Inter national J our nal of Electrical and Computer Engineering (IJECE) V ol. 11, No. 6, December 2021, pp. 5367 5378 ISSN: 2088-8708, DOI: 10.11591/ijece.v11i6.pp5367-5378 r 5367 Numerical appr oach of riemann-liouville fractional deri v ati v e operator Ramzi B. Albadar neh 1 , Iqbal M. Batiha 2 , Ahmad Ad wai 3 , Nedal T ahat 4 , A.K. Alomari 5 1,3,4 Department of Mathematics, The Hashemite Uni v ersity , Zarqa, Jordan 2 Department of Mathematics, F aculty of Science and T echnology , Irbid National Uni v ersity , Irbid, Jordan 2 Nonlinear Dynamics Research Center (NDRC), Ajman Uni v ersity , Ajman, U AE 5 Department of Mathematics, Y armouk Uni v ersity , Irbid, Jordan Article Inf o Article history: Recei v ed Oct 30, 2020 Re vised May 19, 2021 Accepted Jun 4, 2021 K eyw ords: Fifth k e yw ord F ourth k e yw ord Fractional calculus Riemann-liouville fractional deri v ati v e operator W eighted mean v alue theorem ABSTRA CT This article introduces some ne w straightforw ard and yet po werful formulas in the form of series solutions together with their residual errors for approximating the Riemann-Liouville fractional deri v ati v e operator . These formulas are deri v ed by uti- lizing some of forthright computations, and by utilizing the so-called weighted mean v alue theorem (WMVT). Undoubtedly , such formulas will be e xtremely useful in es- tablishing ne w approaches for se v eral solutions of both linear and nonlinear fractional- order dif ferential e quations. This assertion is confirmed by addressing se v eral linear and nonlinear problems that illustrate the ef fecti v eness and the practicability of the g ained findings. This is an open access article under the CC BY -SA license . Corresponding A uthor: Ramzi B. Albadarne Department of Mathematics The Hashemite Uni v ersity Zarqa 13133, Jordan Email: rbadarneh@hu.edu.jo 1. INTR ODUCTION The principle of fractional calculus has been endorsed as distinguished mathematical tools to charac- terize man y real-w orld phenomena in the recent decades [1]-[5]. It has been increasingly considered by man y researchers in numerous areas of engineering and science, some of these areas are and not limited to con- trol engineering [6], electrochemis try [7], electromagnetism [8], bioscience [9], and dif fusion processes [10]. Se v eral dif ferent fractional deri v ati v es and inte grals definitions ha v e been formulated and accepted, and the y are di vided into dif ferent cate gories. It’ s w orth mentioning that there are tw o fractional deri v ati v e definitions; the first definition is the deri v ati v e of a function’ s con v olution with a po wer la w k ernel, as suggested by Rie- mann and Liouville, the second is caputo’ s proposal of con v olution of the local deri v ati v e of a gi v en function with a po wer la w function [11]. In vie w of dif ferent suggestions of man y applied mathematicians, the caputo fractional deri v ati v e operator is acceptable for man y real-w orld problems because it allo ws for the use of spec- ified initial conditions when taking fractional deri v ati v es, for instance, the Laplace transform [1], [12], [13]. Atang ana et al. in [1] as serted that when a fractional inte gral operates, the initial function does not reco v er well, according to the mathem atical definition of the caputo operator . As a result, although the caputo deri v ati v e is e x- tremely useful and practical, it may not be appropriate for mathematical purposes [1]. The Riemann-Liouville operator , on the other hand, satisfies the mathematical principle in the fractional calculus sense. Furthermore, J ournal homepage: http://ijece .iaescor e .com Evaluation Warning : The document was created with Spire.PDF for Python.
