-(1) Consider the fractional operator of Caputo-Fabrizio of order α, where 0 < a < 1, which is given by: CF Daf (t)= M(a) St 1-a a α exp[-(1-7)]f'(7) dr 1-α (1) where t≥ 0 and M(a) is a normalization function such that M(0) = M(1) = 1, and f belongs to the Sobolev space H¹(a, b) with b> a. a) For a = 0, show that (1) can be written as: CF Daf (t) = M(a) f'(t) * exp 1-a α (2) a b) Calculate the Laplace transform of (1) using (2), that is, conclude that C (CFD f(t)) = (a) (+) 1-a c) Use expression (3), a, ß < 1, a + ẞ ± 1, and show that s+ (CF by CF D³)f(t), (3) a is written as a linear combination of CF Daf (t) and CF D³ f(t) (4) d) Show that L (CF Da+³ f(t)) = M(a+B) 1-α- β (5) 8+ a+B 1-a-B e) Use items c) and d) to conclude that (CF Da CF D³) f(t) CF Da+ẞf(t).
-(1) Consider the fractional operator of Caputo-Fabrizio of order α, where 0 < a < 1, which is given by: CF Daf (t)= M(a) St 1-a a α exp[-(1-7)]f'(7) dr 1-α (1) where t≥ 0 and M(a) is a normalization function such that M(0) = M(1) = 1, and f belongs to the Sobolev space H¹(a, b) with b> a. a) For a = 0, show that (1) can be written as: CF Daf (t) = M(a) f'(t) * exp 1-a α (2) a b) Calculate the Laplace transform of (1) using (2), that is, conclude that C (CFD f(t)) = (a) (+) 1-a c) Use expression (3), a, ß < 1, a + ẞ ± 1, and show that s+ (CF by CF D³)f(t), (3) a is written as a linear combination of CF Daf (t) and CF D³ f(t) (4) d) Show that L (CF Da+³ f(t)) = M(a+B) 1-α- β (5) 8+ a+B 1-a-B e) Use items c) and d) to conclude that (CF Da CF D³) f(t) CF Da+ẞf(t).
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.6: Additional Trigonometric Graphs
Problem 77E
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