AN INTRODUCTION IN FRACTIONAL DERIVATIVES

1 AN INTRODUCTION IN FRACTIONAL DERIVATIVES 1. Introduction Each student of Mathematics is introduced very early in the significance of the differential gears symbolisms of the form d d2 d3 etc., and certain of them think, indubitably, why , , dx dx 2 dx 3 it is essential each differential gear has entire an order? For which reason we can not have symbols of the form d 2 d1 2 or still ? dx 1 2 dx  2 However, the continuous use of derivatives with an entire order on the one side and the obvious geometrical explanations of the derivative as bent and the integral as area on the other side, create a gradual polarisation in the thought of hatched mathematician, so that never returns in his initial concerns. The answer in this question is affirmative. It is really possible to be fixed a derivative of an integral explicit, non-explicit or even of a transcendental order. This extension of order of a differential’s gear symbolism in a transcendental level beyond the obvious theoretical interest that presents has enormous importance in the applied mathematics, as we are going to see below. It is remarkable that the make of explicit sequence of logic has, if not the growth, at least the age of a usual differential gear of sequence of logic that was given birth by the work of British Isaac Newton and of the German G. Leibniz. It is known that Leibniz used first the symbolism dny for an n -order derivative of a dx n function y  x  This was perhaps an unsophisticated game of symbols that made L’Hospital to ask Leibniz (with personal correspondence in 1695) what would happen in the case where n was a fraction. "What happens when n is 1 " asked L’Hospital. The 2 insufficient but prophetic answer of Leibniz was: "This would lead 2 in a certain paradoxical. From this paradoxical, however, we will come out one day with very useful consequences". The first time where the significance of derivative of an accidental order on the differential and absolute sequence of logic was presented in 1819 in the book of P. F. Lacroix. Lacroix constituted by 700 pages. Beginning from the y  x   x n function dedicates less than two pages in this subject, his book, however, is (1) where n is a positive entire number, he showed that the m order of a derivative function y  x  was the dmy n!  x n m m dx n  m  ! function (2). Then, by using the symbol of Legendre Gamma (  , the acquaintance that is to say  –function), that generalises the growth of factors, and replacing m with 1 and n with any positive 2 real number  , Lacroix "manufactured" the type (3) that expresses the derivative of order of 1 of a function. We remind to the reader 2 that the  –function is fixed with the each integral   z    e t t z 1dt  0 (4) for a number of z in the transcendental level apart from the points z  0, 1, 2, 3, Certain of the basic attributes of Gamma (  ) function is the function   z  1  z   z   n  1  n ! (6) as well as the function (5), the function 1  1  1  1 ,      ,      2  2  2 (7) Type (3) was a formal product of the way with which worked the mathematicians of that period. Making use of relations (5)–(7) Lacroix calculated, as an application, the derivative of order the function y  x   x and took the function d1/2x 2   x 1/2 dx  1 of 2 (8). 3 This value of this half derivative of x that answers in the interrogative proposal of the title of this article, coincides with the corresponding value that gives the admissible today definition of an explicit derivative at Riemann-Liouville. It deserves to be stressed in this article that since then that L’Hospital placed the question of the explicit derivative (1695), needed to pass 124 years in order to be presented a formal answer in the book of Lacroix (1819), and 289 years in order to be written the first book that was dedicated in this subject (1974). Euler and Fourier even if they cited derivative explicit of order, did not give any application neither any example. Thus, the value for the first application of significance of explicit derivative belongs to Niels Henrik Abel, which applied in 1823 the explicit sequence of logic in solution of an absolute equation that is presented in the study of problem of simultaneous time. This problem, that sometimes is also named problem of equal time, is reported in the finding of form of orbit that is found on a vertical level, so that the time that is needed a material point in order to slip up to the more inferior point of orbit, is independent from the point of starting line of movement. The solution of Abel was so much elegant that existed, probably, the reason of attracting of attention of Liouville, which made the first serious effort to give a reasonable definition of the explicit derivative. The confrontation of the problem from Liouville, that published three works of many pages in 1832 and much works later (1833-1835), became with the following way. For m as an entire positive number is in effect the equation Dme  x   me  x (9) where D  d Liouville generalised the type (9) for not its entire dx values m that is to say Dve  x   e  x (10) with  as an accidental number. Then he developed the function f  x  in a form of 4 exhibitor f  x    c e  n 0 nx n (11) and fixed the derivative of any order of the function f  x  with the value D v f  x    c nnv e n x (12).  