5368 r ISSN: 2088-8708 when using the Laplace transform, the initial condition with fractional e xponent is tak en into account, which is both practical and mathematically realistic [1]. In the same conte xt, T aylor’ s series including fractional ones is suggested by man y authors as one of the most ef ficient po wer series [14]. In 1847, the idea of the fractional generalized T aylor series w as re v ealed, when Riemann used a series structure to formulate an analytic function [15]. The proof of the v alidity of such e xpansion for some classes of functions w as gi v en by Hardy [16]. Recently , the related mean v alue theorem problem w as discussed by T rujillo et al. [17], and the result from t he Riemann-Liouville case to the Caputo case w as e xtended by Odibat et al. [18]. Another vie w to sho w fractional calculus, including Gr ¨ unw ald-Letnik o v and Riemann-Liouville definitions has been suggested by Oldham and Spanier [19], also a po wer series in- v olving inte ger deri v ati v es of the analytic function w as constructed [20]. Afterw ards, to describe fractional deri v ati v e, a ne w series w as proposed by Samk o et al. [21]. In reference [22], the T aylor -Riemann series using Osler’ s theorem w as in v estig ated to obtain certai n double infinite series e xpansions of some elementary functions. Some classical po wer series theorems ha v e been generalized for fractional po wer series, and a ne w construction of the generalized T aylor’ s po wer series has been introduced in [14]. In reference [1], a numer - ical approximation of the Riemann-Liouville fractional deri v ati v e operator w as presented. Analogues of the T aylor’ s theorem and the mean v alue theorem for fractional dif ferential operators were established in reference [23]. More recently , W ei et al. ha v e de v eloped a general structure for T aylor series in fr actional case by e x- panding an analytic function at the current time or at the initial instant [20]. This structure tak es into account the Caputo definition, the Riemann-Liouville definition, the v ariable order and the constant order [20]. In this paper , a ne w straightforw ard formula in a series form for approximating the fractional deri v ati v e operator in the sense of Riemann-Liouville , D y ( t ) ; 0 < 1 , is introduced. Based on the weighted mean v alue theorem (WMVT) and some direct computations; this formula is deri v ed. Because the solutions of some linear and nonlinear fractional dif ferential equations are e xtremely dif ficult to obtain; such formula will be v ery useful to establish ne w approaches for them. These solutions will be in series forms that could be used in order to determine the analytic solutions in man y cases. Ho we v er the rest of this article is or g anized as follo ws: The Riemann-Liouville dif ferenti al and inte gral operators are presented in section 2 with basic definitions and theorems. The theoretical frame w ork is presented in section 3. Section 4 pro vides some e xamples to demonstrate the method. The final part of the paper is the conclusion. 