数学专业外文翻译----分数阶导数的儿童乐园(编辑修改稿)

数学专业外文翻译----分数阶导数的儿童乐园(编辑修改稿)内容摘要：

onal championships in the late sixties at distances from 25 kilometers to 50 miles. He is the author of two running books. Introduction We are all familiar with the idea of derivatives. The usual notation ()dfxdx or 1 ()Df x ， 2 2()d f xdx or 2 ()Df x is easily understood. We are also familiar with properties like [ ( ) ( ) ] ( ) ( )D f x f y D f x D f y   But what would be the meaning of notation like 1/21/2()d f xdxor 1/2 ()D f x。 Most readers will not have encountered a derivative of ―order ‖ before, because almost none of the familiar textbooks mention it. Yet the notion was discussed briefly as early as the eighteenth century by Leibnitz. Other giants of the past including L’Hospital, Euler, Lagrange, Laplace, Riemann, Fourier, Liouville, and others at least toyed with the idea. Today a vast literature exists on this subject called the ―fractional calculus.‖ Two text books on the subject at the graduate level have appeared recently, [9] and [11]. Also, two collections of papers delivered at conferences are found in [7] and [14]. A set of very readable seminar notes has been prepared by Wheeler [15], but these have not beenpublished. It is the purpose of this paper to introduce the fractional calculus in a gentle manner. Rather than the usual definition—lemma—theorem approach, we explore the idea of a fractional derivative by first looking at examples of familiar nth order derivatives like Dn ax n axe a e and then replacing the natural number n by other numbers like In this way, like detectives, we will try to see what mathematical structure might be hidden in the idea. We will avoid a formal definition of the fractional derivative until we have first explored the possibility of various approaches to the notion. (For a quick look at formal definitions see the excellent expository paper by Miller [8].) As the exploration continues, we will at times ask the reader to ponder certain questions. The answers to these questions are found in the last section of this paper. So just what is a fractional derivative? 9 Let us see. . . . Fractional derivatives of exponential functions We will begin by examining the derivatives of the exponential function axe because the patterns they develop lend themselves to easy exploration. We are familiar with the expressions for the derivatives of axe . 1 2 2 3 3,ax ax ax ax ax axD e ae D e a e D e a e  , and, in general, n ax n axD e a e when n is an integer. Could we replace n by 1/2 and write 1/2 1/2ax axD e a e Why not try? Why not go further and let n be an irrational number like 2 or a plex number like1+i ? We will be bold and write ax axD e a e ， (1) for any value of  , integer, rational, irrational, or plex. It is interesting to consider the meaning of (1) when is  a negative integer. We naturally want 1( ( ))ax axe D D e .Since 1( ( ))ax axe D ea ,we have 1 ()ax axD e e dx  .Similarly, 2 ()a x a xD e e dx dx   ,so is it reasonable to interpret D when  is a negative integer –n as the nth iterated integral. D represents a derivative if  is a positive real number and an integral if  is a negative real number. Notice that we have not yet given a definition for a fractional derivative of a general function. But if that definition is found, we would expect our relation (1) to follow from it for the exponential function. We note that Liouville used this approach to fractional differentiation in his papers [5] and [6]. Questions Q1 In this case does 1 2 1 21 2 1 2()a x a x a x a xD c e c e c D e c D e   ? Q2 In this case does ax axD D e D e    ? Q3 Is 1 ()ax axD e e dx  , and is 2 ()a x a xD e e dx dx   ,(as listed above) really true, or is there something missing? Q4 What general class of functions could be differentiated fractionally be means of the idea contained in (1)? Trigonometric functions: sine and cosine. We are familiar with the derivatives of the sine function: 0 1 2sin sin , sin c o s , sin sin ,D x x D x x D x x    10 This presents no obvious pattern from which to find 1/2sinDx. However, graphing the functions discloses a pattern. Each time we differentiate, the graph of sin x is shifted /2 to the left. Thus differentiating sin x n times results in the graph of sin x being shifted /2n to the left and so sin sin ( )2n nD x x . As before, we will replace the positive integer n with an arbitrary  . So, we now have an expression for the general derivative of the sine function, and we can deal similarly with the cosine: s in s in ( ) , c o s c o s ( ) .22D x x D x x      (2) After finding (2), it is natural to ask if these guesses are consistent with the results of the previous section for the exponential. For this purpose we can use Euler’s expression, cos si nixe x i x Using (1) we can calculate ( / 2 ) c o s ( ) s in ( )22i x i x i i xD e i e e e x i x           which agrees with (2). Question Q5 What is sin( )D ax ? Derivatives o。