Hindawi Publishing Corporation Advances in Difference Equations Volume 2011, Article ID 303472, 15 pages doi:10.1155/2011/303472 Research Article On Efficient Method for System of Fractional Differential Equations Najeeb Alam Khan,1 Muhammad Jamil,2, Asmat Ara,1 and Nasir-Uddin Khan1 Department of Mathematics, University of Karachi, Karachi 75270, Pakistan Abdul Salam School of Mathematical Sciences, GC University, Lahore, Pakistan Department of Mathematics, NEDUET, Karachi 75270, Pakistan Correspondence should be addressed to Najeeb Alam Khan, njbalam@yahoo.com Received 14 December 2010; Accepted February 2011 Academic Editor: J J Trujillo Copyright q 2011 Najeeb Alam Khan et al 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 The present study introduces a new version of homotopy perturbation method for the solution of system of fractional-order differential equations In this approach, the solution is considered as a Taylor series expansion that converges rapidly to the nonlinear problem The systems include fractional-order stiff system, the fractional-order Genesio system, and the fractional-order matrix Riccati-type differential equation The new approximate analytical procedure depends only on two components Comparing the methodology with some known techniques shows that the present method is relatively easy, less computational, and highly accurate Introduction Fractional differential equations have received considerable interest in recent years and have been extensively investigated and applied for many real problems which are modeled in different areas One possible explanation of such unpopularity could be that there are multiple nonequivalent definitions of fractional derivatives Another difficulty is that fractional derivatives have no evident geometrical interpretation because of their nonlocal character However, during the last 12 years fractional calculus starts to attract much more attention of scientists It was found that various, especially interdisciplinary, applications 2– can be elegantly modeled with the help of the fractional derivatives The homotopy perturbation method is a powerful devise for solving nonlinear problems This method was introduced by He 7–9 in the year 1998 In this method, the solution is considered as the summation of an infinite series that converges rapidly 2 Advances in Difference Equations This technique is used for solving nonlinear chemical engineering equations 10 , timefractional Swift-Hohenberg S-H equation 11 , viscous fluid flow equation 12 , FourthOrder Integro-Differential equations 13 , nonlinear dispersive K m, n, equations 14 , Long Porous Slider equation 15 , and Navier-Stokes equations 16 It can be said that He’s homotopy perturbation method is a universal one, which is able to solve various kinds of nonlinear equations The new homotopy perturbation method NHPM was applied to linear and nonlinear ODEs 17 In this paper, we construct the solution of system of fractional-order differential equations by extending the idea of 17, 18 This method leads to computable and efficient solutions to linear and nonlinear operator equations The corresponding