Sumudu Decomposition Method for Solving Fractional Delay Differential Equations

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1 vol. 1 (2017), Aricle ID , 13 pages doi: /2017/ AgiAl Publishing House hp:// Research Aricle Sumudu Decomposiion Mehod for Solving Fracional Delay Differenial Equaions Hassan Elayeb 1 and Elayeb Abdeldaim 2 1 Mahemaics Deparmen College of Science, King Saud Universiy 2 Omdurman Islamic Universiy College of Science and echnology, Sudan Absrac. In his paper, The Sumudu ransform decomposiion mehod is applied o solve he linear and nonlinear fracional delay differenial equaions (DDEs). Numerical examples are presened o suppor our mehod. Keywords: Sumudu ransform decomposiion, linear and nonlinear fracional delay differenial equaions Mahemaics Subjec Classificaion: 44A05, 44A10, 26a33 Corresponding Auhor Hassan Elayeb hgadain@ksu.edu.sa Edior Maria Alessandra Ragusa Daes Received 21 November 2016 Acceped 19 April 2017 Copyrigh 2017 Hassan Elayeb and Elayeb Abdeldaim. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. 1. Inroducion In he lieraure here are a kind of inegral ransforms used in physics and engineering, he inegral ransforms were exensively used o solve he differenial equaions, several works on he heory and applicaion of inegral ransforms such as Laplace, Fourier, Mellin and Hankel. Waugala [1] inroduce a new inegral ransform named he Sumudu ransform and applied i o soluion of ordinary differenial equaion in conrol engineering problems for properies of Sumudu ransform see [2], [3], [4] and [5]. In [18] Maria Ragusa proved a sufficien condiion for commuaors of fracional inegral operaors. The Sumudu ransform is defined over he se of he funcions: By he following formula: τ A = f() M, τ { 1, τ 2 > 0, f() < Me j, if ( 1) j [0, ). } F(u) = S [f()] = 1 u 0 e u f()d, u ( τ 1, τ 2 ). Delay differenial equaions arise when he rae of change of a ime dependen process in is mahemaical modeling is no only deermined by is presen sae bu also a cerain pas esae known as is hisory. Inroducion of delays in models enriches he dynamics of such models and allow a precise descripion of real life phenomena. DDEs arise frequenly in single processing, digial images, conrol sysem [8], lasers, raffic models [6], meal cuing, populaion dynamic [9], chemical kineics [7], and in many physical phenomena. How o cie his aricle: Hassan Elayeb and Elayeb Abdeldaim, Sumudu decomposiion mehod for solving fracional delay differenial equaions, Research in Applied Mahemaics, vol. 1, Aricle ID , 13 pages, doi: /2017/ Page 1

2 Theorem 1. If F n (u) is he Sumudu ransform of n-h order derivaive of f n () hen for n 1, For more deails see [4]. F n (u) = F(u) u n n 1 k=0 f k (0) u n k. Analysis of he Mehod In his paper we will consider a class of nonlinear delay differenial equaion of he form: wih he iniial condiion: d n y + R(y) + N( τ) = f(), (1) dn u k (0) = u k 0, (2) where y = y(), R is a linear bounded operaor and f() is a given coninuous funcion N is a nonlinear bounded operaor and dn y is he erm of he highes order derivaive. d n The Sumudu decomposiion mehod consiss of applying he Sumudu ransform firs on boh side of (1) o give: By Theorem 1, we have where C = n 1 k=0 f k (0), d n y S [ d n + S [R(y)] + S [N( τ)] = S [f()]. ] S (y()) C + S [R(y)] + S [N( τ)] = S [f()], u n un k S (y()) = u k C u n S [R(y)] u n S [N( τ)] + u n S [f()]. (3) The sandard Sumudu decomposiion mehod defines he soluion y() by he series: he nonlinear operaor is decomposed as: y() = y n (), (4) N( τ) = A n, (5) where A n is he a domain polynomial of y 0, y 1, y 2,, y n ha are given by: A n = 1 d n n! dλ n [ N( λ n y n, n = 0, 1, 2, (6) ] λ=0 doi: /2017/ Page 2

