So with this simple example, we show what fractional calculus is and how it is connected with as well as how it generalizes the classical version. One might easily verify that the antiderivative of the square function can be obtained via two similar half-order integration procedures (substituting –1/2 in the previously shown formulas). This is the first derivative of the square function! It is obtained via two “half-order fractional differentiation” procedures. First, let’s calculate the n th-order ordinary derivative of the square function: Let’s take the square function and derive the formula for the fractional derivatives using some simple algebraic manipulations. We will talk about them in this blog post.
Three of these definitions are the most popular and important in practice. Hence, there are different approaches on how to define a fractional “differintegration” operation. After several algebraic manipulations, this integral equation might be rewritten in the form, which is what we call now the Caputo fractional derivative.ĭuring the last two centuries, scientists from different areas and backgrounds worked on the theory of fractional calculus (considering it from different points of view). However, speaking about derivatives or antiderivatives or integrals, we assume the order n is integer.Ībel considered the generalized version of the tautochrone problem (also known as Abel’s problem) on how to determine the equation for the curve KCA along the slope from the prescribed transit time T = f( x) given as a function from the distance x = AB.Ībel obtained the integral equation for the unknown function φ( x), the determination of which makes it possible to find the equation for the curve itself. The foundations of calculus were developed by Newton and Leibniz back in the seventeenth century, with differentiation and integration being the two fundamental operations of this subject.Įvery student of calculus knows that the first derivative of the square function is x, while the result of integrating it is and that integration is essentially the inverse operation of differentiation (the integral of order n may be regarded as a derivative of order –n). Realizing the importance and potential of this topic, we have added support for fractional derivatives and integrals in the recent release of Version 13.1 of the Wolfram Language. This branch is becoming more and more popular in fluid dynamics, control theory, signal processing and other areas. Fractional calculus studies the extension of derivatives and integrals to such fractional orders, along with methods of solving differential equations involving these fractional-order derivatives and integrals.
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