solutions of the integer-order equations are found to follow as special cases of those of fractional-order equations We consider the system of fractional-order equations of the form Dαi yi t Fi t, y1 , y2 , y3 , , yn fi t , ci , yi t0 < αi ≤ 1, i 1, 2, , n 1.1 Basic Definitions We give some basic definitions, notations, and properties of the fractional calculus theory used in this work Definition 2.1 The Riemann-Liouville fractional integral operator J μ of order μ on the usual Lebesgue space L1 a, b is given by J μf x x Γ μ x−t μ−1 f t dt, μ > 0, 0 J f x 2.1 f x It has the following properties: i J μ exists for any x ∈ a, b , ii J μ J β J μ β, iii J μ J β J βJ μ, iv J α J β f x v Jμ x − a J βJ αf x , γ Γγ /Γ α γ x−a μ γ , where f ∈ L1 a, b , μ, β ≥ and γ > −1 Definition 2.2 The Caputo definition of fractal derivative operator is given by Dμ f x J m−μ Dn f x Γ m−μ t x−τ m−μ−1 f m τ dτ, 2.2 Advances in Difference Equations where m − < μ ≤ m, m ∈ N, x > It has the following two basic properties for m − < μ ≤ m and f ∈ L1 a, b : Dμ J μ f x J μ Dμ f x f x − f x , x−a k , k! m−1 f k k 2.3 x > Analysis of New Homotopy Perturbation Method Let us consider the system of nonlinear differential equations Ai yi fi t , t ∈ Ω, 3.1 where Ai are the operators, fi are known functions and yi are sought functions Assume that operators Ai can be written as Ai yi Li yi Ni yi , 3.2 where Li are the linear operators and Ni are the nonlinear operators Hence, 3.1 can be rewritten as follows: Li yi Ni yi fi t , t ∈ Ω 3.3 We define the operators Hi as Hi Yi ; p ≡ − p Li Yi − Li yi,0 p Ai Y − fi , where p ∈ 0, is an embedding or homotopy parameter, Yi t; p : Ω × 0, → are the initial approximation of solution of the problem in 3.3 can be written as Hi Yi ; p ≡ Li Yi − Li yi,0 pLi yi,0 p Ni Yi − fi 3.4 Ê and yi,0 3.5 Clearly, the operator equations Hi v, 0 and Hi v, are equivalent to the equations and Ai Y − fi t 0, respectively Thus, a monotonous change of Li Yi − L yi,0 parameter p from zero to one corresponds to a continuous change of the trivial problem Li Yi − Li yi,0 to the original problem Operator Hi Yi , p is called a homotopy map Next, we assume that the solution of equation Hi Yi , p can be written as a power series in embedding parameter p, as follows: Yi Yi,0 pYi,1 , i 1, 2, 3, , n 3.6 Now, let us write 3.5 in the following form: Li Yi yi,0 t p fi − Ni Yi − yi,0 t 3.7 Advances in Difference Equations By applying the inverse operator, L−1 to both sides of 3.7 , we have i Yi L−1 yi,0 t i p L−1 f − L−1 Ni Yi − L−1 yi,0 t i i i 3.8 Suppose that the initial approximation of 3.3 has the form ∞ yi,0 t ai,n Pn t , i 1, 2, 3, , n, 3.9 n 0, 1, 2, are unknown coefficients and Pn t , n 0, 1, 2, are specific where ai,n , n functions on the problem By substituting 3.6 and 3.9 into 3.8 , we get Yi,0 pYi,1 ∞ L−1 i ai,n Pn t p L−1 fi − L−1 Ni Yi,0 i i pYi,1 − L−1 i n ∞ ai,n Pn t n 3.10 Equating the coefficients of like powers of p, we get the following set of equations: coefficient of p0 : Y0 ∞ L−1 ai,n Pn t , 3.11 n coefficient of p1 : Y1 L−1 fi i L−1 Yi,1 − L−1 Ni Yi,0 i i Now, we solve these equations in such a way that Yi,1 t solution may