n 0 Type (12) is known as the "first definition of Liouville" and it has the obvious disadvantage that values which should have taken the type were of order of the result of proliferation in factors that are checked by the convergence of line. "The second definition of Liouville" is reported in functions of the form x  ,  0 It considers the integral      u 1e  xudu (13)  0 and it applies the transformation xu  t (14) in order to take the value x      1    (15). Then is also applied by the mathematic symbol D v in the two members of equation (15), and with the use of equation D v x a  (10) we take 1 1 u a 1(D v e ux )du  u a 1(1)v uv e ux du    ( ) 0 ( ) 0  equation  (1)v (1)v v a 1  ux u e du I (v  a )   ( ) 0 ( )  the (15) and (16) give now the definition D v x a  (16). Equations (1)v  (  v ) a v (17). x ( ) However, and this second definition (17), had the disadvantage that it was applied in a very limited age-group of functions. In the period between 1835 and 1850 existed a conflict round the rule of the explicit derivative. George Peacock and certain other mathematicians were supporting the definition of Lacroix, while another team were supporting the definition of Liouville. Augustus de Morgan had the opinion that the two definitions were rather right if they were considered as special cases of more general definition. In 1850 William Center observed that the difference between these two definitions of explicit derivative was found in the value 5 that was predicted for the explicit derivative of a constant factor of a function. According to the definition of Lacroix, the explicit derivative constant factor of a function was not annihilated, while constant factor of function was zero because   0    according to the definition (17) of Liouville each explicit derivative In 1847 Riemann followed a way in order to fix the functional (explicit) integral. Supported in the observation of d n y  x  / dx n  0 (18) is the equation (19), that is also generalised Liouville, according to which the general solution of the equation for an n accidental positive number, attempted to fix explicit integration with type D v f  x   1 (x  t )v 1 f (t )dt  (x ) (20). Where (v ) 0   is an accidental positive number and the integral is emanated function   x  is a function of integration (no anymore constant from the type of Cauchy for multiple integrals, but also the factor of integration) for which it will be in effect that Dv   x   0 (21), that is to say  is the solution of equation (18) with n   Cauchy observed in 1880 that also the explicit integral of Riemann (20) had a form that was not determined. The development, however, of the mathematic ideas, seldom becomes without faults. Thus, therefore, also in explicit sequence of logic the growth of significances followed the path of faults, even if in their way met mathematicians of the class of Liouville and Riemann. All above paradoxical, today have been untied and has been also found the cause of their creation that is owed basically in that no one of these mathematicians was thought to ask the consequences of all above definitions in the transcendental level. In this article we shall give the definitions of growth of derivatives and integration of an accidental real order initially and an accidental transcendental order afterwards and also we shall make a short analysis of significances. 6 We shall finish with the apposition of certain examples and applications of explicit logic that also include the solution of irregular absolute equation of the type of Volterra. 2. Definition of explicit integration and the growth of derivatives We are beginning with the determination of symbolism that is owed mainly in Harold T. Davis. The symbol of the type (22) will  of a function f  x  in the closed interval c , x  (we shall consider henceforth represent the explicit integration of an accidental order each integration as negative growth of derivatives). We now consider the mathematic problem of the rule of explicit integration and the growth of explicit derivatives. All mathematicians that we reported above tried – with formal processes – to resolve a problem which they had fully understand following: for which function of f  z  , z  x  iy there is enough but they had not placed completely. What we really ask is the big age-group of functions, and for which number  maxim, non- explicit or transcendental we should replace such a g function where: c Dzv f  z   g  z  or c Dxv f  x   g  x  when the