Abstract and Appied Anaysis Voume 203, Artice ID 6700, 9 pages http://d.doi.org/0.55/203/6700 Research Artice New Iterative Method: An Appication for Soving Fractiona Physica Differentia Equations A. A. Hemeda Department of Mathematics, Facuty of Science, Tanta University, Tanta 3527, Egypt Correspondence shoud be addressed to A. A. Hemeda; aahemeda@yahoo.com Received 2 September 202; Accepted 7 March 203 Academic Editor: Soon Y. Chung Copyright 203 A. A. Hemeda. This is an open access artice distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the origina work is propery cited. The new iterative method with a powerfu agorithm is deveoped for the soution of inear and noninear ordinary and partia differentia equations of fractiona order as we. The anaysis is accompanied by numerica eampes where this method, in soving them, is used without inearization or sma perturbation which confirm the power, accuracy, and simpicity of the given method compared with some of the other methods.. Introduction Considerabe attention has been devoted to the study of the fractiona cacuus during the past three decades and its numerous appications in the area of physics and engineering. The appications of fractiona cacuus used in many fieds such as eectrica networks, contro theory of dynamica systems, probabiity and statistics, eectrochemistry, chemica physics, optics, and signa processing can be successfuy modeed by inear or noninear fractiona differentia equations. So far there have been severa fundamenta works on the fractiona derivative and fractiona differentia equations [ 3]. These works are to be considered as an introduction to the theory of fractiona derivative and fractiona differentia equations and provide a systematic understanding of the fractiona cacuus such as the eistence and uniqueness [4, 5]. Recenty, many other researchers have paid attention to eistence resut of soution of the initia vaue probem and boundary probem for fractiona differentia equations [4 6]. Finding approimate or eact soutions of fractiona differentia equations is an important task. Ecept for a imited number of these equations, we have difficuty in finding their anaytica soutions. Therefore, there have been attempts to deveop new methods for obtaining anaytica soutions which reasonaby approimate the eact soutions. Severa such techniques have drawn specia attention, such as Adomain s decomposition method [7], homotopy perturbation method [8 0], homotopy anaysis method [, 2], variationa iteration method [3 7], Chebyshev spectra method [8, 9], and new iterative method [20 22]. Among them, the new iterative method provides an effective procedure for epicit and numerica soutions of a wide and genera cass of differentia systems representing rea physica probems. The new iterative method is more superior than the other noninear methods, such as the perturbation methods where this method does not depend on sma parameters, such that it can find wide appication in noninear probems without inearization or sma perturbation. The motivation of this paper is to etend the appication ofthenewiterativemethodproposedbydaftardar-gejji and Jafari [20 22] to sove inear and noninear ordinary and partia differentia equations of fractiona order. This motivation is based on the importance of these equations and their appications in various subjects in physica branches [0,, 4, 23 25]. There are severa definitions of a fractiona derivative of order α>0[3, 26]. The two most commony used definitions are Riemann-Liouvie and Caputo. Each definition uses Riemann-Liouvie fractiona integration and derivative of whoe order. The difference between the two definitions
