Fractional Versions of the Fundamental Theorem of Calculus

Home , Riemann integral, Riesz potential, Weyl integral

Applied Mathematics, 2013, 4, 23-33 http://dx.doi.org/10.4236/am.2013.47A006 Published Online July 2013 (http://www.scirp.org/journal/am)

Eliana Contharteze Grigoletto, Edmundo Capelas de Oliveira Department of Applied Mathematics, University of Campinas, Campinas, Brazil Email: [email protected], [email protected]

Received April 27, 2013; revised May 27, 2013; accepted June 7, 2013

Copyright © 2013 Eliana Contharteze Grigoletto, Edmundo Capelas de Oliveira. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT The concept of fractional integral in the Riemann-Liouville, Liouville, Weyl and Riesz sense is presented. Some properties involving the particular Riemann-Liouville integral are mentioned. By means of this concept we present the fractional derivatives, specifically, the Riemann-Liouville, Liouville, Caputo, Weyl and Riesz versions are discussed. The so-called fundamental theorem of fractional calculus is presented and discussed in all these different versions.

Keywords: Fractional Integral; Fractional Derivative; Riemann-Liouville Derivative; Liouville Derivative; Caputo Derivative; Weyl Derivative and Riesz Derivative

1. Introduction much research on its applications as there is on the calculus of integer order. Several applications in all areas of Fractional calculus, a popular name used to denote the knowledgement are collected, presented and discussed in calculus of non integer order, is as old as the calculus of different books as follow [7-12]. integer order as created independently by Newton and The main objective of this paper is to explain what is Leibniz. In contrast with the calculus of integer order, meant by calculus of non integer order and collect any fractional calculus has been granted a specific area of different versions of the fractional derivatives associated mathematics only in 1974, after the first international with a particular fractional integral. Specifically, we re- congress dedicated exclusively to it. Before this congress there were only sporadic independent papers, without a cover the concepts of fractional integral and fractional consolidated line [1,2]. derivative in different versions and present a new version During the 1980s fractional calculus attracted re- of the so-called fundamental theorem of fractional cal- searchers and explicit applications began to appear in culus (FTFC), which is interpreted as a generalization of several fields. We mention the doctoral thesis, published the classical fundamental theorem of calculus. We men- as an article [3], which seems to be the first one in the tion three recent works where FTFC is discussed, Tara- subject and the classical book by Miller and Ross [1], sov’s book [12], a paper by Tarasov [13] and a paper by where one can see a timeline from 1645 to 1974. After Dannon [14] in which a particular case of the parameter the decade of 1990, completely consolidated, there ap- associated with the derivative is presented. The paper is peared some specific journals and several textbooks were written as follows: in section two, we first review the published. These facts lent a great visibility to the subject concept of fractional integral in the Riemann-Liouville and it gained prestige around the world. An interesting sense, which can be interpreted as a generalization of the timeline from 1645 to 2010 is presented in references integral of integer order and in the Liouville sense, which [4-6]. We recall here that an important advantage of us- is a particular case of the Riemann-Liouville one. We ing fractional differential equations in applications is review also the concept of fractional integral in the Weyl their non-local property. The use of fractional calculus is sense and in the Riesz sense. Section two present also the more realistic and this is one reason why fractional cal- concepts of derivative as proposed by Riemann-Liouville, culus has become more popular. Liouville, Caputo, Weyl and Riesz, showing the real im- Nowadays, fractional calculus can be considered a portance and applications. Some properties are also pre- frontier area in mathematics in the sense that there is as sented, among which one associated with the semigroup

