An Analytical Scheme for Multi-order Fractional Differential Equations

Transcription

1 Tamsui Oxford Journal of Mathematical Sciences 26(3) (2010) Aletheia University An Analytical Scheme for Multi-order Fractional Differential Equations H. M. Jaradat Al Al Bayt University, Jordan Fadi Awawdeh Hashemite University, Jordan and E. A. Rawashdeh Dhofar University, Oman Received November 18, 2009, Accepted December 11, Abstract In this work, on the basis of homotopy analysis method, we propose a numerical procedure for solving multi-order fractional differential equations by transforming the original multi-order fractional differential equation into a system of single-order equations. To our knowledge this paper presents the first viable numerical method for an analytic solution of multi-order fractional differential equations. This approach provides a simple way to ensure the convergence of series of solution so that one can always get accurate enough approximations. Keywords and Phrases: Analytic solution, Multi-order fractional differential equations Mathematics Subject Classification. Primary 65H20, 65H05, 45J05. Corresponding author. husseinjaradat@yahoo.com awawdeh@hu.edu.jo edirash@yahoo.com

2 306 H. M. Jaradat, Fadi Awawdeh, and E. A. Rawashdeh 1. Introduction This paper concerns the numerical solution of multi-order fractionaldifferential equations with general form D α y(t) =f(t, y(t),dβ 1 y(t),dβ 2 y(t),..., Dβ n y(t)), (1) where D γ y(t) is used to represent the Caputo-type fractional derivative of order γ>0. We assume that α>β n >β n 1 > >β 1 and α β n 1, β j β j 1 1 for all j and 0 <β 1 1. An initial value problem consists of (1) equipped with initial conditions y (k) (0) = a k, k =0, 1,..., [α] 1. (2) The notation [α] is used to denote the integer closest to and not less than α. Multi-order fractional differential equations have been used to model various types of visco-elastic damping (see [3]). Model equations proposed so far are almost always linear. Indeed, some writers have proposed that the use of fractional differential equations in a model can avoid altogether the need to introduce nonlinearity. So our experiments in this paper will focus on equations of the linear form D α y(t) =λ 0y(t)+ n j=1 λ j D β j y(t)+f(t). (3) The use of fractional differentiation for the mathematical modeling of real world physical problems has been widespread in recent years, e.g. the modeling of earthquake, the fluid dynamic traffic model with fractional derivatives, measurement of viscoelastic material properties, etc. Numerical methods for the solution of linear fractional differential equations involving only one fractional derivative are well established (see for example [4], [5] and [9]-[11]). There have been some attempts to solve linear problems with multiple fractional derivatives (the so-called multi-order equations [6],[7],[8],[12]) but a complete analysis has not been given so far. In 2004, Diethelm and Ford [6] introduced the first scheme for the solution of multi-order FDEs of the form (1). They converted equations with commensurate multiple fractional derivatives into a very simple linear system of

3 An Analytical Scheme for Multi-order Fractional Differential Equations 307 fractional differential equations of low order and then the application of any single-term equation solver solves the problem. We are concerned with providing a good quality algorithms for the solution of multi-order fractional differential equations. In this paper, we will suggest an approach to the search for an explicit analytic solution for the initial value problem consists of (1) and equipped with initial conditions (2) by homotopy analysis method (HAM). For more details about HAM the reader can refer to [1],[2] and [13]-[21]. 2. Preliminaries and Notations In this section, let us recall essentials of fractional calculus. The fractional calculus is a name for the theory of integrals and derivatives of arbitrary order, which unifies and generalizes the notions of integer-order differentiation and n-fold integration. For the purpose of this paper the Caputo s definition of fractional differentiation will be used, taking the advantage of Gaputo s approach that the initial conditions for fractional differential equations with Caputo s derivatives take on the traditional form as for integer-order differential equations. Definition 1. Caputo s definition of the fractional-order derivative is defined as D α 1 t f(t) = (t τ) n α 1 f (n) (τ)dτ, Γ(n α) a where n 1 <α n, n N, α is the order of the derivative and a is the initial value of function f. For the Caputo s derivative we have: D α C = 0, C is constant, { D α t β 0 β α 1 = Γ(β+1) Γ(β α+1) tβ α β>α 1 For establishing our results, we also necessarily introduce the following Riemann Liouville fractional integral operator.