2 Abstract and Appied Anaysis is in the order of evauation. Riemann-Liouvie fractiona integration of order α is defined as I α f( = Γ (α (ξ α f (ξ dξ, 0 α>0, >0. The net two equations define Riemann-Liouvie and Caputo fractiona derivatives of order α, respectivey,as D α f( = dm d m (Imα f(, ( (2a From (5and(6, (4is equivaent to u i =fn(u 0 { { N( i= { i j=0 i u j N( j=0 We define the foowing recurrence reation: u 0 =f, u j } }. } (7 D α f( =Imα ( dm f(, dm (2b where m<α<m, m N. Caputo fractiona derivative first computes an ordinary derivative foowed by a fractiona integra to achieve the desired order of fractiona derivative. Riemann-Liouvie fractiona derivative is computed in the reverse order. Therefore, Caputo fractiona derivative aows traditiona initia and boundary conditions to be incuded in the formuation of the probem. From properties of D α and Iα, it is important to note that D α β = Γ(ββα, β α, (3a Γ(βα where D α is Caputo derivative operator of order α, I α β = Γ(ββα Γ(βα. 2. Basic Idea of New Iterative Method (3b For the basic idea of the new iterative method, we consider the foowing genera functiona equation [20 22]: u ( =f( N(u (, (4 where N is a noninear operator from a Banach space B B and f is a known function. We have been ooking for a soution of (4having the series form u ( = u i (. (5 The noninear operator N canbedecomposedas N( u i =N(u 0 { { N( i= { i j=0 i u j N( j=0 u j } }. } (6 Then, u =N(u 0, u n =N(u 0 u u n N(u 0 u u n, n =, 2,.... (u u 2 u n =N(u 0 u u n, u= u i =fn( n=,2,..., u i. If N( N(y < k y, 0<k<,then u n = N(u 0 u n N(u 0 u n k u n kn u 0, n=0,,2,..., (8 (9 (0 and the series u i absoutey and uniformy converges to asoutionof(4[27], which is unique, in view of the Banach fied point theorem [28]. The n-term approimate soution of (4and(5isgivenbyu( = n u i. 2.. Convergence of the Method. Now we anayze the convergence of the new iterative method for soving any genera functiona equation (4. Let e=u u,whereu is the eact soution, u is the approimate soution, and e is the error in the soutionof (4; obviousy e satisfies (4, that is, e ( =f( N(e ( ( and the recurrence reation (8becomes e 0 =f, e =N(e 0, e n =N(e 0 e e n N(e 0 e e n, n=,2,... (2
Abstract and Appied Anaysis 3 If N( N(y k y, 0<k<,then e 0 =f, e = N(e 0 k e 0, e 2 = N(e 0 e N(e 0 k e k2 e 0, e 3 = N(e 0 e e 2 N(e 0 e. k e 2 k3 e 0, e n = N(e 0 e n N(e 0 e n k e n kn e 0, n=0,,2,... (3 Thus e n 0as n, which proves the convergence of the new iterative method for soving the genera functiona equation (4. For more detais, you can see [29]. 3. Suitabe Agorithm In this section, we introduce a suitabe agorithm for soving noninear partia differentia equations using the new iterative method. Consider the foowing noninear partia differentia equation of arbitrary order: D α tu (, t =A(u, u B(, t, m<α<m, m N, (4a k t u (, 0 =h k k (, k=0,,...,m, (4b where A is a noninear function of u and u (partia derivatives of u with respect to and t andb is the source function. In view of the new iterative method, the initia vaue probem (4aand(4b isequivaentto theintegraequation where u (, t = m k=0 h k ( tk k! Iα t BIα t f= m k=0 h k ( tk k! Iα t B, N (u =I α t A. A=fN(u, (5 (6a (6b Remark. When the genera functiona equation (4is inear, therecurrencereation(8 can be simpified in the form u 0 =f, u n =N(u n, n = 0,, 2,.... (7 Proof. From the properties of integration and by using (8 and (6b, we have u n =N(u 0 u n u n N(u 0 u n =I α t [u 0 u n u n ] I α t [u 0 u n ] =I α t [u 0] I α t [u n] I α t [u n]i α t [u 0]I α t [u n] =I α t [u n]=n(u n, n = 0,, 2,.... (8 Therefore, we get the soution of (5 by empoying the recurrence reation (8or(7. 