2. RIEMANN LIOUVILLE DIFFERENTIAL AND INTEGRAL OPERA T ORS Calculus of inte grals and deri v ati v es of an y arbitrary real or comple x order is the topic of fr actional calculus [24], [25]. As one of the most important fractional deri v ati v es operators, the Riemann-Liouville operator satisfies all mathematical principles within the frame w ork of fractional calculus [1], [26], [27]. T o help researchers better unders tand ho w this operator generalizes ordinary dif ferential operators, some definitions and properties related to this operator will be e xhibited. Let u s , firstly , assume that [ a; b ] is a finite interv al, where a; b 2 R and 1 < a < b < 1 . The left-sided Riemann-Li ou vi lle fractional inte gral of order 2 R + is defined as [21], [28]: J a + f ( x ) = 1 ( ) Z x a f ( ) ( x ) 1 d ; x > a; (1) and the right-sided Riemann Liouville fractional inte gral of order 2 R + is [28]: J b f ( x ) = 1 ( ) Z b x f ( ) ( x ) 1 d ; x < b: (2) Observ e that we ha v e limited the v alues of the fractional order to the real positi v e numbers [28], which is necessary for some practical applications, b ut one may find that belongs to comple x numbers in references [21], [28]. On the other hand, the left-sided Riemann-Liouville fractional deri v ati v e of order 2 R + is defined by [28]: D a + f ( x ) = 1 ( n ) d n dx n Z x a f ( ) ( x ) n +1 d ; x > a; (3) and the right-sided Riemann-Liouville fractional deri v ati v e of order 2 R + is [28]: Int J Elec & Comp Eng, V ol. 11, No. 6, December 2021 : 5367 – 5378 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Elec & Comp Eng ISSN: 2088-8708 r 5369 D b f ( x ) = ( 1) n D n J n b f ( x ) = ( 1) n ( n ) d n dx n Z b x f ( ) ( x ) n +1 d ; x < b; (4) where n = d e , and where de denotes to the ceil ing function. The popular forms of the Riemann-Liouville fractional inte gral and deri v ati v e of order 2 R + coincide with the left-sided Riemann-Liouville definitions. Actually , these forms can be defined as [29], [30], [31]: J a f ( x ) = 1 ( ) Z x a f ( ) ( x ) 1 d ; x > a; (5) and D a f ( x ) = 1 ( n ) d n dx n Z x a f ( ) ( x ) n +1 d ; x > a: (6) It should be mentioned that the inte gral operators J a + ; J b and J a in (1), (2) and (5), res pecti v ely , are defined on L p ( a; b ) The space of inte grable functions, where p 2 [1 ; 1 ) . At the same time, the dif ferential operators D a + ; D b and D a in (3), (4) and (6), respecti v ely , are defined on C [ a; b ] The space of continuous functions [28]. Ne xt, some important properties of the inte gral operator are stated for completeness. Theorem 1: [32] Let ; 0 and 2 L 1 [ a; b ] . Then, J a J a = J + a holds almost e v erywhere on [ a; b ] . If additionally 2 C [ a; b ] or + 1 , then the identity holds e v erywhere on [ a; b ] . Corollary 2: [32] Let ; 0 and 2 L 1 [ a; b ] . Then, J a J a = J a J a . Theorem 3: [30] The Riemann-Liouville fractional inte gral J a of the po wer function satisfies: J a ( x a ) = ( + 1) ( + + 1) ( x a ) + ; > 0 ; > 1 : Ha ving stated some fundamental properties of the Riemann-Liouville inte gral operator , we are no w ready to state some properties of the corresponding dif ferential operator . Theorem 4: [32] Let 0 . Then for e v ery f 2 L 1 [ a; b ] , we ha v e D a J a f ( x ) = f ( x ) almost e v erywhere. Theorem 5: [32] Let > 0 , If there e xists some 2 L 1 [ a; b ] such that f = J a , then, J a D a f ( x ) = f ( x ) almost e v erywhere. Theorem 6: [32] Let > 0 and n 1 < n; n 2 N . Assume that f is such t hat J ( n ) a f 2 A n [ a; b ] The set of all functions with an absolutely continuous ( n 1) th deri v ati v e. Then, J a D a f ( x ) = f ( x ) n 1 X k =0 ( x a ) k 1 ( k ) lim z ! a + D ( n k 1) J ( n ) a f ( z ) : (7) In particular , for 0 < < 1 , we ha v e: J a D a f ( x ) = f ( x ) ( x a ) 1 ( ) lim z ! a + J (1 ) a f ( z ) : (8) Theorem 7: [32] The Riemann-Liouville fractional deri v ati v e D a of the po wer function satisfies: D a ( x a ) = ( + 1) ( + 1 ) ( x a ) ( ) ; if = 2 N : 3. THE THEORETICAL FRAMEW ORK This section illustrates the theoretical frame w ork of the present study . Actually , it introduces tw o no v