Copyright © 2013 SciRes. AM 24 E. C. GRIGOLETTO, E. C. DE OLIVEIRA property. Our main result appears in section three, in and if Re   0 or   0 , then which we present and demonstrate the many faces of the    IIaaIa an d IIbb I b  . FTFC, in all different versions and which are interpreted (Riemann-Liouville integrals) Let1 f xaL,1  b , as a generalization of the fundamental theorem of calcu- with ab  . The fractional integrals of Rie- lus. Applications are presented in section four. mann-Liouville of order   with Re  0 , on the left and on the right, are defined respectively by 2. Fractional Calculus  1 x f  The integral and derivatives of non integer order have I:a f xx 1 d , with a ,   a several applications and are used to solve problems in  x  different fields of knowledge, specifically, involving a and fractional differential equation with boundary value con- 1 b f  I: f xx d , with b . ditions and/or initial conditions [7,9,11,12]. They can be b   1  x   x seen as generalizations of the integral and derivatives of 00 integer order. On the other hand, we mention two papers, If   0 , we have IIab :=I, where I is the iden- by Heymans & Podlubny [15] and Podlubny [16] that tity operator. provide an interesting geometric interpretation, and discuss applications of fractional calculus, with integral and 2.2. Fractional Derivative of Riemann-Liouville derivatives of non integer order. Also, we mention a re- After we introduce the fractional integral in the Rie- cent paper in which the authors discuss a fractional dif- mann-Liouville sense, we define the fractional derivative ferential equation with integral boundary value condi- of Riemann-Liouville, which is the most used by mathe- tions [17]. We remember that, there are several ways to maticians, particularly, into the problems in which the introduce the concepts of fractional integral and frac- initial conditions are not involved. tional derivatives, which are not necessarily coincident (Riemann-Liouville derivative) Let   , with with each other [18]. The so-called Grünwald-Letnikov Re   0 and n  Re   1 , where  denotes derivative, which will be not discussed in this paper, is        the integer part of  , the fractional derivatives in the convenient and useful to affront problems involving a Riemann-Liouville sense, on the left and on the right, are numerical treatment [19]. defined by In this section, the concept of fractional integral in the n Riemann-Liouville, Liouville, Weyl and Riesz sense is d n D:Iaaf xf n x (1) presented. Some properties involving the particular Rie- dx  mann-Liouville integral are mentioned. By means of this and concept we present the fractional derivatives, specifically, n n d n the Riemann-Liouville, Liouville, Caputo, Weyl and Ri- D:1Ibbf xf   n x, (2) dx  esz versions are discussed. respectively. Note that the derivatives in Equations (1) and (2) exist 2.1. Fractional Integral of Riemann-Liouville for2 f  xa ACn  ,b . The fractional integral of Riemann-Liouville is an inte- If, in particular,  n  , then gral that generalizes the concept of integral in the classi- n n cal sense, and which can be obtained as a generalization Da f xfx  of the Cauchy-Riemann integral. As we have already said, and before we define the fractional integral Riemann-Liou- D=(1)nnf  xf  ()n()x. ville.  b  Definition 1 (Spaces IL and IL The ap   ap  2.3. Fractional Integral and Derivative in the   spaces ILap  and ILap  are defined, for Liouville Sense Re  0 and p 1, by An interesting particular case of the fractional integral of p IL:apf xfx :  I a gxgx  ,   L, ab Riemann-Liouville and the corresponding fractional derivative, is the so-called Liouville fractional integral and and

1 p b 1 p p p IL:apf xfx :  I a hxhx  ,  L, ab fx Lab ,if  fx  d x  ,for1 p  . a  respectively. 2Space of complex functions f  x whose derivatives up to order Property 2.1 (Semigroup) If Re  0 or   0 n 1 are absolutely continuous on ab, .

