-(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).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Solve this
![-(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).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F62d7bd71-12a8-4d88-8c42-1257ba6dd969%2F63d98a21-c7c2-4fba-bc67-bb21152962bc%2F71v3hug_processed.png&w=3840&q=75)
Transcribed Image Text:-(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).
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