th Edition, John Wiley & Sons, New Jersey, 2011.
  • [10] Villafuerte L. and Chen-Charpentier B.M., A random differential transform method: Theory and applications, Applied Mathematics Letters, 25-10 (2012) 1490-1494.
  • [11] Khudair A.R., Haddad S.A.M. and Khalaf S.L., Mean square solutions of second-order random differential equations by using the differential transformation method, Open Journal of Applied Sciences, 6-4 (2016) 287-297.
  • [12] Villafuerte L. and Cortés J.-C., Solving Random Differential Equations by Means of Differential Transform Methods, Advances in Dynamical Systems and Applications, 8-2 (2013) 413-425.
  • [13] Calbo G., Cortés J.-C., Jódar L. and Villafuerte L., Solving the random Legendre differential equation: mean square power series solution and its statistical functions, Comput. Math. Appl., 61-9 (2011) 2782–2792.
  • [14] Villafuerte L., Braumann C.A., Cortés J.-C. and Jódar L., Random differential operational calculus: Theory and applications, Computers and Mathematics with Applications, 59-1 (2010) 115-125.
  • [15] Gökdoğan A., Merdan M. and Yildirim A., The Modified Algorithm for the Differential Transform Method to Solution of Genesio Systems, Communications in Nonlinear Science and Numerical Simulation, 17-1 (2012) 45-51.
  • [16] Benhammouda B. and Vazquez-Leal H., Analytical solutions for systems of partial differential–algebraic equations, SpringerPlus, 3-1 (2014) 137.
  • [17] Sungu, I.C. and Demir, H., A Computational Method for the Time-Fractional Navier-Stokes Equation, Cumhuriyet Science Journal, 39-4 (2018) 900-911.
  • [18] Merdan, M., Bekiryazici, Z., Kesemen, T. and Khaniyev, T., Analyzing the dynamics of Ebola transmission with random effects. Communications in Mathematical Biology and Neuroscience, 2018 (2018) Article-ID 22.
  • [19] Mehdi Rashidi, M. andErfani, E., The modified differential transform method for investigating nano boundary-layers over stretching surfaces. International Journal of Numerical Methods for Heat & Fluid Flow, 21-7 (2011) 864-883.
  • [20] Onwubuoya, C., Nwanze, D. E., Erejuwa, J. S. and Akinyemi, S. T., An Approximate Solution of a Computer Virus Model with Antivirus using Modifed Differential Transform Method, International Journal of Engineering Research & Technology, 7-4 (2018) 154-161.
  • [21] Bekiryazici, Z., Bazi Kompartmanli Modellerin Rastgele Ektiler Altinda Incelenmesi, PhD Thesis, Karadeniz Technical University, Trabzon, 2017.
There are 21 citations in total.

Details

Primary Language English
Journal Section Research Article
Authors

Pınar Oral 0000-0002-6149-437X

Mehmet Merdan 0000-0002-8509-3044

Zafer Bekiryazıcı 0000-0001-5671-9995

Submission Date September 18, 2018
Acceptance Date May 15, 2019
Publication Date June 30, 2019
DOI https://doi.org/10.17776/csj.460984
IZ https://izlik.org/JA48DB56AM
Published in Issue Year 2019 Volume: 40 Issue: 2

Cite

APA Oral, P., Merdan, M., & Bekiryazıcı, Z. (2019). Analysis of a Random Zeeman Heartbeat Model with Differential Transformation Method. Cumhuriyet Science Journal, 40(2), 285-298. https://doi.org/10.17776/csj.460984

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