In this paper, the existence and uniqueness problem of the initial and boundary value problems of the linear fractional Caputo-Fabrizio differential equation of order σ ∈ (1, 2] have been investigated. By using the Laplace transform of the fractional derivative, the fractional differential equations turn into the classical differential equation of integer order. Also, the existence and uniqueness of nonlinear boundary value problem of the fractional Caputo-Fabrizio differential equation has been proved. An application to mass spring damper system for this new fractional derivative has also been presented in details.

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. ConformableKesirli Diferansiyel Denklemlerinin Taylor ve Sonlu Farklar Metodu ile Sayısal Çözümleri Anahtar Kelimeler Kesirli diferansiyel denklemler, Kesirli euler metodu, Kesirli adams metodu, Riemann-liouville ve caputo türevi, Conformable kesirli türev, Taylor metodu Özet: 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 oldugunu 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 olmadıgı için, metotlar daha dogru sonuç verecek ve bilgisayar programlaması daha kolay olacaktır. Bu öngörülen metotların kararlılık ve yakınsaklıkları ispatlanmış olup, teorik sonuçları destekleyen sayisal örnekler verilmiştir.

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In this study, Lyapunov-type inequalities for fractional boundary value problems involving the fractional Caputo Fabrizio differential equation with mixed boundary conditions when the fractional order of β ∈ (1, 2] and Dirichlet-type boundary condition when the fractional order of σ ∈ (2, 3] have been derived. Some consequences of the results related to the fractional Sturm-Liouville eigenvalue problems have also been given. Additionally, we examine the nonexistence of the solution of the fractional boundary value problem.

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