3 The firs a domain polynomials are given by: A 0 = f(y 0 ) A 1 = y 1 f 1 (y 0 ) A 2 = y 2 f 1 (y 0 ) + 1 2! y2 1 f 2 (y 0 ) (7) A 3 = y 3 f 1 (y 0 ) + y 1 y 2 f 2 (y 0 ) + 1 3! y3 1 f 3 (y 0 ). Apply (4) and (5) ino (3) we have: S y [ n ] = uk C u n S R y [ n ] un S A [ n ] + un S [f()], (8) comparing boh side of (8): S [y 0] = u k C + u n S [f()] (9) S [y 1] = u n S [Ry 0] u n S [A 0] (10) S [y 2] = u n S [Ry 1] u n S [A 1]. (11) In general he recursive relaion is given by: S [y n] = u n S [Ry n 1] u n S [A n 1], n 1 (12) applying inverse Sumudu ransform o (9) (12) hen: y 0 = H() (13) y n = S 1 [u n S [Ry n 1] + u n S [A n 1]], n 1 (14) Where H() is a funcion ha a rises from he source erm and prescribed iniial condiions. Numerical Examples Example 1. Consider he nonlinear delay differenial equaion of firs order: y () = 1 2y 2 ( 2 ), 0 1, y(0) = 0. (15) Apply Sumudu ransform o boh side of equaion (15): S [y ()] = S [ 1 2y 2 ( 2 )]. doi: /2017/ Page 3

4 Using Theorem 1 and iniial condiion we have: Y(u) y(0) u = 1 S [ 2y 2 ( Y(u) u = 1 S [ 2y 2 ( Applying he inverse Sumudu ransform o (16) we have: S [y()] = u us [ 2y 2 ( 2 )]. (16) y() = S 1 [u] S 1 [ us ( 2y2 ( 2 ))], y 0 () = S 1 [u] =, y 0 ( 2 ) = 2, (17) From equaion (7) y n+1 () = S 1 [us (2A n)]. (18) A 0 = y 2 0 ( 2) A 1 = 2y 0 ( 2) y 1 ( 2) (19) A 2 = 2y 2 ( 2) y 0 ( 2) + y2 1 ( 2), A n = 0 in equaion (18): subsiuing equaion (19) in (20) we ge: y 1 () = S 1 [us (2A 0)], (20) y 1 () = S 1 [ us ( 2y2 0 ( 2 ))], y 1 () = S 1 [ us ( 2( 4 ) )] = S 1 [ us ( 2 A n = 1 in equaion (18) we have: y 1 () = S 1 [u (u 2 )] = S 1 [u 3 ] = 3 3!, 3 y 1 ( 2 ) = ( 2) 3! = (21) y 2 () = S 1 [us (2A 1)], (22) doi: /2017/ Page 4

5 subsiuing equaions (19) ino (22): y 2 () = S 1 [ us ( 2(2y 0 ( 2 ) y 1 ( 2 ))], y 2 () = S 1 [ us 4 3 ( 2 ( 48 ))] = S 1 [ us 4 ( 24)], y 2 () = S 1 [ us 4! u 4 ( 24 )] = S 1 [u 5 ] = 5 5! = 5 120, A n = 2 in equaion (18) we have: y 2 ( 2 ) = , (23) Subsiuing equaions (19) and (24): y 3 () = S 1 [us (2A 2)] (24) y 3 () = S 1 [ us ( 2(2y 2 ( 2 ) y 0 ( 2 ) + y2 1 ( 2 ))], y 3 () = S 1 [ 2uS 2 5 ( 3840 ( 2 ) ( 48 ) )], y 3 () = S 1 [ 2uS ( )], 6 y 3 () = S 1 [ 2u u 6.6! ( u6.6! 2304)], The series soluion is given by: y 3 () = S 1 [u 7 ] = 7 7! = y() = y 0 () + y 1 () + y 2 () +, The exac soluion is y() = y() = sin() Example 2. Consider he linear delay differenial equaion of firs order y () y ( 2 ) = 0, 0 < α 1, 0 < 1, (25) doi: /2017/ Page 5

6 wih iniial condiion y(0) = 1, apply Sumudu ransform o boh side of (25) using Theorem 1 and iniial condiion: S [y ()] = S [ y ( Y(u) y(0) u = S [ y ( Y(u) 1 u = S [ y ( Y(u) = 1 + us [ y ( Applying he inverse Sumudu ransform o (26): S [y()] = 1 + us [ y ( 2 )]. (26) y() = S 1 [1] + S 1 [ us [ y ( 2 )]], y 0 () = S 1 [1] = 1, y 0 ( 2 ) = 1, (27) a n = 0 in equaion (28): y n+1 () = S 1 [ us [ y n ( 2 )]], (28) y 1 () = S 1 [ us [ y 0 ( 2 )]], y 1 () = S 1 [us [1]], y 1 () = S 1 [u] =, a n = 1 in equaion (28): y 1 ( 2 ) = 2, (29) y 2 () = S 1 [ us [ y 1 ( 2 )]], y 2 () = S 1 [ us [ 2 ]], y 2 () = S 1 u 2 [ 2 ] = 2 4, doi: /2017/ Page 6

