Monte Carlo based stochastic approach for the first order nonlinear ODE systems

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.

Keywords

• Monte Carlo method,Stochastic method,Population models

Birinci mertebeden lineer olmayan adi diferansiyel denklem sistemleri için Monte Carlo temelli stokastik yaklaşım

Öz

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.

Anahtar Kelimeler

• Monte Carlo yöntemi,Stokastik yöntem,Populasyon modelleri

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