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

Year 2019, Volume: 23 Issue: 3, 850 - 863, 25.12.2019

Şuayip Toprakseven

https://doi.org/10.19113/sdufenbed.579361

Cited By: 7

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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