By R Hilfer
Fractional calculus is a suite of quite little-known mathematical effects bearing on generalizations of differentiation and integration to noninteger orders. whereas those effects were collected over centuries in numerous branches of arithmetic, they've got till lately chanced on little appreciation or software in physics and different mathematically orientated sciences. this example is commencing to switch, and there are actually more and more learn parts in physics which hire fractional calculus. This quantity offers an advent to fractional calculus for physicists, and collects simply obtainable evaluation articles surveying these parts of physics during which functions of fractional calculus have lately turn into well-liked.
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Extra resources for Applications of Fractional Calculus in Physics
One sometimes resorts to alternative definitions of fractal dimension. For subsets of Rn , the following definition is useful. Definition 69. Let F ∈ Rn and for each δ > 0 let Nδ denote the number of δ-mesh cubes that intersect F . The lower and upper box-counting dimensions of F are given by: dimB (F ) = limδ↓0 log Nδ (F ) , − log δ dimB (F ) = limδ↓0 log Nδ (F ) , − log δ (105) Local Fractional Derivatives 33 respectively. The box-counting dimension of F is: dimB (F ) = lim δ↓0 log Nδ (F ) , − log δ (106) whenever the limit exists.
T|2α+1 (151) exists, we have by Lemma 83. that 1 dα f + 1 (x ) = lim dxα Γ(1 + α)Γ(1 − α) h↓0 (1 − t)−α 0 f (ht + x) − f (x) dt. hα (152) We will now prove that 1 lim h↓0 f (ht + x) − f (x) dt = 0, hα (1 − t)−α 0 (153) which completes the proof. Since f is locally α-H¨older continuous in [a, b], there is constant M > 0 such that |f (x) − f (y)| ≤ M |x − y|α for y sufficiently close to x. For 0 < ǫ < 1, we have: 1 0 (1 = 1 1−ǫ (1 − t)−α f (ht+x)−f (x) hα − t)−α f (ht+x)−f (x) hα dt + 1−ǫ (1 0 dt = − t)−α f (ht+x)−f (x) hα dt (154) := I1 + I2 .
Since 2 δ ≥ m ≥ δ −1 and δ < δ0 , we conclude from Proposition 70. that Nδ ≥ mδ −1 C ψ(δ) By definition we conclude that dimB ≥ limδ↓0 C ψ(δ) . log(ψ(δ)) log(δ) . Using the previous two theorems, we can easily obtain the box dimension of the graph of the Weierstrass function. Lemma 73. Let λ > 1 and 1 < s < 2. The graph Γλ,s of the Weierstrass function Wλ,s has box dimension dimB (Γλ,s ) = s. Proof. From Proposition 18. we realize that the conditions of Theorems 71. and 72. are verified for ψ(x) = xs .
Applications of Fractional Calculus in Physics by R Hilfer