Calculus

# Download An Introduction to the Fractional Calculus and Fractional by Kenneth S. Miller PDF 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.

Read Online or Download An Introduction to the Fractional Calculus and Fractional Differential Equations PDF

Best calculus books

Nonlinear Dynamics and Chaos

This textbook is aimed toward rookies to nonlinear dynamics and chaos, specially scholars taking a primary direction within the topic. The presentation stresses analytical equipment, concrete examples and geometric instinct. the idea is constructed systematically, beginning with first-order differential equations and their bifurcations, through section aircraft research, restrict cycles and their bifurcations, and culminating with the Lorenz equations, chaos, iterated maps, interval doubling, renormalization, fractals, and unusual attractors.

Introduction to Complex Hyperbolic Spaces

Because the visual appeal of Kobayashi's e-book, there were numerous re­ sults on the simple point of hyperbolic areas, for example Brody's theorem, and result of eco-friendly, Kiernan, Kobayashi, Noguchi, and so on. which make it worthy to have a scientific exposition. even if of necessity I re­ produce a few theorems from Kobayashi, I take a special course, with diversified functions in brain, so the current e-book doesn't great­ sede Kobayashi's.

Additional resources for An Introduction to the Fractional Calculus and Fractional Differential Equations

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.