Download Fractional Differentiation Inequalities by George A. Anastassiou PDF
By George A. Anastassiou
Fractional differentiation inequalities are by way of themselves a huge region of study. they've got many functions in natural and utilized arithmetic and plenty of different technologies. some of the most very important functions is in developing the individuality of an answer in fractional differential equations and platforms and in fractional partial differential equations. additionally they supply higher bounds to the strategies of the above equations.
In this ebook the writer provides the Opial, Poincaré, Sobolev, Hilbert, and Ostrowski fractional differentiation inequalities. effects for the above are derived utilizing 3 varieties of fractional derivatives, specifically by means of Canavati, Riemann-Liouville and Caputo. The univariate and multivariate instances are either tested. every one bankruptcy is self-contained. the idea is gifted systematically in addition to the functions. the applying to info idea can also be examined.
This monograph is appropriate for researchers and graduate scholars in natural arithmetic. utilized mathematicians, engineers, and different utilized scientists also will locate this booklet useful.
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Extra info for Fractional Differentiation Inequalities
Example text
5 we obtain the following inequality involving fractional derivatives of three orders. 46 4. 8. Let 1/p+1/q = 1 with p, q > 1, let γ ≥ 0, ν ≥ γ +2−1/p, and let f ∈ L1 (0, x) have an integrable fractional derivative Dν f ∈ L∞ (0, x) such that Dν−j f (0) = 0 for j = 1, . . , [ν] + 1. 15) 0 where Ω2 (x) = x2(rp+1)/p , 2(Γ(r + 1))2 (rp + 1)2/p r = ν − γ − 1. 16) Proof. Write Φ(t) = |Dν f (t)| and r = ν − γ − 1. 4 and from the definition of the fractional integral we obtain |Dγ f (x)| ≤ U (x) := I r+1 Φ(x), |Dγ+1 f (x)| ≤ I r Φ(x) = U (x) .
13. Let ν > γ ≥ 0, and let f ∈ L1 (0, x) have an integrable fractional derivative Dν f ∈ L∞ (0, x) such that Dν−j f (0) = 0 for j = 1, . . , [ν] + 1. 22) t∈[0,x] 0 where Ω5 (x) = xm(ν−γ)+1 . 3 Applications (i) Uniqueness of solution to fractional initial value problem ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ Let γ i ≥ 0, ν > γ i + 1/2, i = 1, . . , r ∈N. Let f ∈ L1 (0, x) have an integrable fractional derivative Dν f ∈ L∞ (0, x) such that Dν−j f (0) = αj ∈R, j = 1, . . , [ν] + 1. ⎪ ⎪ ⎪ Furthermore, let ⎪ ⎪ ⎪ ⎩ Dν f (t) = F (t, {Dγ i f (t)}r ) for all t ∈ [0, x].
0 Next we integrate over [x0 , x ˜] ⊆ [x0 , b], k x ˜ x ˜ (Dxν0 g)(w) · (Dxν−1 g)(w) · dw + 0 x0 γ qj (w) · (Dx0j g)(w) · (Dxν−1 g)(w)dw 0 j=1 x0 x ˜ + x0 qk+1 (w) · g(w) · (Dxν−1 g)(w)dw = 0. 11 we get that k ((Dxν−1 g)(˜ x))2 0 x ˜ = −2 j=1 x0 γ qj (w) · (Dx0j g)(w) · (Dxν−1 g)(w) · dw 0 x ˜ −2 x0 qk+1 (w) · g(w) · (Dxν−1 g)(w) · dw. 4 Applications 37 Therefore we have k x ˜ ≤ 2 g)(˜ x))2 ((Dxν−1 0 x0 j=1 γ |qj (w)| · |(Dx0j g)(w)| · |(Dxν−1 g)(w)| · dw 0 x ˜ +2· x0 |qk+1 (w)| · |g(w)| · |(Dxν−1 g)(w)| · dw =: (∗).