By Kenneth S. Miller

Commences with the ancient improvement of fractional calculus, its mathematical theory—particularly the Riemann-Liouville model. quite a few examples and theoretical purposes of the idea are provided. positive factors issues linked to fractional differential equations. Discusses Weyl fractional calculus and a few of its makes use of. comprises chosen actual difficulties which result in fractional differential or essential equations.

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**Example text**

5t if t ≥ 0 19. f (t) = 3 2 − t2 16. g(t) = −2t if t < 0 3t if t ≥ 0 20. g(t) = 4 − 2t if t < 3 −2 if t ≥ 3 17. f (t) = 3 − t if t ≤ 1 t − 2 if t > 1 21. f (t) = −t2 t2 18. g(t) = 5 + t if t ≤ 2 10 − t if t > 2 22. g(t) = (t + 1)2 4 − t2 if t < 1 if t ≥ 1 if t < 0 if t ≥ 0 if t ≤ −1 if t > −1 In exercises #23–26 you should graph the function f ◦ g. 23. f from #17 and g(t) = t2 25. f from #17, and g from #22 24. f from #18 and g(t) = 6 − t2 26. ” 30. Graph the function f (t) = 1 − u(t). 27. Graph the function u(t).

1, write a formula for calculating f (t). Solution: When t < 0, the graph of this function is linear and has a slope of −1. Specifically, it’s the line y = −t. The behavior of the function changes when t > 0. The graph is still linear, but has a slope of 1. Specifically, it’s the line y = t. 5 You might recognize this as the absolute value function, which is perhaps the most important piecewise defined function. 6. Suppose the heart has just beat, and the voltage on its AV node is x. By the time the next signal from the pacemaker arrives, the voltage on the AV node will have decayed to a fraction of that, say mx where m ∈ (0, 1).

The graph is still linear, but has a slope of 1. Specifically, it’s the line y = t. 5 You might recognize this as the absolute value function, which is perhaps the most important piecewise defined function. 6. Suppose the heart has just beat, and the voltage on its AV node is x. By the time the next signal from the pacemaker arrives, the voltage on the AV node will have decayed to a fraction of that, say mx where m ∈ (0, 1). If mx is less than a so-called critical voltage, the heart will beat. Otherwise it continues to relax.