7 a n = 1 in equaion (28): y 3 () = S 1 [ us [ y 2 ( 2 )]] = S 1 [ us 2 [ 16]] = u 3 S 1 [ 8 ] = The series soluion is given by: y() = y 0 () + y 1 () + y 2 () +, The exac soluion is y() = k=0 y() = ( 1 2 ) 1 2 k(k 1) Fracional Delay Differenial Equaion k! k. In his secion we apply he Sumudu decomposiion mehod o solve linear and nonlinear fracional delay differenial equaion. Definiion 1. The Sumudu ransform of he Capuo fracional derivaive is defined as follows: n 1 S [D α f()] = u α S [f()] u α+k f k (0), k=0 n 1 < α n, for more deails see [16]. 2. Analysis of he Mehod of Fracional Order Here we will consider a class of nonlinear delay differenial equaion of he form: wih he iniial condiion: D α y() + R(y) + N( τ) = f(), τ R, < τ, n 1 < α n, (30) u k (0) = u k 0, (31) where R is a linear bounded operaor and f() is a given coninuous funcion N is a nonlinear bounded operaor and D α y() is he erm of he fracional order derivaive. The Sumudu decomposiion mehod consiss of applying he Sumudu ransform firs on boh side of (30) o give: by Definiion 1, Where C = n 1 k=0 f k (0) S [D α y()] + S [R(y)] + S [N( τ)] = S [f()], S (y()) C + S [R(y)] + S [N( τ)] = S [f()]. u α uα k S (y()) = u k C + u α S [f()] u α S [R(y)] u α S [N( τ)]. (32) doi: /2017/ Page 7

8 The sandard Sumudu decomposiion mehod define he soluion y() by he series: he nonlinear operaor is decomposed as: y() = y n (), (33) N( τ) = A n (34) Where A n as in (6). The firs a domain polynomials are given as in (7). Apply (33) and (34) in (32) we have: S y [ n ] = uk C + u α S [f()] u α S R y [ n ] uα S A [ n ] (35) Comparing boh side of (35): S [y 0] = u k C + u α S [f()], (36) S [y 1] = u α S [Ry 0] u α S [A 0], (37) S [y 2] = u α S [Ry 1] u α S [A 1]. (38) In general he recursive relaion is given by: S [y n] = u α S [Ry n 1] u α S [A n 1], n 1, (39) applying inverse Sumudu ransform o (36) (39) hen: y 0 = H(), (40) y n = S 1 [u α S [Ry n 1] + u α S [A n 1]], n 1, (41) where H() is a funcion ha a rises from he source erm and prescribed iniial condiions. Example 3. Consider he nonlinear delay differenial equaion of firs order: D α y() = 1 2y 2 ( 2 ), 0 1, 0 < α 1, (42) apply Sumudu ransform o boh side of equaion (42): y(0) = 0, (43) S [D α y()] = S [ 1 2y 2 ( doi: /2017/ Page 8

9 by using Definiion 1 and iniial condiion (43) we have: Y(u) y(0) u α = 1 S [ 2y 2 ( Y(u) u α = 1 S [ 2y 2 ( Applying he inverse Sumudu ransform o (44) we have: S [y()] = u α u α S [ 2y 2 ( 2 )]. (44) y() = S 1 [u α ] S 1 [ uα S ( 2y 2 ( 2 ))], y 0 () = S 1 [u α ] = y 0 ( 2 ) = ( 2) α Γ(α + 1) = α Γ(α + 1), α 2 α Γ(α + 1), (45) From equaion (7), we have y n+1 () = S 1 [u α S (2A n)], (46) A 0 = y 2 0 ( 2) A 1 = 2y 0 ( 2) y 1 ( 2) (47) A 2 = 2y 2 ( 2) y 0 ( 2) + y2 1 ( 2), a n = 0 in equaion (46): subsiuing equaion (47) in (48) we ge: y 1 () = S 1 [u α S (2A 0)], (48) y 1 () = S 1 [ uα S ( 2y 2 0 ( 2 ))], y 1 () = S 1 [ uα S 2 ( ( 2 α Γ(α + 1)) α 2)] = S 1 [ uα S ( 2α 2 2α 1 (Γ(α + 1)) y 1 () = S 1 u 2α Γ(2α + 1) [ uα ( 2 2α 1 (Γ(α + 1)) y 1 () = S 1 u 3α Γ(2α + 1) [ 2 2α 1 (Γ(α + 1)) 2 ], doi: /2017/ Page 9