4 308 H. M. Jaradat, Fadi Awawdeh, and E. A. Rawashdeh Definition 2. The Riemann Liouville fractional integral operator of order α 0, of a function f C μ, μ 1, is defined as J α f(t) = 1 Γ(α) t (t τ) α 1 f(τ)dτ a We mention only some properties of the operator J α :Forf C μ, μ, γ 1, α, β 0: J 0 f(t) = f(t), J α J β f(t) =J α+β f(t) = J β J α f(t), J α t γ Γ(γ +1) = Γ(γ + α +1) tγ+α Also, we need here two of its basic properties. If m 1 <α m, m N, and f Cμ m, μ 1, then D α J α f(t) =f(t), J α D α f(t) =f(t) m 1 i=0 f (i) (0 + ) ti i!, t > 0. Let M be the least common multiple of the denominators of α, β 1,..., β n, and set γ = 1/M and N = Mα. We have the following theorem on equivalence of a nonlinear system: Theorem 1 (Diethelm and Ford [6]). Equation (1), equipped with the initial conditions (2), is equivalent to the system of equations D γ y 0 (t) = y 1 (t), D γ y 1(t) = y 2 (t), D γ y 2(t) = y 3 (t), D γ y N 2 (t) = y N 1 (t), together with the initial conditions. (4) D γ y N 1 (t) = f(t, y β1 /γ(t),...,y βn /γ(t)), { ak if j = km with some k N y j (0) = 0 otherwise, (5) in the following sense:

5 An Analytical Scheme for Multi-order Fractional Differential Equations 309 (1) Whenever Y =(y 0,..., y N 1 ) T with y 0 C α [0,b] for some b>0 is the solution of the system (4), equipped with the corresponding initial conditions, the function y = y 0 solves the multi-order equation (1), and it satisfies the initial conditions (2). (2) Whenever y C [α] [0,b] is a solution of the multi-order equation (1) satisfying the initial conditions (2), the vector-valued function Y =(y 0,..., y N 1 ) T =(y 0,D γ y, D2γ satisfies the system (4) and the initial conditions (5). y,..., D(N 1)γ y) T It follows that we can rewrite Eq. (3) as a system of N single-term equations D γ Y (t) =G(t, Y (t)) (6) with initial conditions Y (0) = Γ 0 (7) where Γ 0 =(y 0 (0),y 1 (0),..., y N 1 (0)) T. For more information about the mathematical properties of fractional derivatives and integrals, one can consult [4] and [22]. 3. Numerical Schemes In this section, we present a complete description of a new approach for the solution of multi-order fractional differential equations (1)-(2) or what amount to the same, D γ Y (t) = G(t, Y (t)) Y (0) = Γ 0 where Y (t) =(y 0 (t),y 1 (t),..., y N 1 (t)) T. According to (4), y i (t) can be expressed by the base functions {t γn : n 1, n N} as y i (t) = b n t γn, n=1

6 310 H. M. Jaradat, Fadi Awawdeh, and E. A. Rawashdeh where b n is a coefficient. It provides us the so-called Solution Expression of y i (t). We define the fractional operator F F[φ(t; q)] = D γ φ(t; q) G(t, φ(t; q)), where q [0, 1] denotes the embedding parameter. Let Y 0 (t) denotes an initial guess of the exact solution Y (t) which satisfies the initial condition (7). Also, h 0 an auxiliary parameter and L an auxiliary linear operator. All of Y 0 (t), L and h will be chosen later with great freedom. Then, we construct a oneparameter family of fractional differential equations subject to the boundary conditions (1 q)l[φ(t; q) Y 0 (t)] = qhf[φ(t; q)] (8) k φ(t; q) t k t=0 =Γ k, k =0, 1,..., [α] 1. (9) Obviously, when q = 0, because of the property L(0) = 0 of any linear operator L, Eqs. (8) and (9) have the solution φ(t;0)=y 0 (t), (10) and when q =1,sinceh 0, Eqs. (8) and (9) are equivalent to the original ones, (1) and (2), provided φ(t;1)=y (t). (11) Thus, according to (10) and (11), as the embedding parameter q increases from 0to1,φ(t; q) varies continuously from the initial approximation Y 0 (t) tothe exact solution Y (t). This kind of deformation φ(t; q) is totally determined by the so-called zeroth-order deformation equations (8) and (9). By Taylor s theorem, φ(t; q) can be expanded in a power series of q as follows φ(t; q) =Y 0 (t)+ Y m (t)q m, (12) where m=1 Y m (t) =D m [φ(t; q)] = 1 m φ(t; q) m! q q=0. (13) m D m is called the mth-order homotopy-derivative of φ.