variable z is f  z  was an analytic function of a real? This replacement should satisfy the following criteria: A. If the function transcendental variable z then the derivative c an analytic function of variable z and variable  Dzv f  z  should be B. The symbolism c Dxv should give the same results such as the usual differential gear when  is a positive entire number. When  is a negative entire number, e.g.   n then the symbolism c Dxn should coincide with the derivative of the integral of order n and the derivative until c n  1 order of the function Dxn f  x  which should be annihilated in x  c 7 function inalterable that is to say c Dx0 f  x   f  x  (23). C. The growth of derivatives of null order should leave the in the function c Dxv [af  x    g(x )]  ac c Dxv f  x   c Dxv g(x ) (24). D. The explicit symbolisms should be linear, as that is to say E. It should be in effect the function in form of exhibitor attribute for explicit integrations of arbitrary order, that is to say 1 the function c D f  x   (x  t )v 1 f (t )dt (25).  (v ) c v x x The rule that satisfies criteria A–E and that is considered today as the acceptable rule of the explicit integrations is the type c Dxv f  x   x  t      1 x c v 1 f t  dt (26)   x  the null function. As c  0 was coinciding with the rule of Type (26) is coinciding with the rule (20) if we consider for Riemann and as c   was proved that was equivalent with the rule of Liouville, type (26) took the name Liouville–Riemann. There are existing various ways in order to take the type (26) making use of type that expresses a multiple integral with one simple theory of linear differential equations, or theory of transcendental functions etc. Technically these methods do not allow their explanation in this article. They are also in effect for the verification of criteria AE from type (26). Type (26) is reported in integration of any order. For growth of derivatives, however, accidental order cannot be applied directly. This difficulty can be overcome with the following simple subterfuge. If  is an accidental positive number, it is smaller entire that is also bigger than  , that is to say m     1 p  m  , p   0,1 , then derivative accidental order  function f  x  is fixed with the type of the 8 Dxv f  x   c Dxm c Dx p f  x   x (27) dm 1 p 1 x t f t dt       dx m   p  c c In other words the subterfuge says that if we ask the 1 3 order of a function’s derivative, we take the integral of order 2 3 and then by developing its derivatives at time we supplement, that is to say with integration, the required order of growth of derivatives, so the growth of derivatives is reported in the immediately bigger entire order, and develops derivative at known. The mathematic justification of type (27) is based on the make that the symbolism D m  p is the continuation of the analytic symbolism D v Then we shall give certain simple examples of all above rules. Example 1 v,m, p  are the factors, as more a constant f  x   k Basing on the rule (27) we have that explicit derivative order v m  p Thus we have the type specifically the type have the type 0 0 0 D1/2 x k  Dx1/2k    0 , k 1 2k  Dxvk  k x v m 1  m    (28) and more x 1/2 (29). From the type (26) we k  x 1/2 (30). The type (29) is the type (3) for Example 2 With 0 the application 3/2 D1/2  3 /4  x (31), x x similarly the of 1/2  types 0 Dx x  type (27) we take (   1) x  1/2 ,   1 (   1/2) the type (32) and 9 0 D1/2 x sin x  1  0 ( x ) (33), where  0 is the function of Bessel, 2 that is of first type and null order. Example 3. In the case of growth of derivatives of non-explicit order, we have 3   0 Dx2 0 Dx(2 0 Dx x 3) x d2 1 (x  t )2 2  dx (2  3) 0 x 3 1 d 1 (x  t ) (x  t ) x [ ]tt 0x  2 dx (2  3) 3  3 2 3 3 3 2 2 3 d2 1 1 1  ( ) 2 x 3 (2  3) 2  3 3  3 dx 1 x 1 (2  3) 3 tdt   (x  t )2 x 3 1 tdt  0  3 they type (34). 3. The case of a transcendental order In the case where the order c of a growth of derivatives (integration) is a transcendental number, we have the following definitions: a) For each transcendental number c with Re c 0 (real part of c positive) the integral order of a transcendental number c of a function f is fixed with the type (35). The type (35) is in effect for each point x   0,   for which the integral exists. Its basic theorem of existence Dxc f  x  claims that if Re c 0 and function f Lévesque), is Dxc f  x  exists almost everywhere (the Lévesque metre of total of locally 0 0 integrated (with the significance of then points that does not exist is zero) and is locally integrated, and b) For each transcendental number c with Re c 0 the derivative order c of the transcendental function f is fixed as a locally integrated function  that verifies type (36). The 10 uniqueness of function  is reported in the age-group of equivalence that determines the equality almost everywhere. Eric Russell Love had fixed and studied the symbolisms of growth of derivatives