Applications of Fractional Calculus in Physics by R Hilfer

By R Hilfer

Fractional calculus is a suite of particularly little-known mathematical effects referring to generalizations of differentiation and integration to noninteger orders. whereas those effects were accrued over centuries in quite a few branches of arithmetic, they've got till lately chanced on little appreciation or software in physics and different mathematically orientated sciences. this example is commencing to swap, and there at the moment are progressively more study components in physics which hire fractional calculus. This quantity presents an advent to fractional calculus for physicists, and collects simply obtainable evaluation articles surveying these parts of physics during which purposes of fractional calculus have lately turn into fashionable.

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The faster the string moves, the higher the frequency and the higher the note that it produces. For the fundamental frequency, only the end points are at rest. If the string has a node at its centre, then it produces a note at exactly double the frequency (heard by the human ear as one octave higher). This is a sound wave. There are peaks and troughs, or highs and lows. These highs and lows define the amplitude of a sound wave. istockphoto. com/Phil Morley 56 7 Differential Equations 7 The more nodes there are, the higher the frequency of the note.

For example, consider the simplest case, in which the mass m and force F are constant, as is the case for a body falling under terrestrial gravity. Then equation (4) can be written as x″(t) = F⁄m. (5) Integrating (5) once with respect to time gives x′(t) = Ft⁄m + b (6) where b is an arbitrary constant. Integrating (6) with 2 respect to time yields x(t) = Ft ⁄2m + bt + c with a second constant c. The values of the constants b and c depend upon initial conditions. Indeed, c is the initial position, and b is the initial velocity.

Moreover, the derivative f ′(x) vanishes at any (local) minimum value inside the interval. Indeed, it can sometimes vanish at places where the value is neither a maximum nor a minimum. An example is f (x) = x 3 for −1 ≤ x ≤1. Here f ′(x) = 3x 2 so f ′(0) = 0, but 0 is neither a maximum nor a minimum. For x < 0 the value of f(x) gets smaller than the value f(0) = 0, but for x > 0 it gets larger. Such a point is called a point 41 7 The Britannica Guide to Analysis and Calculus 7 of inflection. In general, solutions of f ′(x) = 0 are called critical points of f.

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