 the Liouville fractional derivative. This case is obtained W   inx Iefx  cinn   by substitution a  and b  in the expres- n sions associated with the fractional integral in the Rie- n0 and mann-Liouville sense.    inx (Liouville integral and derivative) The fractional inte- Wefx  cinn   , n grals in the Liouville sense on the real axis, on the left n0 and on the right, for x   and Re  0 , are defined respectively, with   and Re  0 . by In the particular case 0  1, if fxL0,21   ,  1 x f  W   Ifx d (3) then IIf  xfx     and  WDx f xf   x ,    1  x    where I and D are defined in Equations (3) and (5), and respectively. If we define the Fourier series of f  x in 1  f  the form I fx d , (4)   x 1    x  1 2  f x ceinx , with c einx fxx d ,   nn0  n 2 respectively. n0 The corresponding fractional derivatives in the Liou- then we obtain for 01   , in the same way, the frac- ville sense are given by tional integral in the Weyl sense on the right and the n fractional derivative on the right, defined by d n  D:If xf n x (5) dx  W  IIf  xfx  and n and n d  n   D:1Ifx   n fx, (6) x WDf  xfx . dx   respectively. where n Re 1 , Re  0 and x   . 2.5. Fractional Derivative in the Caputo Sense 2.4. Fractional Integral and Derivative in the Weyl Sense The differential operator of non integer order in the Caputo sense is similar to the differential operator of non The operations involving the fractional integral and the integer order in the Riemann-Liouville sense. The capital fractional derivative, as above defined by means of the difference is that: in the Caputo sense, the derivative acts Riemann-Liouville operators are convenient for a func- first on the function, after we evaluate the integral and in tion represented by a series of power but not for func- the Riemann-Liouville sense, the derivatives acts on the tions defined, for example, by means of Fourier series, integral, i.e., we first evaluate the integral and after we because f x is a periodic function with period 2 , calculate the derivative. The derivative in the Caputo  the Ia f x cannot be periodic. For this reason, it is sense is more restritive than the Riemann-Liouville one. convenient to introduce the fractional integral and the We also note that, both derivatives are defined by means fractional derivatives in the so-called Weyl sense. First, of the Riemann-Liouvile fractional integral. The impor- some remarks on the Fourier series are presented. tance associated with this derivative is that, the derivative Let f x be a periodic function with period 2 , in the Caputo sense can be used, for example, in the case defined on the real axis, with null average value, i.e., of a fractional differential equation with initial conditions which have a well known interpretation, as in the calcu- 1 2 fxd0 x , lus of integer order [20,21]. 0  2 (Caputo derivative) Let f xabACn  ,  , with and ab  ,   for Re  0 and  1 2 f xc einx , with c einx fx dx ,   nn0  n  Re   1 . The f raction al derivat ives on the left, n 2   D , and on the right, D , in the Caputo sense, are representing the Fourier series of f x where c are Ca Cb  n defined in terms of the fracti onal integral operator of the corresponding Fourier coefficients. Note that, by hy- Riemann-Liouville as potesis, as the function has null average value, we have Df xf:I n n x (7) c0  0 .  Ca a (Weyl integral and derivative) We define the fractional and integral and the respective fractional derivatives in the n n D:1If xf n  x. Weyl sense by  Cb     b 

00 For   0 we have CaDD: Cb I. In p articular, form by  if  n  , then we have  2 1   n n fff  ,withRe  0. CaD  f xfx  and The so-called fractional integral of order  in the  n n n Riesz sense, denoted by  , which is also known by CbD1 f xf   x. Riesz potential and is defined by the Fourier convolution product 2.6. Fractional Integral in the Riesz Sense   ffxxnKd , First of all, we introduce the fractional integral and the  corresponding fractional derivative in the Euclidean with Re   0 , and space n , but for our purpose we discuss specifically  n the case f :  , only. The operations of fractional  x ,0 n ,2,,  integral an d fractional derivative in the Euclidean space 1  K: x    n 1 n  x log , 0, 2, ,  are fractional powers of the Laplacian operator, n   n  2  x  . For  0 and functions f x , “suffi-  n ciently good” [9], with x  , the fractional operator is the Riesz kernel and  n   is defined in [9], as fol-   2 f x is defined in terms of the Fouri er trans- lows

1 n  , with n  0, 2, 4, , n  2 n   n  2  1   n 22 2 1 2 1 , with n 0, 2, 4,  2

(Riesz integral) As we have already said, we take in Applying the Fourier transform in both sides of Equa- particular, f :  , then tion (9), we get 1     fcfg  2   1   fxf x d (8)  1     and 2 2  2  ˆ ˆ  fcfg  ˆ ,   1,3,5, which is the Riesz fractional integral. ˆ We can also write Equation (8) in terms of the Liou- where the functions f  and gˆ  are the Fourier ville integrals. For this end, we introduce a convenient transforms of the functions f x and g x , respec- convolution product, i.e., we rewrite Equation (8) in the tively. Thus, we can write, following form 1    f xcfgx , (9) 2 2     2  gˆ   . 1 1    1 2 2 with g xx , c  1 ,   1,3,5,   2 2  Then, rewriting Equation (8) in terms of the Liouville 2 integrals, we get