be obtained as yi t ∞ L−1 Yi,0 t Therefore, the approximate ai,n Pn t 3.12 n Applications Application Consider the following linear fractional-order 2-by-2 stiff system: α Dt u t k −1 − ε u t α Dt v t k 1−ε u t k 1−ε v t , k −1 − ε v t 4.1 with the initial conditions u0 1, v 3, 4.2 Advances in Difference Equations where k and ε are constants To obtain the solution of 4.1 by NHPM, we construct the following homotopy: α − p Dt U t − u0 t α p Dt U t − k −1 − ε U t − k − ε V t 0, α − p Dt V t − v0 t α p Dt V t − k − ε U t − k −1 − ε V t 4.3 α Applying the inverse operator, Jtα of Dt both sides of the above equation, we obtain U t U Jtα u0 t − pJtα u0 t − k −1 − ε U t − k − ε V t , V t V Jtα v0 t − pJtα v0 t − k − ε U t − k −1 − ε V t 4.4 The solution of 4.1 to has the following form: U t U0 t pU1 t , V t V0 t pV1 t 4.5 Substituting 4.5 in 4.4 and equating the coefficients of like powers of p, we get the following set of equations: Jtα u0 t , U0 t U V0 t U1 t Jtα −u0 t k −1 − ε U0 t V1 t Jtα −v0 t k − ε U0 t Jtα v0 t , V k − ε V0 t , 4.6 k −1 − ε V0 t 20 20 tk , U u , and V Assuming u0 t n an Pn , v0 t n bn P n , P k solving the above equation for U1 t and V1 t lead to the result U1 t a1 tα 2a2 tα 6a3 tα 24a4 t2α 2k − 4εk − a0 tα − − − − Γα Γα Γα Γ α Γα V1 t b1 t α 2b2 tα 6b3 tα 24b4 t2α −2k − 4εk − b0 tα − − − − Γ α Γα Γα Γ α Γα v and ··· , 4.7 Vanishing U1 t and V1 t lets the coefficients , bi , i values: a0 2k − 2ε , a3 a6 a9 −8k4 − 2ε4 , 8k7 − 2ε7 , 45 −8k10 − 2ε10 , 2835 a1 a4 0, 1, 2, by taking α −4k2 − 2ε2 , 4k5 − 2ε5 , a2 a5 a7 −16k8 − 2ε8 , 315 a8 a10 8k11 − 2ε11 , 14175 a11 ··· the following 4k3 − 2ε3 , −8k6 − 2ε6 , 15 4k9 − 2ε9 , 315 −16k12 − 2ε12 , 155925 Advances in Difference Equations a12 8k13 − 2ε13 , 467775 a15 −16k16 − 2ε16 , 155925 a18 8k19 − 2ε19 , 97692469875 b0 b9 b12 b15 b18 −16k14 − 2ε14 , 6081075 a14 16k15 − 2ε15 , 42567525 4k17 − 2ε17 , 638512875 a17 −8k18 − 2ε18 , 10854718875 −16k20 − 2ε20 , 1856156927625 a20 8k21 − 2ε21 , 9280784638125 a16 a19 4k2 2ε2 , −4k3 b2 2ε3 , 2ε , b1 8k4 2ε4 , b4 −4k5 2ε5 , b5 8k6 2ε6 , 15 −8k7 2ε7 , 45 b3 b6 −2k a13 b7 16k8 2ε8 , 315 b8 −4k9 2ε9 , 315 b11 8k10 2ε10 , 2835 b10 −8k11 2ε11 , 14175 −8k13 2ε13 , 467775 b13 16k14 2ε14 , 6081075 16k16 2ε16 , 155925 b16 −4k17 2ε17 , 638512875 −8k19 2ε19 , 97692469875 b19 16k20 2ε20 , 1856156927625 16k12 2ε12 , 155925 b14 b17 b20 −16k15 2ε15 , 42567525 8k18 2ε18 , 10854718875 −8k21 2ε21 9280784638125 4.8 Therefore, we obtain the solutions of 4.1 as 2k − 2ε tα 4k2 − 2ε2 tα − Γα Γα u t 3− 2k 2ε tα Γα 4k2 2ε2 tα Γα v t 8k3 − 2ε3 tα Γα − 8k3 2ε3 tα Γα − 16k4 − 2ε4 tα Γα − 16k4 2ε4 tα Γα ··· , ··· 4.9 Our aim is to study the mathematical behavior of the solution u t and v t for different values of α This goal can be achieved by forming Pade’ approximants, which have the advantage of manipulating the polynomial approximation into a rational function to gain more information about u t and v t It is well known that Pade’ approximants will converge on the entire real axis, if u t and v t are free of singularities on the real axis It is of interest to note that Pade’ approximants give results with no greater error bounds than