el theorems that of fer tw o po werful e xpressions formulated in the form of po wer series to approximate the Riemann-Liouville fractional deri v ati v e operator . In summary , here are the main results of this w ork. Theorem 8: Let y 2 C n +1 [ a; b ] , 0 < 1 , and a 0 . Then for e v ery t 2 ( a; b ] , there e xist 2 ( a; b ) such that the Riemann-Liouville fractional deri v ati v e operator D a y ( t ) can be written, in terms of fractional series Numerical appr oac h of riemann-liouville fr actional derivative oper ator (Ramzi B. Albadarneh) Evaluation Warning : The document was created with Spire.PDF for Python.
5370 r ISSN: 2088-8708 and its reminder term, in the follo wing form: D a y ( t ) = 1 (1 )   y ( a )( t a ) + n X k =1 y ( k ) ( a )( t a ) k Q k j =1 ( j ) + y ( n +1) ( )( t a ) n +1 Q n +1 j =1 ( j ) ! : (9) The Riemann-Liouville fractional deri v ati v e operator for 0 < 1 is kno wn as: D a y ( t ) = 1 (1 ) d dt Z t a y ( x )( t x ) dx: (10) Using inte gration by part to (10) yields: D a y ( t ) = 1 (1 ) y ( a )( t a ) + Z t a y 0 ( x )( t x ) dx : (11) Ag ain, applying inte gration by part inducti v ely n -times to (11) leads to the follo wing assertion: D a y ( t ) = 1 (1 ) y ( a )( t a ) + y 0 ( a )( t a ) 1 1 + y 00 ( a )( t a ) 2 (1 )(2 ) + + 1 (1 ) y ( n ) ( a )( t a ) n Q n k =1 ( k ) + 1 Q n k =1 ( k ) Z t a y ( n +1) ( x )( t x ) 1 1 dx : (12) Observ e that y 2 C n +1 [ a; b ] and ( t x ) n does not change its sign in [ a; t ] . Therefore, one can conclude using the WMVT that there e xist 2 ( a; b ) such that: Z t a y ( n +1) ( x )( t x ) n Q n k =1 ( k ) dx = y ( n +1) ( ) Q n k =1 ( k ) Z t a ( t x ) n dx = y ( n +1) ( )( t a ) n +1 ( n + 1 ) Q n k =1 ( k ) = y ( n +1) ( )( t a ) n +1 Q n +1 k =1 ( k ) ; (13) which consequently implies the desired result. Theorem 9: Let y 2 C n + m [ a; b ] , a 0 , and m 1 < < m , where m is positi v e inte ger . Then for e v ery t 2 ( a; b ] , there e xist 2 ( a; b ) such that the Riemann-Liouville fractional deri v ati v e operator D a y ( t ) can be written, in terms of fractional series and its reminder term, in the follo wing form: D a y ( t ) = 1 (1 )   y ( a )( t a ) + n X k =1 y ( k ) ( a )( t a ) k Q k j =1 ( j ) + y ( n +1) ( )( t a ) n +1 Q n +1 j =1 ( j ) ! : (14) The Riemann-Liouville operator gi v en in (6) can be re written in the follo wing form: D a y ( t ) = 1 ( m )( m ) y ( a )( m )( m 1) ::: (1 )( t a ) + d m dt m Z t a y 0 ( x )( t x ) m dx : (15) But, based on the follo wing assertion: d m dt m Z t a y 0 ( x )( t x ) m dx = Z t a y 0 ( x )( m )( m 1) ::: (1 )( t x ) 1 dx; Int J Elec & Comp Eng, V ol. 11, No. 6, December 2021 : 5367 – 5378 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Elec & Comp Eng ISSN: 2088-8708 r 5371 in (15) will be as (16): D a y ( t ) = 1 (1 ) y ( a )( t a ) + Z t a y 0 ( x )( t x ) dx : (16) Finally , using the same proof of Theorem 8 yields also the desired result. 4. ILLUSTRA TIVE NUMERICAL EXAMPLES In this part, the ef fecti v eness and ef ficienc y of our findings are numerically v erified through solving some linear and nonlinear fractional dif ferential equations. Example 10: Consider the function y ( t ) = exp( t ) + t 8 + cos t 2 3 6 t . The tw o formulas (9) and (10) gi v en, respecti v ely , in theorem 8 and theorem 9 are emplo yed via mathematica package in order to g ain the approximate v alues of the Riemann-Liouville fractional deri v ati v e operator D a y ( t ) v ersus the time t for dif ferent v alues of and n . Ho we v er , T ables 1, 2, 3, and 4 sho w the error terms between the approximate and the e xact v alues. One can observ e that these errors can be reduced