11    22 11x  1  fxf x dd  f x f  x  d 11   x     2222   22   1   2 1 IIfx fx II. fx fx 1        2 2 2cos  2 2

l Finally, we can write the fractional integral in the Ri- where d,n l   and  f x are defined in [9]. The esz sense, in terms of a sum of two Liouville integrals derivative in terms of the Fourier tra nsform is 1  2 1   f xfIIxf x (10) ff    f.   2cos In the particular case, f :  , we have 2  fx fx   with   1,3,5, . For the best of our knowledged this  fx k d, (12)    1 is a new result.   with 01   and 2.7. Fractional Derivative in the Riesz Sense   1 The fractional derivative in the Riesz sense has been in- 21  sin 22 2 troduced in problems that can be treat as a Fourier con- k  3 . volution product. In this section, we introduce this frac- 2 tional derivative and express it in terms of the Liouville Thus, considering 01   , we can write Equation derivative. An example of a specific convolution product (12) in terms of the fractional derivative in the Liouville will be proved. As we have already mentioned, we pre- sense, as follows sent the general definition but we are interested in a par- 1 ticular case involving the parameter. f xfDDxf x , (13)    (Riesz derivative) The fractional derivative of f x 2cos in the Riesz sense, with x n , is defined for 2 Re  0 , by means of with 01   . l f x In what follow we express the Riesz derivative  1     flx :d,n n   (11)  fx  in terms of a convolu tion product. Using Equa- d,l   n   tion (13) we get

1 fxDDfx fx    2cos 2

11dx 1d  fx  ddf x    1dxx 1 dx 2cos 2 1dx d  fx d 1 fx  d  ddxx x 12cos  2     11 d   11     fx   dd f x     dx    x 12cos  12cos  2 2     11  11 11   fx    dd.  fx       12cos  2cos 22 

Thus, the convenient convolution product is 01   .

 Applying the Fourier transform in both sides of Equa-  f xdfhx  , (14) tion (14), we obtain the Fourier transform of the function hx   x 1 :    11 11   1   where hx x , d  and  ff h     2cos  2cos 2 2

 1  Note that, the coefficient in Equation (16) is  11 cd  ffˆˆ hˆ,    complex, because 01   .  2cos 2 ˆ ˆ 3. The Fundamental Theorem of Fractional where f  and h are the Fourier transforms of Calculus f x and hx, respectively. Thus,   After the presentation of different versions of the frac-  2cos tional integral operator and the corresponding fractional 2  hˆ  , (15) derivative it is natural to introduce the corresp onding 11  FTFC associated with these different versions. Then, we  present in this section the so-called FTFC, in the Rie- with 01 . mann-Liouville, Caputo, Liouville, Weyl and Riesz ver- Using this result we prove a theorem involving the sions. The results that are known we mention the refer- Fourier convolution of two particular functions. ence where one can see the proof, otherwise, we present Theorem 1 The Fourier convolution product of the the proof. As we have already said, in all cases we first functions g xx  1 and hx   x 1, with 01   write the theorem in general form, consider a particular is given by case and finally, we recover, as a convenient limit, the 1 fundamental theorem in the corresponding classical ver- g hx x, (16)   sion. cd Theorem 2 (Riemann-Liouville) Consider a function 1 f  x such that fab:,    , with  ab    ;    11 2  let   with Re   0 and n Re 1 . If with c  1 , d  and      n n 2 2   2cos f  xaAC ,b or C, ab then, for every x ab,  2 2 we have:   2  1) DI fx fx and DI fx fx. 4c os  aa    bb    1 2   nn  2) For IAC,a fx   ab we have cd 11  where  ()x is the Dirac delta function. IDaafx  Proof. Evaluating the Fourier transform of the func-  j 1 n1 xa (17) tion g hx, and using Equation (N) and Equation  nj1 n   fx DIa fx (),  xa  (15), we have j0  j    gh  gˆ  hˆ nn  and in the case IAC,b fx   ab , we have   11     . IDbbfx   fx  cdcd cd      nj11  j n1 1 bx (18)    nj1 n   DIb fx . Thus, the Fourier transform of the convolution product  xb j0  j  can be written as  For fx  ILap   we have, 1 gh .   ID fx  fx (19) cd  aa To recover the convolution product, we apply the cor-  and for f xIb Lp we have, responding inverse Fourier transform in both sides of the  last equation, and we get IDbbf  x fx. (20 ) 1 ghx   x In particular, if 0Re   1 in Equations (17) and cd  (18), the n

