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.