Experienced researchers will find the text useful as a reference for discrete fractional calculus and topics of current interest. This text provides the first comprehensive treatment of the discrete fractional calculus. Fractional nonlocal Maxwell's equations and the corresponding fractional wave equations are considered A fractional generalization of exterior differential calculus of differential forms is discussed. The proofs of these theorems are realized for simplest regions. The fractional Green's, Stokes' and Gauss's theorems are formulated. We define the differential and integral vector operations. We solve some problems of consistent formulations of FVC by using a fractional generalization of the Fundamental Theorem of Calculus. The main approaches to formulate a FVC, which are used in the physics during the past few years, will be briefly described in this paper. The history of fractional vector calculus (FVC) has only 10 years. The theory of derivatives and integrals of non-integer order goes back to Leibniz, Liouville, Grunwald, Letnikov and Riemann. International Nuclear Information System (INIS) Two forms of initialization, terminal and side are developed.įractional vector calculus and fractional Maxwell's equations Two basis calculi are considered the Riemann-Liouville and the Grunwald fractional calculi. This definition set allows the formalization of an initialized fractional calculus. This paper demonstrates the need for a nonconstant initialization for the fractional calculus and establishes a basic definition set for the initialized fractional differintegral. Further, based on our definition we generalize hypergeometric functio. We define a kind of fractional Taylor series of an infinitely fractionally-differentiable function. Based on this theorem, in this paper we introduce fractional series expansion method to fractional calculus. In a recently paper (Annals of Physics 323 (2008) 2756-2778), the Fundamental Theorem of Fractional Calculus is highlighted. Series expansion in fractional calculus and fractional differential equationsįractional calculus is the calculus of differentiation and integration of non-integer orders. Note that a derivative can be expressed symbolically using the “prime” notation, as r ′. Any rates that are given in the problem should be expressed as derivatives with respect to time. Be as explicit as you can at this stage, so you do not risk confusing yourself later on. Determine variables for each and write these down. After you understand the problem, you should write down the information that you know, as well as the information that you don't know. Mark the radius as the distance from the center to the circle. In the case, you are to assume that the balloon is a perfect sphere, which you can represent in a diagram with a circle.
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