and an integration of clearly imaginary order. He also made an example of a function that had derivative of anyone of clearly imaginary order that is to say of order c with Re c  0 It deserves to be marked at this point that the definition of functional integration can be connected with the Fourier’s transformations of (sine and cosine) via a derivative of clearly imaginary order, with the following way. For Re  we 0 have the type (37) that by this transformation in type (38) results the type (39), where L we symbolize the transformation of Laplace. Placing then   in and assuming that the function c Dxin f  x  exists, we take the type (40) from which results the type (41), that is the transformation of Fourier of – cosine of function f , as well as the type (42), that is the transformation of Fourier – sine of function f We clearly observe, consequently, a relation between the explicit sequence of logic of an imaginary order and the harmonious analysis. 4. Certain applications In this last part of article we will try to give certain applications of explicit sequence of logic, for three mainly reasons. Firstly, for the creation of motives for a deeper research of this sector, secondly, in order to give a limited description of the way of confrontation of the corresponding problems, and finally, in order to be vindicated the existence explicit reasonable as self-existent mathematic sector. A first example is reported in the calculation of not elementary certain integrals. 11 From the relation (43) with the transformation (44) we take the type (45), and for x  1 the type (46). Placing  q  z , p  1 (47) the type (46) is also written as type q  f 1  z  dz    1  p 1 1 p  0 D1 f  x  (48) p  p 0 The type (48) is very useful for the calculation of integrals as the type (49) or the type (50). A second example is reported in the finding of a simple way We consider for example the problem of finding the function f  x  for the solution of certain absolute equations of type of Volterra. that would satisfy the equation xf  x    x 0 x  t  1 2   f t  dt     D 1 2 f  x  (51) 2 The absolute equation (51) is also a type of Volterra of third class with irregular core, that is to say an equation that without fail cannot be characterized of the simplest. Explicit sequence of logic, however, makes the solution really simple. We observe that from the rule of explicit integral we have the type 1 xf  x      0 Dx1 2 f  x  (52) 2 Symbolizing the 0 Dx1 2 with D 1 2 for simplification we have the type (53) and taking the mathematic derivative of order 1 of 2 the two members of type (53), it results to the type (54). According to the rule of Leibniz we have the type (55), and the type (54) is written as type (56) because derivative’s superior orders x are annihilated. We replace the type (53) in the type (56) We can however calculate the function D1 2 f  x  by taking its and we have the type (57). derivative’s first class of the type (53), and then we are resulted to the type (58) or the type (59). 12 Erasing then the function D1 2 f  x  from the types (57) and (59) we are leaded to the usual differential equation  3x     f  x   0 (60) x2 f  x     2  f  x   cx 3 2e  x (61) This equation has its solution as the type where c is a regularly integration. The function (61) constitutes the solution of an irregular absolute equation (51). We note here that the difficulty of solution of the equation (51) is owed in the non-explicit character of the core x  t  1 2 and that the accountant profit from the use of explicit sequence of logic is found in the make that the symbolism of derivative of order 1 2 changes the character of core in explicit number. Even if the study of functional (explicit) sequence of logic belongs to the region of classic analysis, its methods could be characterized as applicable mathematics. Naturally, we cannot claim that the confrontation of a problem with explicit sequence of logic is better than certain other. It is however certain that for a lot of problems the explicit sequence of logic provides short and elegant solutions, and generally constitutes the physiologic frame of confrontation of a problem. Physicians, engineers and the applied mathematicians found explicit sequence of logic very useful in various applied sectors, because is used at the last decade in regions as the quantitative biology, the electrochemistry, and the theory of scattering, the theory of diffusion, the theory of probabilities, the elasticity, the theory of potential, and the theory of locomotion. Nevertheless, a lot of mathematicians ignore the possibilities of the explicit sequence of logic, which it means that the level of applications is found much more behind the theory. Thus this theory is found in the disposal of scientists that will exploit its methods in order to represent various natural 13 phenomena with the compact elegance of the explicit sequence of logic.