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

Commences with the old improvement of fractional calculus, its mathematical theory—particularly the Riemann-Liouville model. a variety of examples and theoretical functions of the idea are awarded. gains themes linked to fractional differential equations. Discusses Weyl fractional calculus and a few of its makes use of. contains chosen actual difficulties which result in fractional differential or imperative equations.

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

Example text

As we go on, the methods of approximation will become more involved. Let us take a look at some crude ones. We will approximate in three ways. Example 33— Approximate this integral with three equal subdivisions, using the right end of each one. Here's the picture: The approximate area is f(w1)∆x1 + f(w2)∆x2 + f(w3)∆x3. Each ∆x = 2, and W1 = 1, w2 = 3, and w3 = 5, the right ends of each interval. The approximation is (∆x)[f(1) + f(3) + f(5)] = 2(3 + 11 + 27) = 82. Example 34— Same picture, same intervals, the minimum approximation, the smallest value in each interval, S 3.

Which rectangular beam that can be cut from a circular log of radius 10 inches will have maximum strength? If we let x be the width and y be the depth, we can write the equation without a picture. The strength S = kxy 2 ; k is an unknown constant. To find a relationship between x and y, we need a picture of the log. One of the things we always look for is the Pythagorean relationship. In this case x2 + y2 = 400 (the square of the diameter). In the original equation, it is easier to solve for y 2, because if we solved for x we would have a square root, which would make the derivative much more difficult and sometimes impossible to finish.

Oblique (Slanted Line) Asymptote This occurs when the degree of the top is exactly 1 more than the bottom. Example 10— Degree of the top = 2; degree of the bottom = 1. Oblique asymptote. We must, unfortunately, long divide the bottom into the top. If you know it, use synthetic division. As x goes to infinity, the remainder 21/(x + 4) goes to 0. The oblique asymptote is y = x - 6. Note 1 If the degree of the top is more than the bottom but not 1, there are no oblique asymptotes. Note 2 At most there is one oblique asymptote or one horizontal asymptote, but not both.