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

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