2   1 4c os xa 2 IDf x fx  I1 fx   x .■ aa a    xa 11  and

 1 j n1 bx 1   xa j IDbbfxfx  I.b fx IDaCafx fx  f x  xb        j0 j! xa On the other hand, if  1 , we have and  11 IDbCbfx  IDaaf xfxf  a jj n1 1 bx  j and fx f x .   xb  11 j0 j! IDbbf xfxfb  , In particular, if 0Re    1, then also3  IDf xfxfa, 11  aC a    abID a f xfbfa     IDbC bf xfxfb  , and  11 and, if  1 , then abID b f xfafb  . 11 Proof. (1) Both results follow from Lemma 2.4 in [9].  abI CDf a  x fb fa (2) To prove Equation (19) and Equation (20), we use and  11 Definition 1 and the case (1). If fxILap , then I Df x fa f b.   abCb    f xgx Ia . Thus, we can w rite, Proof. (1) It follows from Lemma 2.21 in [9]. (2) It IDf xg I DI x aa a aa  follows from Lem ma 2.22 i n [9].■    Theorem 4 (L iouvil le) Let f (x) be a fu nction de- I.a g xfx  fined on r eal axis,   with Re  0 and Now, if fx IL , it follows in an analogous n n  bp  n  Re   1. If fx AC ,  or C, way that then, for x  , we have:    IDbbf xfx .  1) DIf  xfx    and DIf xfx   . The Equations (17) and (18) follow from [11]. In the 2) If 0Re    1 and particular case 0Re 1 we must substitute n 1 limf  xf 0 lim x, in Equations (17) and (18). xx  In the case  1 we recover then

11 1 1   IDfx  I D fx IDf  xfx    and IDf xfx   . ab a a a    xb f xfa   fbfa  Proof. (1) Using Part (1) of the Theorem 2, follows xb    DIf  x  limDIaa fxf lim xf x, and aa   11 1 1 IDfx  I D fx in the same way, we have DIf xfx . ab b b b xa (2) Using Part (2) of the Theorem 2, we have that: if f xfb   faf b.■ xa 0Re    1, then  We will now show that the Theorem 2 in which we IDfx consider the fractional operator in the Cap uto s ense. xa  1 Theo rem 3 (Caputo) Let f  x be a function 1 limf (xf ) Ia xfx a xa fab:,    , with  ab    and let    with Re () >0 and n Re  1 . If f  x   and ACn ab ,  or C,n ab , then, for x ab, : IDfx 1) For Re   or    , we have    1   bx DI f xfx  and DI f xfx   . 1  Caa Cbb lim f xIb fx fx, b xb   2) We have with the functio n fx 0 when x  or .■ 3 11 1   IIxx I  x. 1 ab  a b  xbxa Theorem 5 (Weyl) Let fxL0,2  be a peri-

Copyright © 2013 SciRes. AM 30 E. C. GRIGOLETTO, E. C. DE OLIVEIRA odic function with period 2 , defined on the real axis,   fx cdfghx with null average valu e, and let   with Re   0 then, at x  in which the Fourier series of f  x is cd f g h x convergent, we have 1 WW WW  cd f  x DIf ()xfxand IDfxfx . cd Proof. Let  fx.■

 1 2 Considering 01   , and using Equation ( 10) and fx ce,withinx c einx fx d, x   nn0  n 2 Equation (13), we can write n0  1    be the Fourier series of f x , with the corresp onding  fx IIfx fx    Fourier coefficients cn , then 2cos 2    W I fx cin einx 2         n    DIfxI fx  DI fx I fx  n   n0 2     and  DIf xfxf DI   DI  xfx DI   W   inx Defx  cinn   , 1 n where   . n0  2cos with this, we hav e 2  By means of the Theo rem 4, for 01 , we can      WW W  inx write DIf  xfxf  DI  x. IDfx I  cinn  e  n  Thus, we can write  n0     inx inx  fx   cininnn ee c fx. nn  nn00 2      2DIDIf xfxfx  . In the same way, we have WWDIf (xfx)   .■   On the other hand, us ing Theorem 6, for 01   , Theorem 6 (Riesz) Let fx , where  denote we have f  x  fx  . In this case, we can w rite, the so-called th e space of Lizorkin functions, defined in 1 [9], and let   0 , then   f xf2DxIfxfDI, x  2  1)  f xfx . 4cos  2 2) For 01 , f  xfx .  or in the following form, Proof. (1) See Property 2.35 in [9]. (2) For 01   , using Equation ( 9) we have   2  DIf xfx DI   2cos  1 fx 2     f xc   fg x , with 01   . 1 Evaluating f  x , we get in the same way, that   1 2 where g xx , and c  1 .   2   2  IDf xfx ID   2cos 1 fx 2  2 2 Thus, using Equation (14), we get with 01   .