10 y 1 () = A 3α Γ(3α+1), where A = Γ(2α+1) a n = 1 in equaion (46) we have: subsiuing equaions (47) in (50): 3α Γ(2α + 1) y 1 () = 2 2α 1 (Γ(α + 1)) 2 Γ(3α + 1), 2 2α 1 (Γ(α+1)) 2, y 1 ( 2 ) = A ( 2) 3α Γ(3α + 1) = A 3α 2 3α Γ(3α + 1), (49) y 2 () = S 1 [u α S (2A 1)], (50) y 2 () = S 1 [ uα S ( 2(2y 0 ( 2 ) y 1 ( 2 ))], y 2 () = S 1 [ uα S ( 4y 0 ( 2 ) y 1 ( 2 ))], y 2 () = S 1 [ uα S 4 α ( ( 2 α Γ(α + 1)) ( A 3α 2 3α Γ(3α + 1) ))], The series soluion is given by: 4α y 2 () = S 1 [ uα S A ( 2 4α 2 Γ(3α + 1) )], y 2 () = S 1 [ uα ( A u4α Γ(4α + 1) 2 4α 2 Γ(3α + 1) )], y 2 () = S 1 [ A u5α Γ(4α + 1) 2 4α 2 Γ(3α + 1) ], 5α Γ(4α + 1) y 2 () = A 2 4α 2 Γ(3α + 1)Γ(5α + 1). y() = y 0 () + y 1 () + y 2 () + y() = α Γ(α + 1) A 3α Γ(3α + 1) + A 5α Γ(4α + 1) 2 4α 2 Γ(3α + 1)Γ(5α + 1) +. In paricular caseα = 1hen we have: y() = Γ(2) 3 Γ(4) + 5 Γ(5) 2 2 Γ(4)Γ(6) +, y() = The exac soluion when α = 1 is given by y() = sin() doi: /2017/ Page 10

11 Example 4. Consider he nonlinear delay differenial equaion of firs order wih iniial condiion y(0) = 1. D α y() y ( 2 ) = 0, 0 < α 1, 0 < 1, (51) Apply Sumudu ransform o boh side of (51) Using Definiion 1 and iniial condiion: S [D α y()] = S [ y ( Y(u) y(0) u α = S [ y ( Y(u) 1 u α = S [ y ( Y(u) = 1 + u α S [ y ( Applying he inverse Sumudu ransform o (52): S [y()] = 1 + u α S [ y ( 2 )]. (52) y() = S 1 [1] + S 1 [ uα S [ y ( 2 )]], y 0 () = S 1 [1] = 1, y 0 ( 2 ) = 1, (53) a n = 0 in equaion (54): y n+1 () = S 1 [ uα S [ y n ( 2 )]], (54) y 1 () = S 1 [ uα S [ y 0 ( 2 )]], y 1 () = S 1 [u α S [1]], y 1 () = S 1 [u α ] = α Γ(α + 1), y 1 ( 2 ) = α 2 α Γ(α + 1), (55) doi: /2017/ Page 11

12 a n = 1 in equaion (54): y 2 () = S 1 [ uα S [ y 1 ( 2 )]], y 2 () = S 1 [ uα S [ 2 α Γ(α + 1)]], α The series soluion is given by: y 2 () = S 1 u 2α Γ(α + 1) [ 2 α ] = 2α Γ(α + 1) 2 α Γ(2α + 1). y() = y 0 () + y 1 () + y 2 () +, y() = 1 + In paricular caseα = 1hen we have: The exac soluion is given by y() = k=0 Conclusion α Γ(α + 1) + 2α Γ(α + 1) 2 α Γ(2α + 1) +. y() = k(k 1) ( 2 ) k! In his paper he Sumudu decomposiion mehod has been successfully applied o solve delay and fracional delay differenial equaions. The mehod is very powerful and efficien in finding he exac soluion. k. Compeing Ineress The auhors declare ha hey have no compeing ineress. References [1] G. K. Waugala, Sumudu ransform: a new inegral ransform o solve differenial equaions and conrol engineering problems, Inernaional Journal of Mahemaical Educaion in Science and Technology, vol. 24, no. 1, pp , [2] M. A. Asiru, Furher properies of he Sumudu ransform and is applicaions, Inernaional Journal of Mahemaical Educaion in Science and Technology, vol. 33, no. 3, pp , [3] S. Tuluce Demiray, H. Bulu, and F. B. Belgacem, Sumudu ransform mehod for analyical soluions of fracional ype ordinary differenial equaions, Mahemaical Problems in Engineering, Aricle ID , 6 pages, [4] F. B. Belgacem and A. Karaballi, Sumudu ransform fundamenal properies invesigaions and applicaions, Journal of Applied Mahemaics and Sochasic Analysis. JAMSA, Aricle ID 91083, 23 pages, [5] H. Elayeb, A. Kilicman, and B. Fisher, A new inegral ransform and associaed disribuions, Inegral Transforms and Special Funcions. An Inernaional Journal, vol. 21, no. 5-6, pp , doi: /2017/ Page 12

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