Conformable Kesirli Diferansiyel Denklemlerinin Taylor ve Sonlu Farklar Metodu ile Sayısal Çözümleri

Abstract

Bu çalışmada yeni tanımlanan conformable kesirli türevli denklemler için güvenilir ve etkili bir metot türettik. Kesirli Taylor açılımından ilk önce Euler ve Taylor metodunu geliştirdik. Bu Taylor açılımı başlangıç noktasından farklı bir noktada açılmış genelleştirilmiş Taylor serisiridir. Öngörülen metotlar daha etkili ve hızlı olduğunu birinci dereceden kesirli diferansiyel denklemlere ve ikinci dereceden salınımlı kesirli diferansiyel denklemlere  uygulayarak gösterdik. İkinci metodumuz ise kesirli diferansiyel denklemi zayıf tekil integral denklemine dönüştürüp, çarpım intagrasyon kuralını uygulayarak çözmek olacaktır. Bu yeni tanımda özel tanımlı fonksiyonlar olmadığı için, metotlar daha doğru sonuç verecek ve bilgisayar programlaması daha kolay olacaktır. Bu öngörülen metotların kararlılık ve yakınsaklıkları ispatlanmış olup, teorik sonuçları destekleyen sayısal örnekler verilmiştir.

Keywords

• Kesirli diferansiyel denklemler,Kesirli euler metodu,Kesirli adams metodu,Riemann-louville ve caputo türevi,Conformable kesirli türev,Taylor metodu

Numerical Solutions of Conformable Fractional Differential Equations by Taylor and Finite Difference Methods

Abstract

We drive efficient and reliable finite difference methods for fractional differential equations (FDEs) based on recently defined conformable fractional derivative. We first derive fractional Euler and fractional Taylor methods based on the fractional Taylor expansion. This fractional Taylor series are the generalized fractional Taylor series that are independent of initial point. We show that the proposed methods are more efficient and faster by applying these methods on first order FDEs and second order oscillatory FDEs. Our second approach is based on inverting FDEs to a weakly singular integral equation that is approximated by product integration rule. This new definition has no special functions and thus the proposed numerical methods will be more accurate and easier to implement than existing methods for FDEs. We prove the stability and convergence of the proposed methods. Numerical examples are given to support the theoretical results.

Keywords

• Fractional differential equations,Fractional euler methods,Fractional adams methods,Riemman-liouville and caputo derivative,Conformal fractional derivative,Taylor methods

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