by suf ficiently increasing the v alue of n or by reducing the round of error that could be occurred when applying mathematica package. T able 1. Error terms for e xample 10 t =0.5,n=20 =0.7,n=20 =0.9,n=20 =0.99,n=20 0.1 0.00 8.88 10 16 0.00 8.88 10 16 0.2 0.00 1.78 10 15 1.78 10 15 8.88 10 16 0.3 8.88 10 16 1.78 10 15 8.88 10 16 1.78 10 15 0.4 8.88 10 16 1.78 10 15 0.00 8.88 10 16 0.5 1.78 10 15 2.66 10 15 4.44 10 15 1.07 10 14 0.6 1.09 10 13 2.26 10 13 4.75 10 13 6.62 10 13 0.7 4.09 10 12 8.28 10 12 1.68 10 11 2.30 10 11 0.8 9.42 10 11 1.86 10 10 3.66 10 10 4.96 10 10 0.9 1.50 10 9 2.88 10 9 5.55 10 9 7.44 10 9 1.0 1.78 10 8 3.35 10 8 6.31 10 8 8.38 10 8 T able 2. Error terms for e xample 10 t =0.5,n=50 =0.7,n=50 =0.9,n=50 =0.99,n=50 0.1 0.00 1.33 10 15 0.00 0.00 0.2 0.00 2.22 10 15 1.78 10 15 0.00 0.3 0.00 1.78 10 15 0.00 1.78 10 15 0.4 8.88 10 16 1.78 10 15 0.00 1.78 10 15 0.5 0.00 0.00 1.78 10 15 8.88 10 16 0.6 0.00 3.55 10 15 1.78 10 15 8.88 10 16 0.7 0.00 3.55 10 15 8.88 10 16 1.78 10 15 0.8 8.88 10 16 3.55 10 15 1.78 10 15 0.00 0.9 1.78 10 15 6.22 10 15 8.88 10 16 1.33 10 15 1.0 8.88 10 16 1.02 10 14 1.89 10 15 4.44 10 16 Example 11: Consider the follo wing nonlinear fractional IVP: D 0 y ( t ) = t 1 (1 y ) 2 ; y (0) = 0 : (17) The e xact solution of this problem is y ( t ) = t 1+ t , for = 1 . Ho we v er , in order to emplo y our proposed scheme to solv e such problem, one can firstly tak e the T aylor’ s series around t = 0 for the left-hand side of (17), and then use the result reported in theorem 8 In other w ords, D 0 y ( t ) can be replaced by the follo wing assertion: D 0 y ( t ) = 1 (1 )   y (0) t + 1 X k =1 y ( k ) (0) t k Q k j =1 ( j ) ! : (18) Numerical appr oac h of riemann-liouville fr actional derivative oper ator (Ramzi B. Albadarneh) Evaluation Warning : The document was created with Spire.PDF for Python.
5372 r ISSN: 2088-8708 Using the initial condition y (0) = 0 implies: t 1 y 0 (0) (2 ) + t 2 y 00 (0) (3 ) + y (3) (0) t 3 (4 ) + : : : = t 1 2 t 2 y 0 (0) + t 3 y 0 (0) 2 y 00 (0) + : : : (19) Equating the coef ficients of t j of (19) yields: y 0 (0) = (2 ) ; y 00 (0) = 2(2 )(3 ) ; y 000 (0) = (2 ) 2 + 2(3 )(2 ) (4 ) ; . . . consequently , substituting each of y (0) ; y 0 (0) ; y 00 (0) ; into the po wer series of y ( t ) around t = 0 leads us to establish the general solution of (17), which w ould be in the follo wing form: y ( t ) = t (2 ) t 2 (2 )(3 ) + 1 6 t 3 (2 ) 2 + 2(3 )(2 ) (4 ) + 1 72 t 4 6(3 )(2 ) 2 (2 ) 2 + 2(3 )(2 ) (4 ) (5 ) + : : : : One can easily v erify that, when = 1 , this solution w ould be the same e xact solution gi v en abo v e, i.e; y ( t ) = t t 2 + t 3 t 4 + t 5 t 6 + t 7 t 8 + t 9 t 10 + ::: = t 1 + t : In particular , the po wer series solution of the IVP gi v en in (17), for 0 < 1 , can be written as: y ( t ) = 1 X m =0 c ( m ) t m m ! ; where c (0) = 0 ; c (1) = (2 ) ; and c ( m ) = ( m + 1)   m 1 X k =0 c ( k ) c ( k + m 1) k !( k + m 1)! 2 c ( m 1) ( m 1)! ! : T able 3. Error terms for e xample 10 t =1.5,n=20 =1.7,n=20 =1.9,n=20 =1.99,n=20 0.1 0.00 2.84 10 14 9.95 10 14 2.66 10 14 0.2 0.00 7.11 10 15 2.13 10 14 7.99 10 15 0.3 3.55 10 15 3.55 10 15 1.42 10 14 5.33 10 15 0.4 0.00 0.00 1.07 10 14 1.33 10 15 0.5 6.93 10 14 1.51 10 13 3.33 10 13 4.57 10 13 0.6 4.28 10 12 8.87 10 12 1.83 10 11 2.54 10 11 0.7 1.37 10 10 2.76 10 10 5.53 10 10 7.55 10 10 0.8 2.76 10 9 5.41 10 9 1.06 10 8 1.43 10 8 0.9 3.91 10 8 7.46 10 8 1.42 10 7 1.90 10 7 1.0 4.17 10 7 7.80 10 7 1.46 10 6 1.93 10 6 Int J Elec & Comp Eng, V ol. 11, No. 6, December 2021 : 5367 – 5378 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Elec & Comp Eng ISSN: 2088-8708 r 5373 