4. Appications To iustrate the effectiveness of the proposed method, severa test eampes are carried out in this section. Eampe 2. In this eampe, we consider the foowing initia vaueprobeminthecaseoftheinhomogeneousbagey- Torvik equation [23, 24]: D 2 u ( D.5 u ( u( =g(, u (0 =, u (0 =, [0, L], (9 where g( =. The eact soution of this probem is u( =. By appying the technique described in Sections 2 and 3, theinitiavaueprobem(9 is equivaent to the integra equation u ( = 2 2 3 6 I2 [D.5 u ( u(]. (20 Let N(u = I 2 [D.5 u( u(]. Inviewofrecurrence reation (7, we have the foowing first approimations: u 0 ( = 2 2 3 6, u ( =N(u 0 = 82.5 5 π 63.5 05 π u 2 ( =N(u = 82.5 5 π 63.5 05 π 2 2 3 6 4 24 5 20, 644.5 945 π 285.5 0395 π 3 6 4 2 5 20 6 720 7 5040, (2
4 Abstract and Appied Anaysis u 2.8.6.4.2 0.2 0.4 0.6 0.8 Figure : Pots of the approimate soution and the eact soution for (9. and so on. In the same manner the rest of components can be obtained. The 6-term approimate soution for (9 is u ( = 5 u i = 6 44 57 5040 9 36288 0 362880 2 47900600 3 6227020800 324.5 945 π 645.5 0395 π 527.5 405405 π 0248.5 689885 π 52 0.5 687465529 π 28.5 976463395 π. (22 Remark 3. In Eampe 2. we have used the recurrence reation (7. If we used the recurrence reation (8 inpaceof (7, we obtain the same resut. In Figure, we have potted the 6-term approimate soution with the corresponding eact soution for (9. It is remarkabe to note that the two soutions are amost equa. Comparing these obtained resuts with those obtained by new Jacobi operationa matri in [23, 24], we can confirm the simpicity and accuracy of the given method. Eampe 4. Consider the foowing fractiona Riccati equation [0]: D α u ( u2 ( =, u(0 =0, >0, 0<α. (23 The eact soution when α=is u( = (e 2 /(e 2. By appying the technique described in Sections 2 and 3, theinitiavaueprobem(23 is equivaent to the integra equation α u ( = Γ (α Iα [u2 (]. (24 Let N(u = I α [u2 (]. In view of recurrence reation (8, we have the foowing first approimations: u 0 ( = α Γ (α, Γ (2α 3α u ( =N(u 0 = Γ (3α Γ(α 2, u 2 ( =N(u 0 u N(u 0 = 2Γ (2α Γ (4α 5α Γ (3α Γ (5α Γ(α 3 Γ(2α 2 Γ (6α 7α Γ(3α 2 Γ (7α Γ(α 4, and so on. The 4-term approimate soution for (23is u ( = = 3 u i α Γ (α Γ (2α 3α Γ (3α Γ(α 2 2Γ (2α Γ (4α 5α Γ (3α Γ (5α Γ(α 3 Γ(2α 2 Γ (6α 7α Γ(3α 2 Γ (7α Γ(α 4 4Γ (2α Γ (4α Γ (6α 7α Γ (3α Γ (5α Γ (7α Γ(α 4 4Γ(2α2 Γ (4α Γ(3α 2 Γ (5α Γ (8α 9α Γ (9α Γ(α 5 2Γ(2α 2 Γ (6α Γ (8α 9α Γ(3α 2 Γ (7α Γ (9α Γ(α 5 4Γ(2α2 Γ(4α 2 Γ ( 0α α Γ(3α 2 Γ(5α 2 Γ ( α Γ(α 6 2Γ(2α 3 Γ(3α 3 Γ (7α Γ (6α Γ ( 0α α Γ ( α Γ(α 6 (25
Abstract and Appied Anaysis 5 4Γ(2α3 Γ (4α Γ (6α Γ(3α 3 Γ (5α Γ (7α Γ ( 2α 3α Γ ( 3α Γ(α 7 Γ(2α 4 Γ(6α 2 Γ ( 4α 5α Γ (3α 4 Γ(7α 2 Γ ( 5α Γ(α 8. (26 u 0.7 0.6 0.5 0.4 0.3 0.2 In Figure 2,we have potted the 4-term approimate soution for (23 for different vaues of α with the corresponding eact soution. It is remarkabe to note that the approimate soution, in case α =, and the eact soution are amost equa (continuous curve whenever the approimate soution, in cases α = 0.9, 0.8, is of high agreement with the eact soution (dashed and dotted curves, resp.. Comparing the obtained resuts with those obtained by homotopy anaysis method, in case h=, in[0], we can confirm the simpicity and accuracy of the given method. Eampe 5. Consider the foowing initia vaue probem with fractiona order [23, 24]: D 3 u ( D2.5 u ( u2 ( = 4, u (0 =u (0 =0, u (0 =2. (27 The eact soution for this probem is u( = 2. As in Eampe 4,theinitiavaueprobem(27isequivaent to the integra equation u ( = 2 7 20 I3 [D2.5 u ( u2 (]. (28 Let N(u = I 3 [D2.5 u( u2 (].Inviewofrecurrence reation (8, we have the foowing first approimations: u 0 ( = 2 7 20, u ( =N(u 0 = 290247.5 42567525 π 7 20 0. 