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Sabatier Agrawal Machado Advances in Fractional Calculus P2

Nhiều chương trình CAD hiện nay cho phép tạo ra các mô hình ba chiều để có thể nhìn từ mọi góc độ. Các chương trình CAD mô hình hóa vật thể đặc tiên tiến là một hệ thống thiết kế hiện thực ảo. Những mô hình đặc như vậy có thể được dùng làm cơ sở cho các phân tích phần tử hữu hạn (FEA) và / hoặc tính toán động lực dòng chảy (CFD) của thiết kế. Cho đến ứng dụng gia công với trợ giúp máy tính (CAM), những mô hình này cũng có thể được dùng. | THE CAPUTO FRACTIONAL DERIVATIVE 29 c Dt v f t cdtv f t w f -v a c t v 0 5 t c a and f t 0 forall t a where y f-v a c t J t-r v1 f r dr as defined in Eq. 4 r v a The generalized fractional derivative for q and p real is defined by cD f t cD cD pf t t c 6 where m is an integer such that m -1 q m and q m - p . Furthermore q 0 and t c 0. In terms of the conventional notation c dm _n dm Df f t c dt p f t w f -p a c t y h m a c t dtm dtm 7 where t c and h aD p f t . It is of course clear that f t may be considered to be a composite function for example a function different than f t may be used for the history period i.e. a t c while fit remains the function to be fractionally differintegrated i.e. t c. It has also been shown 7 that for terminal initialization of the integer derivative tp h m a c t 0 t c 8 and the definition in Eq. 4 is applied for y f -p a c t in Eq. 7 . The next section considers the extention to the Caputo fractional derivative. 3 Initialization of the Caputo Fractional Derivative The Caputo fractional derivative was introduced 8 9 to alleviate some of the difficulties associated with the RL approach to fractional differential equations when applied to the solution of physical problems and is defined by 8 Cdft 1 f t-T m-a-1 f m d-r m -1 a m 9 r m - a a 30 Achar Lorenzo and Hartley As is well known in the solution of fractional differential equations the initial conditions are specified in terms of fractional derivatives in the RL approach but in terms of integer order derivatives with known physical interpretations in the Caputo approach 10 . In view of the popularity of the Caputo formulation in applications of physical interest the key question to be asked is when viewed from the LH general initialization perspective what history is inferred 11 12 for the Caputo derivative 4 Relation Between the Initialized LH and Caputo Fractional Derivatives As has been noted the generalized initialization as applied to RL fractional derivative according to LH is