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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

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

College Algebra
College Algebra
1st Edition
ISBN:9781938168383
Author:Jay Abramson
Publisher:Jay Abramson
Chapter6: Exponential And Logarithmic Functions
Section6.1: Exponential Functions
Problem 60SE: The formula for the amount A in an investmentaccount with a nominal interest rate r at any timet is...
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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
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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