0 0.2 0.4 0.6 0.8 Figure 2: Pots of the approimate soution for different vaues of α and the eact soution for (23. u 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 Figure 3: Pots of the approimate soution and the eact soution for (27. and so on. The 4-term approimate soution and the corresponding eact soution for (27 are potted in Figure 3. It is remarkabe to note that the two soutions are amost equa. Comparing these obtained resuts with those obtained by new Jacobi operationa matri in [23, 24], we can confirm the simpicity and accuracy of the given method. 2 38600 7 79928000, u 2 ( =N(u 0 u N(u 0 =.7008 3 7.5 5.95252 4 8 7.250 6 2 4.28636 6 2.5 9.09455 9 7.39 9 7.5 5.9730 0 8.20298 2 22 2.48834 2 22.5 2.9667 5 27.02289 5 27.5 2.69485 8 32 6.62568 28 37, (29 Eampe 6. Consider the foowing fractiona order wave equation in 2-dimensiona space [4]: D α t u(,y,tc(u u y =0, u(,y,0=sin [π ( y ], 0<α. The eact soution for this probem when α=is (30 u(,y,t=sin [π ( y2ct ]. (3 The initia vaue probem (30 is equivaent to the integra equation u(,y,t=sin [π ( y ] I α t [c (u u y ]. (32
6 Abstract and Appied Anaysis Let N(u = I α t [c(u u y ]. In view of recurrence reation (7, we have the foowing first approimations: u 0 (,y,t=sin [π ( y ], u (,y,t= Γ (α (2cπtα cos [π ( y ], u 2 (,y,t= Γ (2α (2cπtα u 3 (, y, t = Γ (3α (2cπtα 2 3 sin [π ( y ], cos [π ( y ], (33 and so on. The n-term approimate soution for (30is u(,y,t= n u i = sin [π ( y ] [ Γ (2α (2cπtα 2 which is the eact soution for the given probem. When α=,the above n-term approimate soution for (30becomes u(,y,t= n u i = sin [π ( y ] [ 2! (2cπt 2 4! (2cπt 6 6! (2cπt ] cos [π ( y ] [( 2cπt In cosed form, this gives 3 3! (2cπt 5! (2cπt 5 7! (2cπt 4 7 ]. (36 In cosed form this gives: Γ (4α (2cπtα 4 6 Γ (6α (2cπtα ] cos [π ( y ] [ Γ (α (2cπtα Γ (3α (2cπtα Γ (5α (2cπtα 3 5 7 Γ (7α (2cπtα ]. u(,y,t= sin [π ( y ] sin ( 2cπtα cos ( 2cπtα cos [π ( y ], =sin [π ( y2ctα ] (34 (35 u(,y,t= sin [π ( y ] sin ( 2cπt cos ( 2cπt cos [π ( y ], =sin [π ( y2ct ], (37 which is the same resut obtained by variationa iteration method in [4]. Eampe 7. Consider the foowing fractiona order heat equation in 2-dimensiona space []: D β t u(,y,t=α(u u yy, u(,y,0=c[sin ( π sin ( πy (38 ], 0<β. The eact soution for this probem when β=is u(,y,t=ce απ2 t/ 2 [sin ( π sin ( πy ]. (39 The initia vaue probem (38 is equivaent to the integra equation u(,y,t=c[sin ( π sin ( πy ] Iβ t [α (u u yy ]. (40
Abstract and Appied Anaysis 7 Let N(u = I β t [α(u u yy ]. Inviewofrecurrence reation (7, we have the foowing first approimations: u 0 (,y,t=c[sin ( π sin ( πy ], u (,y,t= u 2 (,y,t= u 3 (,y,t= c t β Γ(β (απ2 2 [sin ( π sin ( πy ], c t β 2 Γ(2β (απ2 2 [sin ( π sin ( πy ], c t β 3 Γ(3β (απ2 2 [sin ( π sin ( πy ], and so on. The n-term approimate soution for (38is u(,y,t= n u i =c[sin ( π sin ( πy ] [ t β Γ(β (απ2 2 t β 2 Γ(2β (απ2 2 t β 3 Γ(3β (απ2 2 ]. (4 (42 When β =,Then-term approimate soution for (38 becomes u(,y,t= n u i In cosed form, this gives =c[sin ( π sin ( πy ] [ t! (απ2 2 2 t 2! (απ2 2 3 t 3! (απ2 2 ]. (43 u(,y,t=ce απ2 t/ 2 [sin ( π sin ( πy ] (44 which is the eact soution for the given probem. The obtained resuts in this eampe are the same as these obtained in [] by the homotopy perturbation method, in case β=, but with the simpicity of the given method. Eampe 8. In this ast eampe, we consider the foowing fractiona order noninear wave equation [25]: D α u (, t uu 2α tt = Γ (3α ( 2 t 2, 2 u (0, t = t2 2, u (0, t =0, <α 