Law of Exponents, Laplace Transform of Fractional Operators, and Fractional Initial Value Problem (IVP) (1) In general, the semigroup law does not hold for derivatives of arbitrary order, that is: Dº Dª Dª D°, DºD³ +D+B To show that the semigroup law does not hold, in general, for derivatives in the sense of Riemann- Liouville, calculate the following expressions: a) (DD) (t); b) (D+) (+); c) (DD) (+); d) (DD) (t³); e) (D) (t). (2) Consider the Laplace transform for the fractional derivative in the sense of Caputo and the Laplace transform for the fractional derivative in the sense of Riemann-Liouville, respectively, as given below: m-1 -1-k L{.D°f(t); 8} = 8° L{f(t)} -Σs¹ƒ) (0°), where and f(*) (0*) := lim D*ƒ(t); m-1 -1-k £{D° f(t); } = 8º£{f(t)} - Σg(*) (0*), -0 9) (0+) = lim Dƒ(t), where g(t)J("-") ƒ (t). In both cases, m-1
Law of Exponents, Laplace Transform of Fractional Operators, and Fractional Initial Value Problem (IVP) (1) In general, the semigroup law does not hold for derivatives of arbitrary order, that is: Dº Dª Dª D°, DºD³ +D+B To show that the semigroup law does not hold, in general, for derivatives in the sense of Riemann- Liouville, calculate the following expressions: a) (DD) (t); b) (D+) (+); c) (DD) (+); d) (DD) (t³); e) (D) (t). (2) Consider the Laplace transform for the fractional derivative in the sense of Caputo and the Laplace transform for the fractional derivative in the sense of Riemann-Liouville, respectively, as given below: m-1 -1-k L{.D°f(t); 8} = 8° L{f(t)} -Σs¹ƒ) (0°), where and f(*) (0*) := lim D*ƒ(t); m-1 -1-k £{D° f(t); } = 8º£{f(t)} - Σg(*) (0*), -0 9) (0+) = lim Dƒ(t), where g(t)J("-") ƒ (t). In both cases, 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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Transcribed Image Text:Law of Exponents, Laplace Transform of Fractional Operators, and Fractional
Initial Value Problem (IVP)
(1) In general, the semigroup law does not hold for derivatives of arbitrary order, that is:
Dº Dª Dª D°, DºD³ +D+B
To show that the semigroup law does not hold, in general, for derivatives in the sense of Riemann-
Liouville, calculate the following expressions:
a) (DD) (t);
b) (D+) (+);
c)
(DD) (+);
d) (DD) (t³);
e) (D) (t).
(2) Consider the Laplace transform for the fractional derivative in the sense of
Caputo and the Laplace transform for the fractional derivative in the sense of
Riemann-Liouville, respectively, as given below:
m-1
-1-k
L{.D°f(t); 8} = 8° L{f(t)} -Σs¹ƒ) (0°),
where
and
f(*) (0*) := lim D*ƒ(t);
m-1
-1-k
£{D° f(t); } = 8º£{f(t)} - Σg(*) (0*),
-0
9) (0+) = lim Dƒ(t), where g(t)J("-") ƒ (t).
In both cases, m-1<a<m.
a) Under what conditions are the Laplace transforms of Riemann-Liouville and Caputo the same?
b) What are the expressions for the Laplace transforms when 0 < a <1?
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