T able 4. Error terms for e xample 10 t =1.5,n=50 =1.7,n=50 =1.9,n=50 =1.99,n=50 0.1 0.00 0.00 9.95 10 14 2.66 10 14 0.2 1.42 10 14 7.11 10 15 2.13 10 14 7.99 10 15 0.3 3.55 10 15 3.55 10 15 1.42 10 14 4.88 10 15 0.4 0.00 0.00 8.88 10 15 4.88 10 15 0.5 3.55 10 15 1.78 10 15 7.11 10 15 3.55 10 15 0.6 0.00 0.00 4.88 10 15 2.22 10 16 0.7 1.78 10 15 6.22 10 15 3.16 10 15 7.11 10 15 0.8 8.88 10 16 1.62 10 14 8.88 10 16 2.66 10 14 0.9 4.22 10 15 2.13 10 14 3.55 10 15 7.46 10 14 1.0 1.78 10 15 6.04 10 14 7.11 10 15 1.21 10 13 Example 12: Consider the follo wing nonlinear fractional IVP that describes the cooling of a semi-infinite body by radiation: D a y ( t ) = t 1 (1 y ( t )) 4 ; y (0) = 0 : (20) The e xact solution of (20) for = 1 is of the form: y ( t ) = 1 (1 + 6 t + 9 t 2 ) 1 3 (1 + 3 t ) : (21) F ollo wing the same technique applied to e xample 11 leads us to deduce the general solution of (20). This solution can be written in the follo wing form: y ( t ) = t (2 ) 2 t 2 (2 )(3 ) + 1 3 t 3 3(2 ) 2 + 4(3 )(2 ) (4 ) + 1 24 t 4 4(2 ) 3 24(3 )(2 ) 2 4 3 3(2 ) 2 + 4(3 )(2 ) (4 ) (5 ) + : : : : F or = 1 , the po wer series solution will be as follo ws: y ( t ) = t 2 t 2 + 14 t 3 3 35 t 4 3 + 91 t 5 3 + : : : ; which coincides e xactly with the po wer series of (21). Ho we v er , T able 5 sho ws the approximate solutions together with their residual errors of (21) for dif ferent v alues of n and . Besides, Figure 1 sho ws the e xact and the approximate solution of (21) for dif ferent v alues of at n = 20 . In vie w of these numerical results, it can be asserted that the residual error is decreased when n becomes lar ge. Figure 1. An approximate solution for e xample 12 for dif ferent v alues of Numerical appr oac h of riemann-liouville fr actional derivative oper ator (Ramzi B. Albadarneh) Evaluation Warning : The document was created with Spire.PDF for Python.
5374 r ISSN: 2088-8708 T able 5. Numerical solutions of e xample 12 together with their residual errors for dif ferent v alues of t =0.5, n = 10 = 0 : 5 , n = 20 = 0 : 7 , n = 10 =0.7, n = 20 (r)2-3(r)4-5(r)6-7(r)4-5(r)8-9 t app. Res. app. Res. app. Res. app. Res. 0.1 0.1693669 1 : 74 0.1787661 4 : 44 0.1326126 5 : 31 0.1398065 2 : 76 10 6 10 11 10 7 10 11 0.2 0.2144900 1 : 17 0.2180800 1 : 44 0.1888217 4 : 05 0.1923520 8 : 08 10 5 10 9 10 6 10 12 0.3 0.2391206 4 : 15 0.2416223 1 : 36 0.2236491 1 : 56 0.2261308 3 : 01 10 5 10 8 10 5 10 9 0.4 0.2567284 9 : 36 0.2585229 5 : 23 0.2492026 3 : 71 0.2510679 1 : 38 10 5 10 8 10 5 10 8 0.5 0.2702355 1 : 44 0.2717184 9 : 45 0.2692866 5 : 97 0.2708023 2 : 72 10 4 10 8 10 5 10 8 0.6 0.2813543 1 : 55 0.2825414 8 : 41 0.2858447 6 : 68 0.2871004 2 : 64 10 4 10 8 10 5 10 8 0.7 0.2906744 1 : 18 0.2917126 3 : 32 0.2998796 5 : 25 0.3009584 9 : 88 10 4 10 8 10 5 10 9 0.8 0.2987601 6 : 25 0.2996669 8 : 93 0.3120482 2 : 85 0.3129942 2 : 50 10 5 10 9 10 5 10 9 0.9 0.3059084 2 : 19 0.3066869 2 : 87 0.3227899 1 : 02 0.3236183 2 : 80 10 5 10 8 10 5 10 9 1.0 0.3121975 4 : 58 0.3129633 3 : 13 0.3323502 2 : 19 0.3331162 4 : 32 10 6 10 8 10 6 10 9 Example 13: Consider the follo wing nonlinear fractional IVP; Riccati dif ferential equation: D a y ( t ) = t 1 + 2 t 1 y ( t ) t 1 y ( t ) 2 ; y (0) = 0 : (22) The e xact solution for = 1 is of the form: y ( t ) = e 2 p 2 t 1 p 2 e 2 p 2 t e 2 p 2 t + p 2 + 1 : (23) Lik e wise e xample 11, we found the general solution of (22) as: y ( t ) = t + t 2 (3 ) (2 ) t 3 (4 ) 6 (2 ) 2 2(2 )(3 ) 1 72 t 4 6(3 )(2 ) 2 + (2 ) 2 2(2 )(3 ) (4 ) (5 ) + : F or = 1 , the po wer series solution is of the form: y ( t ) = t + t 2 + t 3 3 t 4 3 7 t 5 15 + ; which coincides e xactly with the po wer