Use the Laplace transform to solve the following fractional differential equations: a). Du(t)= u(t), t≥0, (0+) = 1, b) Dou(t) = u(t), t≥0, with limo+ J₁u(t)=1. (4) Show that for the Mittag-Leffler function the following holds: to¹Ea.a(t) = +E (-1º). (5) Use the relation between Riemann-Liouville and Caputo derivatives, that is, .D. f(t) = (Dof(t) RL Σ f(k)(a) T(k-a+1)( k-0 and show that the Leibniz rule for the Caputo derivative is given by: .D (f(x)g(2)) = ΣDf(z)D*g(2) D* f(x) A-0 1 a)k-a --) (2) (fg)(*)(0)* I(k (6) Let a > 0. If there exists some Є L¹[a, b] such that f(t) show that Jo Do, f(t) = Do Jo f(t) = f(t). = Jav(t), Note: Observe that for the relationship to hold on both sides, it was necessary to take a specific f. (7) Let F(s) be the Laplace transform of f(t). Show that the Laplace transform of the fractional derivative according to Caputo of order a when m−1
Use the Laplace transform to solve the following fractional differential equations: a). Du(t)= u(t), t≥0, (0+) = 1, b) Dou(t) = u(t), t≥0, with limo+ J₁u(t)=1. (4) Show that for the Mittag-Leffler function the following holds: to¹Ea.a(t) = +E (-1º). (5) Use the relation between Riemann-Liouville and Caputo derivatives, that is, .D. f(t) = (Dof(t) RL Σ f(k)(a) T(k-a+1)( k-0 and show that the Leibniz rule for the Caputo derivative is given by: .D (f(x)g(2)) = ΣDf(z)D*g(2) D* f(x) A-0 1 a)k-a --) (2) (fg)(*)(0)* I(k (6) Let a > 0. If there exists some Є L¹[a, b] such that f(t) show that Jo Do, f(t) = Do Jo f(t) = f(t). = Jav(t), Note: Observe that for the relationship to hold on both sides, it was necessary to take a specific f. (7) Let F(s) be the Laplace transform of f(t). Show that the Laplace transform of the fractional derivative according to Caputo of order a when m−1
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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![Use the Laplace transform to solve the following fractional differential
equations:
a). Du(t)=
u(t), t≥0, (0+) = 1,
b) Dou(t) = u(t), t≥0, with
limo+
J₁u(t)=1.
(4) Show that for the Mittag-Leffler function the following holds:
to¹Ea.a(t) = +E (-1º).
(5) Use the relation between Riemann-Liouville and Caputo derivatives, that
is,
.D. f(t) = (Dof(t)
RL
Σ
f(k)(a)
T(k-a+1)(
k-0
and show that the Leibniz rule for the Caputo derivative is given by:
.D (f(x)g(2))
=
ΣDf(z)D*g(2)
D* f(x)
A-0
1
a)k-a
--) (2)
(fg)(*)(0)*
I(k
(6) Let a > 0. If there exists some Є L¹[a, b] such that f(t)
show that
Jo Do, f(t) = Do Jo f(t)
=
f(t).
=
Jav(t),
Note: Observe that for the relationship to hold on both sides, it was necessary to take a specific f.
(7) Let F(s) be the Laplace transform of f(t). Show that the Laplace
transform of the fractional derivative according to Caputo of order a when
m−1<a<m, is given by
C{cD°f(t);x}=» F(x)
−−1 ƒ(*) (0).
-0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F62d7bd71-12a8-4d88-8c42-1257ba6dd969%2F43b27c8e-fbd9-470d-abf0-8df7622e8de7%2Fz8hg6jd_processed.png&w=3840&q=75)
Transcribed Image Text:Use the Laplace transform to solve the following fractional differential
equations:
a). Du(t)=
u(t), t≥0, (0+) = 1,
b) Dou(t) = u(t), t≥0, with
limo+
J₁u(t)=1.
(4) Show that for the Mittag-Leffler function the following holds:
to¹Ea.a(t) = +E (-1º).
(5) Use the relation between Riemann-Liouville and Caputo derivatives, that
is,
.D. f(t) = (Dof(t)
RL
Σ
f(k)(a)
T(k-a+1)(
k-0
and show that the Leibniz rule for the Caputo derivative is given by:
.D (f(x)g(2))
=
ΣDf(z)D*g(2)
D* f(x)
A-0
1
a)k-a
--) (2)
(fg)(*)(0)*
I(k
(6) Let a > 0. If there exists some Є L¹[a, b] such that f(t)
show that
Jo Do, f(t) = Do Jo f(t)
=
f(t).
=
Jav(t),
Note: Observe that for the relationship to hold on both sides, it was necessary to take a specific f.
(7) Let F(s) be the Laplace transform of f(t). Show that the Laplace
transform of the fractional derivative according to Caputo of order a when
m−1<a<m, is given by
C{cD°f(t);x}=» F(x)
−−1 ƒ(*) (0).
-0
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