Lorenzo Carl F, Hartley Tom T
NASA Glenn Research Center, National Aeronautics and Space Administration, 21000 Brookpark Road, Cleveland, Ohio, USA.
Crit Rev Biomed Eng. 2007;35(6):447-553.
This paper provides a formalized basis for initialization in the fractional calculus. The intent is to make the fractional calculus readily accessible to engineering and the sciences. A modified set of definitions for the fractional calculus is provided which formally include the effects of initialization. Conceptualizations of fractional derivatives and integrals are shown. Physical examples of the basic elements from electronics are presented along with examples from dynamics, material science, viscoelasticity, filtering, instrumentation, and electrochemistry to indicate the broad application of the theory and to demonstrate the use of the mathematics. The fundamental criteria for a generalized calculus established by Ross (1974) are shown to hold for the generalized fractional calculus under appropriate conditions. A new generalized form for the Laplace transform of the generalized differintegral is derived. The concept of a variable structure (order) differintegral is presented along with initial efforts toward meaningful definitions.
本文为分数阶微积分中的初始化提供了形式化基础。目的是使工程和科学领域能够轻松理解分数阶微积分。给出了一组经过修改的分数阶微积分定义，其中正式包含了初始化的影响。展示了分数阶导数和积分的概念。列举了电子学中基本元件的物理示例，以及动力学、材料科学、粘弹性、滤波、仪器仪表和电化学方面的示例，以说明该理论的广泛应用，并演示数学方法的使用。结果表明，罗斯（1974年）建立的广义微积分的基本准则在适当条件下适用于广义分数阶微积分。推导了广义分数阶积分的拉普拉斯变换的一种新的广义形式。提出了可变结构（阶数）分数阶积分的概念，并对有意义的定义进行了初步探讨。