series of (23). Ho we v er , T able 6 sho ws the approximate solution together with its residual error of (23) for dif ferent v alues of n and . Besides, Figure 2 sho ws the e xact and the approximate solution of (22) for dif ferent v alues of at n = 20 . Example 14: Consider the follo wing nonlinear fractional IVP: D a y ( t ) = t 2 (3 y 0 ( t )) 2 = 3 ; y (0) = 0 ; y 0 (0) = 0 : (24) The e xact solution of (24) for = 2 is of the form: y ( t ) = 1 108 t 2 t 2 12 3 p 3 t + 54 3 2 = 3 : (25) Int J Elec & Comp Eng, V ol. 11, No. 6, December 2021 : 5367 – 5378 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Elec & Comp Eng ISSN: 2088-8708 r 5375 The same technique applied to e xample 11 can be also applied here to find the general solution of (24) for 1 < < 2 . This solution is, ho we v er , of the form: y ( t ) = 1 2 3 2 = 3 t 2 (3 ) t 3 (3 )(4 ) 3 3 2 = 3 + 1 24 t 4 h 2 9 (3 )(4 ) (3 ) 2 9 i (5 ) + 1 120 t 5 " 4(3 ) 3 81 3 p 3 + 2(4 ) 3 27 3 p 3 (5 ) 2 9 (3 )(4 ) (3 ) 2 9 9 3 p 3 # (6 ) + : Observ e that the po wer series solution, for = 2 , is of the form: y ( t ) = 1 2 3 2 = 3 t 2 t 3 3 3 2 = 3 t 4 108 + : This solution is e xactly equals the po wer series of (25). F or more insight, T able 7 sho ws the approx- imate solution together with its residual error of (25) for dif ferent v alues of n and . From these numerical results, we can ob viously observ e that such error decreases when n suf ficiently increases. T able 6. Numerical solution of e xample 13 together with it’ s residual error for dif ferent v alues of and n (r)2-3(r)4-5(r)6-7(r)4-5(r)8-9 =0.5, n = 10 = 0 : 5 , n = 20 = 0 : 7 , n = 10 =0.7, n = 20 0.1 0.540394230 9 : 72 0.569770720 2 : 73 0.269035900 1 : 07 0.281235570 3 : 98 10 7 10 10 10 7 10 12 0.2 0.897926500 6 : 89 0.917109000 2 : 44 0.525631260 9 : 39 0.537244240 1 : 39 10 12 10 6 10 9 10 10 0.3 1.149028300 2 : 34 1.162603400 6 : 82 0.776353670 3 : 87 0.787393800 1 : 65 10 5 10 9 10 6 10 9 0.4 1.328655900 4 : 96 1.338402600 2 : 72 1.008963800 9 : 42 1.018869100 7 : 33 10 5 10 8 10 6 10 9 0.5 1.460432400 7 : 19 1.467716300 5 : 41 1.213574200 1 : 50 1.222075500 1 : 43 10 5 10 8 10 5 10 8 0.6 1.560239000 7 : 37 1.565813600 4 : 22 1.386626200 1 : 63 1.393695300 1 : 28 10 5 10 8 10 5 10 8 0.7 1.638015900 5 : 36 1.642429100 1 : 82 1.529416700 1 : 24 1.535196300 4 : 97 10 5 10 8 10 5 10 9 0.8 1.700248500 2 : 72 1.703821800 7 : 41 1.645777700 6 : 48 1.650470400 3 : 65 10 5 10 9 10 6 10 9 0.9 1.751172100 9 : 19 1.754111700 7 : 00 1.740261300 2 : 24 1.744068700 1 : 23 10 6 10 8 10 6 10 9 1.0 1.793582400 1 : 86 1.796084600 1 : 25 1.817146700 4 : 63 1.820257000 4 : 75 10 6 10 7 10 7 10 9 Figure 2. An approximate solution of e xample 13 for dif ferent v alues of Numerical appr oac h of riemann-liouville fr actional derivative oper ator (Ramzi B. Albadarneh) Evaluation Warning : The document was created with Spire.PDF for Python.
5376 r ISSN: 2088-8708 T able 7. Numerical solution of e xample 14 together with it’ s residual error for dif ferent v alues of and n (r)2-3(r)4-5(r)6-7(r)4-5(r)8-9 =1.5, n = 10 = 1 : 5 , n = 20 = 1 : 7 , n = 10 =1.7, n = 20 0.1 0.009030744 1 : 11 0.009030744 2 : 22 0.009167872 3 : 33 0.009167872 1 : 44 10 16 10 16 10 16 10 15 0.2 0.035396410 7 : 44 0.035396410 2 : 22 0.036019388 4 : 44 0.036019388 1 : 55 10 15 10 16 10 16 10 15 0.3 0.078049074 3 : 51 0.078049074 4 : 44 0.079604717 1 : 89 0.079604717 1 : 33 10 13 10 16 10 14 10 15 0.4 0.135995280 5 : 15 0.135995280 0 0.139011330 3 : 69 0.139011330 1 : 78 10 12 10 13 10 15 0.5 0.208294260 4 : 07 0.208294260 0 0.213363390 3 : 73 0.213363390 2 : 00 10 11 10 12 10 15 0.6 0.294056080 2 : 17 0.294056080 6 : 66 