approximation by polynomials To consider the behavior of solution for different values of α, we will take advantage of the explicit formula 4.9 available for < α ≤ and consider the following two special cases Advances in Difference Equations Case Setting α form as 1, k 50, ε 0.01 in 4.9 , we obtain the approximate solution in a series u 10,11 t 148.73t 1203.65t2 51963.1t3 · · · , 50.7628t 1227.89t2 18726.5t3 · · · v 10,11 t 69.439t 3823.59t2 57.1463t 1550.5t2 40311.9t3 26447.2t3 4.10 ··· ··· Case In this case, we will examine the linear fractional stiff equation 4.1 Setting α 1/2, k 50, ε 0.01 in 4.9 gives 196t1/2 39992t3/2 − √ √ π π ut v t 204t1/2 3− √ π For simplicity, let t1/2 7999984t5/2 228571424t7/2 − √ √ 15 π 15 π 13336t3/2 2666672t5/2 − √ √ π π 196z 39992z3 √ − √ π π v z 204z 3− √ π 7999984z5 228571424z7 − √ √ 15 π 15 π 13336z3 2666672z5 − √ √ π π 9.58 × 10−8 9.581 × 10−8 0.0000126t1/2 2.0904 × 10−6 t1/2 ··· , 4.12 533333344z7 − ··· √ 35 π Calculating the 10/11 Pade’ approximants and recalling that z v 10,11 4.11 533333344t7/2 − ··· √ 35 π z, then uz u 10,11 ··· , t1/2 , we get 0.0002357t − 0.0001051t3/2 − · · · 4.56399 × 10−6 t 2.66 × 10123 − 1.216 × 10125 t1/2 8.88 × 10122 − 6.45605 × 10123 t1/2 0.000096t2 8.69947 × 10125 t 4.22967 × 10124 t ··· ··· ··· , 4.13 Application Consider the following nonlinear fractional-order 2-by-2 stiff system: α Dt u t α Dt v t −1002u t 1000v2 t , u t − v t − v2 t 4.14 with the initial conditions u0 1, v 4.15 Advances in Difference Equations To obtain the solution of 4.14 by NHPM, we construct the following homotopy: α − p Dt U t − u0 t 1002U t − 1000V t α p Dt U t 0, 4.16 α − p Dt V t − v0 t α p Dt V t − U t V2 t V t α Applying the inverse operator, Jtα of Dt both sides of the above equation, we obtain U t Jtα u0 t − pJtα u0 t U 1002U t − 1000V t , 4.17 V t Jtα v0 t − pJtα v0 t − U t V V t V2 t The solution of 4.14 to have the following form: U t U0 t pU1 t , V t V0 t pV1 t 4.18 Substituting 4.18 in 4.17 and equating the coefficients of like powers of p, we get the following set of equations: U0 t Jtα u0 t , U U1 t V1 t V0 t Jtα −u0 t − 1002U0 t Jtα −v0 t V Jtα v0 t , 1000V02 t , U0 t − V0 t − V02 t 20 20 Assuming u0 t tk , U u , and V n an Pn , v0 t n bn P n , P k solving the above equation for U1 t and V1 t lead to the result U1 t a1 tα 2a2 tα 6a3 tα 24a4 tα − a0 tα − − − − − ··· , Γα Γα Γα Γ α Γα V1 t b1 t α 2b2 tα 6b3 tα 24b4 tα − b0 tα − − − − −··· Γα Γα Γ α Γα Γα Vanishing U1 t and V1 t lets the coefficients , bi , i a0 b0 −2, a1 −1, b1 4, a2 1, b2 −4, a3 −1 , b3 , a4 , b4 4.19 4.20 0, 1, 2, to take the following values: −4 , , a20 −1 , , b20 24 v and −8 , 9280784638125 −1 2432902008176640000 4.21 Advances in Difference Equations Therefore, we obtain the solution of 4.14 as ut 1− 2tα Γ α 4tα 8tα − Γα Γα 16tα 32tα − Γ α Γα ··· , v t tα 1− Γα tα tα − Γ α Γα tα tα − Γα Γ α ··· The exact solution of 4.14 for α is u t e−2t , v t 4.22 e−t Application Consider the following nonlinear Genesio system with fractional derivative: α Dt u t v t , α Dt v t w t , 4.23 −cu t − bv t − aw t α Dt w t u2 t with the initial conditions u0 0.2, v −0.3, w 0.1, 4.24 where a, b, and