  cfgxc d  fg hx, 4. Applications

In this section, using the FTFC, fractional differential  equations are solved, one of them associated with the  1 11 with hx x and d  . Riemann-Liouville case and the other involving the  Caputo case.  2cos 2 Example 1 Consider the following fractional differen- Using the Theorem 1, we ob tain tial equation and its initial condition:

 [9], i.e., from T heor em 5.2 , Equation (5.2. 31) in [9] with C D,and00 yt c y   0,  1 and  1 , to obtain the result with c a complex constant, t  0 and 0Re  1.   TEtt    , (26) Applying the fractional integral op erator I0 to the fractional differential equation and using Theorem 3, item (2), where E  x is the one-parameter Mittag-Leffler func- we can write tion.  Using the initial condition we have   ct ID00C yt I0 c yt   .  1 XT0ex  xx123 x,

The next application, we discuss the same problem and by Equation (26), T0  1, then Xex  x123xx. which has been discussed by Jafari & Momani [22] using another methodology, the so-called modified homotoy Substituting this result in equation involving X x  we perturbation method. We solve the equation using the have 3X x   X  x  , i.e.,  3 . method of separation of variables and the FTFC (Rie- Thus, the solution of the initial value problem, i.e., the mann-Liouville). fractional diffusion equation and the initial condition, is Example 2 Consider the initial value problem involv- given by ing the fractional diffusion equation xx123 x  uxt,e E3. t (27) D,and,0e uuux  xx123 x, (21) C 0  We note that, in the paper by Jafari & Momani [22] its solution is presented with a misprint, i.e., as can be veri- where uu  xt,  , x  xx123,,x, with  xi   , for i 1, 2, 3 , t  0 and  0,1  . fied this solution is not a solution of Equation (21). We remark, in passing, that the solution presented in the pa- Suppose a solution with the form per by Jafari & Momani [22] is different from ours be- uxt,XT  x t. (22) cause it solution is not a solution of Equation (21). As a particular case, we consider the problem associ- Substituting Equation (22) into the fractional diffusion ated with the unidimensional diffusion equation, that is equation, Equation (21), we get D,and,0 uuux e,x DT tx X C 0 C 0 , TXtx where uu  ,xt , with   x   , t  0 and  0,1   . where  is a real constant.  x  In this case, the solution is uxt,eE  t , We first consider the fractional differential equation since,   1 in Equation (26). For   0.8 the gra- DT t phic is as in Figure 1. C 0   . Thus, we obtain Tt 5. Conclusion DT t Tt . (23) C 0   After a brief introduction about the calculus of non-inte- Substituting Equation (7) into Equation (23), we get an ger order, popularly known as fractional calculus, we equivalent equation presented the concept of fractional integral in the Rie- mann-Liouville sense. We then discussed the formulation n n IT0 tt  T  . of fractional derivatives as introduced by Riemann- n Liouville and, interchanging the integral with the deriva- Applying operator D0 on both sides of the last equation we have tive, we introduced the formulation proposed by Caputo. We presented also the fractional integral and fractional nnn n DIT00 tt D0 T . derivatives in the Liouville, Weyl and Riesz sense. As our main result, we colleted and showed the many faces Using Theorem 2, item (1), we can write of the FTFC, associated with the Ri emann-Liouville, n n Caputo, Liouville, Weyl and Riesz version. As applica- TDT.tt  0  (24) tions, we discussed two examples involving fractional As  0,1 , we have n 1 . We can also write differential equations. A natural continuation of this n n TDTtt 0 . Equation (24) can then be written work resides in the fact that we can obtain solutions as- DT11t  D T t . (25) sociated with fractional differential equations involving 00 also fractional derivatives as proposed by Riesz and This is a known equation and can be seen in referenc e Weyl. A study in this direction is being developed [23].

 x  Figure 1. Graphics of uxt ,eE   t x in the case   0.8 .

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