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.

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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 .

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Applications of Fractional Calculus in Physics by R Hilfer

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