0.301821170 2 : 48 0.301821170 1 : 11 10 10 10 16 10 11 10 15 0.7 0.392439810 8 : 79 0.392439810 0 0.403580350 1 : 24 0.403580350 1 : 33 10 10 10 10 10 15 0.8 0.502651700 2 : 90 0.502651700 2 : 22 0.517871410 5 : 03 0.517871410 1 : 11 10 9 10 16 10 10 10 15 0.9 0.623943290 8 : 15 0.623943290 8 : 88 0.643958930 1 : 74 0.643958930 4 : 44 10 9 10 16 10 9 10 16 1.0 0.755609580 2 : 01 0.755609580 8 : 88 0.781140820 5 : 27 0.781140820 1 : 33 10 8 10 16 10 9 10 15 5. CONCLUSION In this paper , tw o ef ficient po wer series formulas together with their error terms ha v e been simply deri v ed for the purpose of approximating the Riemann-Liouville fractional deri v ati v e operator . It has been sho wn through addressing se v eral numerical e xamples that these formulas, which successfully ha v e generated ef fecti v e series solutions, can be emplo yed to s olv e man y linear and nonlinear problems in the field of fractional calculus. REFERENCES [1] A. Atang ana, and J. F . G ´ omez-Aguilar , “ Numerical approximation of Riemann-Liouville definition of fractional deri v ati v e: From Riemann-Liouville to Atang ana-Baleanu, ” Numerical Methods for P artial Dif ferential Equations, v ol. 34, no. 5, pp. 1–22, 2017, doi: 10.1002/num.22195. [2] I. M. Batiha, R. B. Albadarneh, S., Momani, and I. H. Jebril, “Dynamics analysis of fractional-order Hopfield neural netw orks, ” International Journal of Biomathematics, v ol.13, no. 08, pp. 2050083, 2020. [3] I. M. Batiha, R. El-Khazali, A. AlSaedi and S. Momani, “The General Solution of Singular Fractional-Order Linear T ime-In v ariant Continuous Systems with Re gular Pencils, ” Entrop y , v ol. 20, no. 6, 2018, doi: 10.3390/e20060400. [4] M. D. P atil, K. V adirajacharya, and K. Sw apnil, “Design of fractional order controllers using constrained optimization and reference tracking method, ” International Journal of Po wer Electronics and Dri v e Systems, v ol. 11, no. 1, pp. 291, 2020, doi: 10.11591/ijpeds.v11.i1.pp291-301. [5] T . Amieur , M. Sedraoui, and O. Amieur , “Design of Rob ust Fractional-Order PID Controller for DC Motor Using the Adjustable Performance W eights in the W eighted-Mix ed Sensiti vity Prob- lem, ” IAES International Journal of Robotics and Automation, v ol. 7, no. 2, pp. 108-118, 2018, doi: 10.11591/ijra.v7i2.pp108-118. [6] B. M. Duarte, and J. A. T enreiro Machado, “Chaotic phenomena and fractional-order dynamics in the tra- jectory control of redundant manipulators, ” Nonlinear Dynamics, v ol. 29, no. 1, pp. 315-342, 2002, doi: 10.1023/A:1016559314798. [7] K. B. Oldham, “Fractional dif ferential equations in electrochemistry , ” Adv ances in Engineering Softw are, v ol. 41, no. 1, pp. 9-12, 2010, doi: 10.1016/j.adv engsoft.2008.12.012. [8] N. Engheta, “On fractional calculus and fractional multipoles in electromagnetism, ” IEEE T ransactions on Antennas and Propag ation, v ol. 44, no. 4, pp. 554-566, 1996, doi: 10.1109/8.489308. [9] R. L. Magin, “Fractional calculus models of comple x dyna mics in biological tissues, ” Computers and Mathematics with Applications, v ol. 59, no. 5, pp.1586-1593, 2010, doi: 10.1016/j.camw a.2009.08.039. [10] V . Gafiychuk, B. Datsk o and V . Meleshk o, “Mathematical modeling of time fractional reaction–dif fusion systems, ” Journal of Computational and Applied Mathematics, v ol. 220, no. 1, pp. 215-225, 2008, doi: 10.1016/j.cam.2007.08.011. [11] A. Atang ana and A. Secer , “ A note on fractional order deri v ati v es and table of fractional deri v ati v es of some special Int J Elec & Comp Eng, V ol. 11, No. 6, December 2021 : 5367 – 5378 Evaluation Warning : The document was created with Spire.PDF for Python.