2. (45 The eact soution for this probem when α=2is u(, t = (/2( 2 t 2 where 0, t. The initia vaue probem (45 is equivaent to the integra equation u (, t = 2 (2 t 2 2α Γ (3α t 2 α 2Γ (α Iα [uu tt]. (46 Let N(u = I α [uu tt]. In view of recurrence reation (8, we have u 0 (, t = 2 (2 t 2 u (, t = 2α Γ (3α t 2 α 2Γ (α, 2α Γ (3α 22α Γ (32α Γ (3α 22α 2Γ (α Γ (32α Γ (32α 23α Γ (α Γ (3α Γ (33α t 2 α ( 2Γ (α 2α Γ (2α Γ (2α 3α 2Γ(α 2 Γ (3α, (47 and so on. The 3-term approimate soution and the corresponding eact soution for (45are potted in Figure 4(a,in case t = /2, for α =.8,.9, 2., in Figure 4(b,in case t =, for α =.8,.9, 2., and in Figure 4(c, incaseα = 2.Itis remarkabe to note that in the first two figures a the soutions are amost equa. Comparing these resuts with those obtained by the modification homotopy perturbation method in [25], we can confirm the accuracy and simpicity of the given method. 5. Concusion In this paper, the new iterative method with suitabe agorithm is successfuy used to sove inear and noninear
8 Abstract and Appied Anaysis 0.6 0.9 u 0.5 0.4 u 0.8 0.7 0.3 0.6 0.2 0.4 0.6 0.8 (a 0.2 0.4 0.6 0.8 (b u 0.5 0 0 0.5 t 0.5 (c 0 Figure 4: (a Pots of the approimate soution for different vaues of α and the eact soution, in case t=/2;for(45. (b Pots of the approimate soution for different vaues of α and the eact soution, in case t=;for(45. (c Pots of the approimate soution, in case α=2 for (45. ordinary and partia differentia equations with fractiona order. It is cear that the computations are easy and the soutions agree we with the corresponding eact soutions and more accurate than the soutions obtained by other methods. Moreover, the accuracy is high with itte computed terms of the soution which confirm that this method with the given agorithm is a powerfu method for handing fractiona differentia equations. References [] K. Mier and B. Ross, An Introduction to the Fractiona Cacuus and Fractiona Diffrentia Equations, JohnWiey&Sons,New York, NY, USA, 993. [2] K.B.OdhamandJ.Spanier,The Fractiona Cacuus, Academic Press, London, UK, 974. [3] I. Podubny, Fractiona Differentia Equations, vo. 98, Academic Press, San Diego, Caif, USA, 999. [4] M. Amairi, M. Aoun, S. Najar, and M. N. Abdekrim, A constant encosure method for vaidating eistence and uniqueness of the soution of an initia vaue probem for a fractiona differentia equation, Appied Mathematics and Computation, vo.27,no. 5, pp. 262 268, 200. [5] J. Deng and L. Ma, Eistence and uniqueness of soutions of initia vaue probems for noninear fractiona differentia equations, Appied Mathematics Letters,vo.23,no.6,pp.676 680, 200. [6] E. Girejko, D. Mozyrska, and M. Wyrwas, A sufficient condition of viabiity for fractiona differentia equations with the Caputo derivative, Journa of Mathematica Anaysis and Appications, vo. 38, no., pp. 46 54, 20. [7] S. S. Ray and R. K. Bera, Soution of an etraordinary differentia equation by Adomian decomposition method, Journa of Appied Mathematics,no.4,pp.33 338,2004. [8] M. Dehghan, J. Manafian, and A. Saadatmandi, Soving noninear fractiona partia differentia equations using the homotopy anaysis method, Numerica Methods for Partia Differentia Equations,vo.26,no.2,pp.448 479,200. [9] I. Hashim, O. Abduaziz, and S. Momani, Homotopy anaysis method for fractiona IVPs, Communications in Noninear Science and Numerica Simuation, vo.4,no.3,pp.674 684, 2009. [0] Z. Odibat, S. Momani, and H. Xu, A reiabe agorithm of homotopy anaysis method for soving noninear fractiona differentia equations, Appied Mathematica Modeing,vo.34, no.3,pp.593 600,200.
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