c are constants To obtain the solution of 4.23 by NHPM, we construct the following homotopy: α − p Dt U t − u0 t α p Dt U t − V t 0, α − p Dt V t − v0 t α p Dt V t − W t 0, α − p Dt W t − w0 t α p Dt W t cU t aW t − U2 t bV t 4.25 α Applying the inverse operator, Jtα of Dt both sides of the above equation, we obtain U t W t Jtα u0 t − pJtα u0 t − V t , V t W t U V Jtα v0 t − pJtα v0 t − W t , Jtα w0 t − pJtα w0 t cU t bV t 4.26 aW t − U2 t The solution of 4.23 to have the following form: U t U0 t pU1 t , V t V0 t pV1 t , W t W0 t pW1 t 4.27 10 Advances in Difference Equations Substituting 4.27 in 4.26 and equating the coefficients of like powers of p, we get the following set of equations: U0 t Jtα u0 t , U V0 t V Jtα v0 t , U1 t Jtα −u0 t Jtα −v0 t Jtα w0 t , W0 t , 4.28 Jtα −w0 t − cU0 t − bV0 t − aV0 t W1 t W V0 t , V1 t W0 t W0 t 20 20 20 Assuming u0 t tk , U u0 , n an Pn , v0 t n bn P n , w t n cn Pn , Pk V v ,W w , a 1.2, b 2.92, and c 6, and solving the above equation for U1 t , V1 t and W1 t lead to the result 3/10 tα − a0 a1 tα 2a2 tα 6a3 tα 24a4 tα − − − − − ··· , Γ α Γα Γα Γ α Γα U1 t b1 t α 2b2 tα 6b3 tα 24b4 tα b0 − 1/10 tα − − − − − ··· , Γα Γα Γ α Γα Γα V1 t − c0 W1 t c1 tα 2c2 tα 6c3 tα 24c4 tα 101/250 tα − − − − − ··· Γα Γα Γα Γ α Γα Vanishing U1 t , V1 t , and W1 t lets the coefficients , bi , ci , i following values: a0 −3 , 10 a1 , 10 a2 −101 , 500 a3 2341 , 7500 a4 , 10 −64170831419403533391899 , 1160098079765625000000000000000000 b1 −101 , 250 b20 c0 −101 , 250 0, 1, 2, to take the −377 , , 6250 a20 b0 4.29 c1 c20 b2 2341 , 2500 b3 −754 , 3125 b4 −5153 , , 75000 33855543777297749556491 , 89238313828125000000000000000000 23411 , 1250 c2 −2262 , 3125 c3 −5153 , 75000 c4 5838803330656480870609733 19334967996093750000000000000000000 −508141 , , 3750000 4.30 Advances in Difference Equations 11 Therefore, we obtain the solutions of 4.23 as u t 3tα − 10Γ α v t −3 10 w t 101tα − 10 250Γ α tα 10Γ α tα 101tα − 10Γ α 250Γ α − 101tα 250Γ α 2341tα 4524tα − 1250Γ α 3125Γ α ··· , 2341tα 4524tα 5153tα − − 1250Γ α 3125Γ α 3125Γ α 2341tα 4524tα 5153tα 508141tα − − − 1250Γ α 3125Γ α 3125Γ α 156250Γ α ··· , ··· 4.31 Application Finally, we consider the following nonlinear matrix Riccati differential equation with fractional derivative: −Y t α Dt Y t Q, Y 0, 4.32 where Q 1/2 −1 100 −1 To find the solution of this equation by NHPM, we will 1 treat the matrix equation as a system of fractional-order differential equations 101 , −u2 t − v t z t α Dt u t 99 , α Dt v t −u t v t − v t w t − α Dt z 99 −u t v t − v t w t − , t α Dt w t 4.33 101 , −u2 t − v t z t with the initial conditions u0 0, v 0, z 0, w 0 4.34 Therefore, we obtain the solution of 4.33 as ut v t w t z t 3tα − 10Γ α −3 tα 10 10Γ α tα 101tα − 10Γ α 250Γ α − 101tα 250Γ α 2341tα 4524tα − 1250Γ α 3125Γ α ··· , 2341tα 4524tα 5153tα − − 1250Γ α 3125Γ α 3125Γ α ··· 4.35 12 Advances in Difference Equations 2.5 u, v 2.5 u, v 1.5 1.5 1 0.1 0.2 0.3 0.4 0.5 0.1 0.2 t 0.3 0.4 0.5 0.3 0.4 0.5 t a b u, v u, v 2.5 2 1.5 −2 0.1 0.2 0.3 0.4 0.5 0.1 0.2 t c t d Figure 1: Solutions of linear stiff system for k 50, ε 0.01, α 1, a Exact, b Numerical, c NHPMPade 10/11 , d NHPM-Pade 10/11 , k 50, ε 0.01, α 0.5 color figure can be viewed in the online issue Concluding Remarks The NHPM for solving system of fractional-order differential equations are based on two component procedure and polynomial initial condition The NHPM applied on fractionalorder Stiff equation, fractional Genesio equation, and the matrix Riccati-type differential equation The Applications in problems 1–4 are plotted in Figures 1, 2, 3, and 4, which show the accuracy of NHPM The computations associated with the applications discussed above, were performed by MATHEMATICA The NHPM is very simple in application and is less computational more accurate in comparison with other mentioned methods By using this method, the solution can be obtained in bigger interval Unlike the ADM 19 , the NHPM is free from the need to use Adomian polynomials In this method, we not need the Lagrange multiplier, correction functional, stationary conditions, and calculating integrals, which eliminate the complications that exist in the VIM 20 In contrast to the HPM and HAM, in this method, it is not required to solve the functional equations in each iteration The efficiency of HAM is very much depended on choosing auxiliary parameter All the applications are taken from 20 with fractional derivatives Advances in Difference Equations 13 0.8 u, v 0.8 u, v 0.6 0.6 0.4 0.4 0.2 0.2 0.2 0.4 0.6 0.8 0.2 0.4 0.6 t 0.8 t a b 1 0.9 0.8 0.7 u, v u, v 0.8 0.6 0.6 0.9 α 0.5 α 0.9 α 1/3 α 1/3 0.4 0.4 0.2 0.3 0.2 0.4 0.6 0.8 0.2 0.4 0.8 0.6 t t c d Figure 2: Solutions of nonlinear stiff system for α 1, a Exact, b Numerical, c NHPM, and c NHPM, α 0.9, 1/3 color figure can be viewed in the online issue 0.4 0.4 w 0.3 0.2 0.2 0.1 u, v u, v 0.3 u w 0.1 u −0.1 −0.1 −0.2 v −0.2 v −0.3 −0.3 0.5 1.5 2.5 0.5 1.5 a 0.4 α 0.3 2.5 b 0.4 w α 0.75 w 0.3 0.2 0.1 0.1 u, v 0.2 u, v t t 0 u −0.1 −0.2 u −0.1 −0.2 v v −0.3 −0.3 0.5 1.5 0.5 c 1.5 t t d Figure 3: Solutions of nonlinear Genesio system for a Numerical, b NHPM α d NHPM, α 0.75 color figure can be viewed in the online issue 1, c NHPM, α 0.5, 14 Advances in Difference Equations u, v u, v 0 −2 −2 −4 −6 −4 0.5 1.5 2.5 0.5 1.5 t 2.5 t a b 10 u, v −5 −10 0.5 1.5 2.5 t c Figure 4: Solutions of matrix Riccati equations u w, v z for α a Numerical, b NHPM-Pade 9/11 , c NHPM-Pade 9/11 , α 0.5 color figure can be viewed in the online issue Acknowledgment M Jamil is highly thankful and grateful to the Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan, the Department of Mathematics & Basic Sciences, NED University of Engineering & Technology, Karachi-75270, Pakistan, and also the Higher Education Commission of Pakistan for generously supporting and facilitating this research work References H M Kilbas, H M Srivastava, and J J Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2007 R L Bagley and R A Calico, “Fractional order state equations for the control of 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fi... various kinds of nonlinear equations The new homotopy perturbation method NHPM was applied to linear and nonlinear ODEs 17 In this paper, we construct the solution of system of fractional- order