![]() This paper presents extensions of traditional calculus of variations for systems containing Riesz fractional derivatives (RFDs). These basic tools are then applied to provide a physical explanation for the fractional advection-dispersion equation for flow in heterogeneous porous media.įractional variational calculus in terms of Riesz fractional derivatives We develop the basic tools of fractional vector calculus including a fractional derivative version of the gradient, divergence, and curl, and a fractional divergence theorem and Stokes theorem. The reader will acquire a solid foundation in the classical topics of the discrete calculus while being introduced to exciting recent developments, bringing them to the frontiers of the.įractional vector calculus for fractional advection dispersion The novel approach taken by the authors includes a simultaneous treatment of the fractional- and integer-order difference calculus (on a variety of time scales, including both the usual forward and backwards difference operators). The presentation of the content is designed to give ample flexibility for potential use in a myriad of courses and for independent study. Several exercises are offered at the end of each chapter and select answers have been provided at the end of the book. Students who are interested in learning about discrete fractional calculus will find this text to provide a useful starting point. 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.
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