(2) Consider the Sumudu transform, which is defined in the space A = {f(t)/³M, T1, T2 > 0,|f(t)|< Me|t|/Ti, se t€ (-1)³ × [0, ∞)}, where the expression applied to a function f(t) is given by: F(u) := S(f(t)) = √ ƒ(ut)e˜dt, u € (−T1, T2) . (6) Show that in the complex plane C, for any R(a) > 0, R(S) O and wЄ C: S [ta-1 E³,a (wt³)] = ua-1 (1 - wu³) -8 'B,a (3) Knowing that the Sumudu transform of the convolution of ƒ and g is given by: S((f + g)(t)) = uF(u)G(u). and that the derivative of K-Caputo is given by: 1 k at (o Da± 4) (x) = - T k ( k − y) √√ ² ( x − t ) X k-Y k - '(t) dt. show that the Sumudu transform of the fractional derivative of the x-Caputo of order 0 < < 1 is given by: (7) k-Y - S[(CD) (x)] = (uk¯¹)**˜¯¹ [F(u) – ƒ(0)] · (4) Consider the ordinary differential equation in terms of the fractional derivative of -Caputo given below: *оy(t) = = (1 − x(t)), – where 0 < y < 1. Show that the solution to equation (8) is given by: 2 - x(t) = (1 − y(0)) E_1 (#) К +4(0). К 入 Tip: Use the Sumudu transform to find the solution.
(2) Consider the Sumudu transform, which is defined in the space A = {f(t)/³M, T1, T2 > 0,|f(t)|< Me|t|/Ti, se t€ (-1)³ × [0, ∞)}, where the expression applied to a function f(t) is given by: F(u) := S(f(t)) = √ ƒ(ut)e˜dt, u € (−T1, T2) . (6) Show that in the complex plane C, for any R(a) > 0, R(S) O and wЄ C: S [ta-1 E³,a (wt³)] = ua-1 (1 - wu³) -8 'B,a (3) Knowing that the Sumudu transform of the convolution of ƒ and g is given by: S((f + g)(t)) = uF(u)G(u). and that the derivative of K-Caputo is given by: 1 k at (o Da± 4) (x) = - T k ( k − y) √√ ² ( x − t ) X k-Y k - '(t) dt. show that the Sumudu transform of the fractional derivative of the x-Caputo of order 0 < < 1 is given by: (7) k-Y - S[(CD) (x)] = (uk¯¹)**˜¯¹ [F(u) – ƒ(0)] · (4) Consider the ordinary differential equation in terms of the fractional derivative of -Caputo given below: *оy(t) = = (1 − x(t)), – where 0 < y < 1. Show that the solution to equation (8) is given by: 2 - x(t) = (1 − y(0)) E_1 (#) К +4(0). К 入 Tip: Use the Sumudu transform to find the solution.
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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![(2) Consider the Sumudu transform, which is defined in the space
A = {f(t)/³M, T1, T2 > 0,|f(t)|< Me|t|/Ti, se t€ (-1)³ × [0, ∞)},
where the expression applied to a function f(t) is given by:
F(u) := S(f(t)) = √ ƒ(ut)e˜dt, u € (−T1, T2) .
(6)
Show that in the complex plane C, for any R(a) > 0, R(S) O and wЄ C:
S [ta-1 E³,a (wt³)] = ua-1 (1 - wu³) -8
'B,a
(3) Knowing that the Sumudu transform of the convolution of ƒ and g is given by:
S((f + g)(t)) = uF(u)G(u).
and that the derivative of K-Caputo is given by:
1
k
at
(o Da± 4) (x) =
-
T k ( k − y) √√ ² ( x − t )
X
k-Y
k
-
'(t) dt.
show that the Sumudu transform of the fractional derivative of the x-Caputo of order 0 < < 1 is
given by:
(7)
k-Y
-
S[(CD) (x)] = (uk¯¹)**˜¯¹ [F(u) – ƒ(0)] ·
(4) Consider the ordinary differential equation in terms of the fractional derivative of -Caputo given
below:
*оy(t) = = (1 − x(t)),
–
where 0 < y < 1. Show that the solution to equation (8) is given by:
2
-
x(t) = (1 − y(0)) E_1
(#)
К
+4(0).
К
入
Tip: Use the Sumudu transform to find the solution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F62d7bd71-12a8-4d88-8c42-1257ba6dd969%2F2bec7999-8edc-4663-9844-12fee0cd292a%2F4fej3or_processed.png&w=3840&q=75)
Transcribed Image Text:(2) Consider the Sumudu transform, which is defined in the space
A = {f(t)/³M, T1, T2 > 0,|f(t)|< Me|t|/Ti, se t€ (-1)³ × [0, ∞)},
where the expression applied to a function f(t) is given by:
F(u) := S(f(t)) = √ ƒ(ut)e˜dt, u € (−T1, T2) .
(6)
Show that in the complex plane C, for any R(a) > 0, R(S) O and wЄ C:
S [ta-1 E³,a (wt³)] = ua-1 (1 - wu³) -8
'B,a
(3) Knowing that the Sumudu transform of the convolution of ƒ and g is given by:
S((f + g)(t)) = uF(u)G(u).
and that the derivative of K-Caputo is given by:
1
k
at
(o Da± 4) (x) =
-
T k ( k − y) √√ ² ( x − t )
X
k-Y
k
-
'(t) dt.
show that the Sumudu transform of the fractional derivative of the x-Caputo of order 0 < < 1 is
given by:
(7)
k-Y
-
S[(CD) (x)] = (uk¯¹)**˜¯¹ [F(u) – ƒ(0)] ·
(4) Consider the ordinary differential equation in terms of the fractional derivative of -Caputo given
below:
*оy(t) = = (1 − x(t)),
–
where 0 < y < 1. Show that the solution to equation (8) is given by:
2
-
x(t) = (1 − y(0)) E_1
(#)
К
+4(0).
К
入
Tip: Use the Sumudu transform to find the solution.
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