Abstract
After the discovery of the effectiveness of the stochastic methods for solving real life problems, these methods have been applied to a wide range of problems in two types; deterministic problems and stochastic problems. The general opinion takes part in applying these methods to stochastic problems since it is preferable for realistic results. Moreover, those methods can also be used in dealing with deterministic models. This study aims to show how stochastic approaches can be applied to deterministic models. Thus, an algorithm based on the Monte Carlo simulation has been presented for solving some systems of nonlinear differential equations. To discuss the behavior of such models, the population equations have been taken into consideration. The considered approach has been seen to produce more accurate results than numerical techniques. A detailed discussion about the results has also been given in this work.
References
- Sadiku MNO. Monte Carlo Methods for Electromagnetics. 1st ed. Taylor and Francis Group, 2009.
- Göncü A. Monte Carlo and Quasi-Monte Carlo Methods in Financial Derivative Pricing. PhD Thesis, Florida State University, Florida, USA, 2009.
- Sobol IM. A Primer for the Monte Carlo Method. 1st ed. Boca Raton, CRC Press, 1994.
- Taylor HM, Karlin S. An Introduction to Stochastic Modeling. 3rd ed. London, UK, Wiley, 1998.
- Resnick S. Adventures in Stochastic Processes. 4th ed. New York, USA, Birkhauser Basel, 1992.
- Kroese DP, Taimre T, Botev ZI. Handbook of Monte Carlo Methods. 1st ed. New Jersey, USA, Wiley, 2011.
- Paul S, Mondal SP, Bhattacharya P. “Numerical solution of lotka volterra prey predator model by using runge–kutta–fehlberg method and laplace adomian decomposition method”. Alexandria Engineering Journal, 55, 613-617, 2015.
- Akhtar MN, Durad MH, Ahmed A. “Solving ınitial value ordinary differential equations by Monte Carlo method”. Proceedings of IAM, 4(2), 149-174, 2015.
- Zhong W, Tian Z. “Solving initial value problem of ordinary differential equations by Monte Carlo method”. 2011 International Conference on Multimedia Technology (ICMT), Hangzhou, China, 26-28 July 2011.
- Fishman G. Monte Carlo: Concepts, Algorithms and Applications. 1st ed. Stanford, USA, Springer, 1999.
- Rubinstein RY, Kroese DP. Simulation and The Monte Carlo Method. 3rd ed. New Jersey, USA, Wiley, 2017.
- Ross SM. A First Course in Probability. 8th ed. New Jersey, USA, Pearson, 2010.
- Uslu H. Behavior of First Order Differential Equations Through a Monte Carlo Based Algorithm. MSc Thesis, Yildiz Technical University, Istanbul, Turkey, 2018.
- Sari M, Uslu H. “Monte Carlo Based Stochastic Approach for Some Nonlinear ODE Systems”. 9th International Conference on Numerical Methods and Applications (NM&A'18), Borovets, Bulgaria, 20-24 August 2018.
- Chapra CS. Applied Numerical Methods with MATLAB for Engineers and Scientists. 4th ed. New York, USA, McGraw Hill, 2017.
- Murray JD. Mathematical Biology. 2nd ed. Seattle, USA, Springer, 2002.
Birinci mertebeden lineer olmayan adi diferansiyel denklem sistemleri için Monte Carlo temelli stokastik yaklaşım
Abstract
Gerçek hayat problemlerini çözmek için stokastik yöntemlerin etkinliğinin keşfinden sonra bu yöntemler, deterministik ve stokastik olmak üzere iki tipteki geniş çaplı problemlere uygulanır oldular. Gerçekçi sonuçlar için tercih edilebilirliğinden dolayı bu yöntemleri stokastik problemlere uygulamak genel kanı olmuştur. Fakat bu yöntemler deterministik modellerle çalışmak için de kullanılabilir. Bu çalışma stokastik yöntemlerin deterministik modellere nasıl uygulanabileceğini göstermeyi amaçlamaktadır. Bu yüzden Monte Carlo simülasyonu temelli bir algoritma, lineer olmayan diferansiyel denklem sistemlerini çözmek için sunulmuştur. Bahsi geçen modellerin davranışlarını tartışmak için popülasyon denklemleri ele alınmıştır. Bu yaklaşımın sayısal tekniklerden daha doğru sonuçlar ürettiği görülmüştür. Bu çalışmada sonuçlar hakkında detaylı bir tartışma yapılmıştır.
References
- Sadiku MNO. Monte Carlo Methods for Electromagnetics. 1st ed. Taylor and Francis Group, 2009.
- Göncü A. Monte Carlo and Quasi-Monte Carlo Methods in Financial Derivative Pricing. PhD Thesis, Florida State University, Florida, USA, 2009.
- Sobol IM. A Primer for the Monte Carlo Method. 1st ed. Boca Raton, CRC Press, 1994.
- Taylor HM, Karlin S. An Introduction to Stochastic Modeling. 3rd ed. London, UK, Wiley, 1998.
- Resnick S. Adventures in Stochastic Processes. 4th ed. New York, USA, Birkhauser Basel, 1992.
- Kroese DP, Taimre T, Botev ZI. Handbook of Monte Carlo Methods. 1st ed. New Jersey, USA, Wiley, 2011.
- Paul S, Mondal SP, Bhattacharya P. “Numerical solution of lotka volterra prey predator model by using runge–kutta–fehlberg method and laplace adomian decomposition method”. Alexandria Engineering Journal, 55, 613-617, 2015.
- Akhtar MN, Durad MH, Ahmed A. “Solving ınitial value ordinary differential equations by Monte Carlo method”. Proceedings of IAM, 4(2), 149-174, 2015.
- Zhong W, Tian Z. “Solving initial value problem of ordinary differential equations by Monte Carlo method”. 2011 International Conference on Multimedia Technology (ICMT), Hangzhou, China, 26-28 July 2011.
- Fishman G. Monte Carlo: Concepts, Algorithms and Applications. 1st ed. Stanford, USA, Springer, 1999.
- Rubinstein RY, Kroese DP. Simulation and The Monte Carlo Method. 3rd ed. New Jersey, USA, Wiley, 2017.
- Ross SM. A First Course in Probability. 8th ed. New Jersey, USA, Pearson, 2010.
- Uslu H. Behavior of First Order Differential Equations Through a Monte Carlo Based Algorithm. MSc Thesis, Yildiz Technical University, Istanbul, Turkey, 2018.
- Sari M, Uslu H. “Monte Carlo Based Stochastic Approach for Some Nonlinear ODE Systems”. 9th International Conference on Numerical Methods and Applications (NM&A'18), Borovets, Bulgaria, 20-24 August 2018.
- Chapra CS. Applied Numerical Methods with MATLAB for Engineers and Scientists. 4th ed. New York, USA, McGraw Hill, 2017.
- Murray JD. Mathematical Biology. 2nd ed. Seattle, USA, Springer, 2002.
Details
| Primary Language | English |
|---|---|
| Subjects | Engineering |
| Journal Section | Research Article |
| Authors | |
| Publication Date | February 20, 2020 |
| Published in Issue | Year 2020 Volume: 26 Issue: 1 |
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