Download PDF by R Hilfer: Applications of Fractional Calculus in Physics

By R Hilfer

ISBN-10: 9810234570

ISBN-13: 9789810234577

Fractional calculus is a set of particularly little-known mathematical effects referring to generalizations of differentiation and integration to noninteger orders. whereas those effects were collected over centuries in a variety of branches of arithmetic, they've got till lately chanced on little appreciation or software in physics and different mathematically orientated sciences. this example is starting to swap, and there are actually more and more study components in physics which hire fractional calculus. This quantity presents an creation to fractional calculus for physicists, and collects simply available evaluate articles surveying these components of physics during which purposes of fractional calculus have lately develop into well-known.

Show description

Read or Download Applications of Fractional Calculus in Physics PDF

Best calculus books

Get Compact Riemann Surfaces (Lectures in Mathematics. ETH PDF

Those notes shape the contents of a Nachdiplomvorlesung given on the Forschungs institut fur Mathematik of the Eidgenossische Technische Hochschule, Zurich from November, 1984 to February, 1985. Prof. okay. Chandrasekharan and Prof. Jurgen Moser have inspired me to put in writing them up for inclusion within the sequence, released through Birkhiiuser, of notes of those classes on the ETH.

Download PDF by Jan R. Magnus, Heinz Neudecker: Matrix Differential Calculus With Applications in Statistics

This article is a self-contained and unified therapy of matrix differential calculus, particularly written for econometricians and statisticians. it will possibly function a textbook for complicated undergraduates and postgraduates in econometrics and as a reference booklet for practicing econometricians.

Download PDF by Paul Turan: On a new method of analysis and its applications

This booklet is among the primary efforts of Turan, an exposition of his strength sum concept. This idea, referred to as "Turan's method," arose as he tried to turn out the Riemann speculation. yet Turan came across functions past these to top numbers. This booklet exhibits the efficacy of the facility sum strategy and encompasses a variety of purposes in its moment half.

Extra info for Applications of Fractional Calculus in Physics

Sample text

Recall that a real-valued function f : Rn → R is continuous at a ∈ Rn if ∀ε > 0, ∃δ > 0 such that ∀x ∈ Bδ (a), |f (x) − f (a)| < ε. 1) The open ball Bδ (a) of radius δ > 0 is a neighborhood of a. Letting V (a) = Bδ (a), the condition on f yields the following two conditions: ∀x ∈ V (a), ∀x ∈ V (a), −ε < f (x) − f (a) f (x) − f (a) < ε ⇒ f (a) − ε < f (x) ⇒ f (x) < f (a) + ε. 2) The first condition says that f (a) is below all limit points of f (x) as x goes to a, while the second one says that f (a) is above, thus yielding the decomposition of the continuity into lower semicontinuity and upper semicontinuity.

If U ⊂ Rn is convex and f : Rn → R ∪{+∞} is convex on U , the set argminf (U ) is convex. 6), it appears that the notion of convexity might be too strong. Indeed, it is sufficient to establish that for all k ∈ R, the set {x ∈ U : f (x) ≤ k} be convex. But to get that property, it is sufficient that ∀x, y ∈ U such that f (x) ≤ k and f (y) ≤ k the property ∀λ ∈ [0, 1], f (λx + (1 − λ)y) ≤ k be verified. This leads to the less restrictive notion of quasiconvexity. 40 Chapter 2. 5. Example of a quasiconvex function that is not convex.

5) is closed in Rn × R. The epigraph epi f is nonempty if and only if dom f = ∅, that is, when f is proper for the infimum. 1. The effective domain dom f of an lsc function is not necessarily closed, as can be seen from the example of the function f (x) = 1/|x| if x = 0 and +∞ if x = 0, where dom f = R \{0}. Proof. If f is lsc on Rn , consider a Cauchy sequence (xn , µn ) ∈ epi f . By definition, µn ≥ f (xn ) and there exists (x, µ) ∈ Rn × R such that xn → x and µn → µ. As f is lsc on Rn , µ = lim µn = lim inf µn ≥ lim inf f (xn ) ≥ f (x) n→∞ n→∞ n→∞ and (x, µ) ∈ epi f .

Download PDF sample

Applications of Fractional Calculus in Physics by R Hilfer


by John
4.1

Rated 4.05 of 5 – based on 12 votes