7 An Analytical Scheme for Multi-order Fractional Differential Equations 311 Fortunately, the homotopy-series (12) contains an auxiliary parameter h, and besides we have great freedom to choose the auxiliary linear operator L, as illustrated by Liao [20]. If the auxiliary linear parameter L and the nonzero auxiliary parameter h are properly chosen so that the power series (12) of φ(t; q) converges at q =1. Then, we have under these assumptions the the so-called homotopy-series solution Y (t) =Y 0 (t)+ Y m (t). According to the fundamental theorems in calculus, each coefficient of the Taylor series of a function is unique. Thus, Y m (t) is unique, and is determined by φ(t; q). Therefore, the governing equations and boundary conditions of Y m (t) can be deduced from the zeroth-order deformation equations (8) and (9). For brevity, define the vectors m=1 Y n (t) ={Y 0 (t),y 1 (t),y 2 (t),...,y n (t)}. Differentiating the zero-order deformation equation (8) m times with respective to q and then dividing by m! and finally setting q =0,wehavethe so-called high-order deformation equation L[Y m (t) χ m Y m 1 (t)] = hr m ( Y m 1 (x)), (14) Y m (0) = 0, where R m ( y m 1 (x)) = D m 1 (N[φ]) = 1 m 1 N[φ(x; q)] (m 1)! q m 1 q=0 (15) and In this line we have that, χ m = { 0, m 1 1, m > 1. R m ( Y m 1 (t)) = D γ Y m 1(t) G(t, Y m 1 (t)). (16) So, by means of symbolic computation software such as Mathematica, Maple, Matlab and so on, it is not difficult to get R m ( Y m 1 (t)) for large values of m.

8 312 H. M. Jaradat, Fadi Awawdeh, and E. A. Rawashdeh Note that the high-order deformation equations (14) are linear FDEs. So, the original nonlinear problem is transferred into an infinite number of linear ODEs. However, unlike perturbation techniques, we do not need any small physical parameters to do such a kind of transformation. Besides, unlike the traditional non-perturbation techniques, we have great freedom to choose the auxiliary linear operator L and the initial guess Y 0 (t). Both the auxiliary linear operator L and the initial guess Y 0 (t) arechosen under the so-called Rule of Solution Expression: the auxiliary linear operator L and the initial guess Y 0 (t) must be chosen so that the solutions of the highorder deformation equations (14) exist and besides they obey the Solution Expression. So, for the solutions to obey the Solution Expression and the initial conditions (7), we choose the initial guess of the solution: Y 0 (t) =Γ 0. Because the original equations (4) are of order γ, we simply choose such an auxiliary linear operator Ly = D γ y, with the property L[C] =0, where C is an integral constant. By taking the inverse of the linear operator L in (14), then we get for m 1, n 1 Y m (t) =χ m Y m 1 (t) χ m k=0 t k k! Y (k) m 1(0) + hj α [R m ( y m 1 (t))], where n 1 <γ n, n N. Inthisway,wegetY m (t) one by one in the order m =1, 2, 3,... Thus, it is easy to get approximations at high enough order, especially by means of the symbolic computation software. Note that we have great freedom to choose the value of the auxiliary parameter h. Mathematically the value of y(t) at any finite order of approximation is dependent upon the auxiliary parameter h, because the zeroth and high order deformation equations contain h. Let R h denotes the set of all values of h which ensure the convergence of the HAM series solution (12) of y(t). According to Liao [20], all of these series solutions must converge to the solution of the original Eqs. (1) and (2). Let h be the variable of the horizontal axis and the limit of the series solution (12) of Y (t) be the variable of vertical

9 An Analytical Scheme for Multi-order Fractional Differential Equations 313 axis. Plot the curve y(t) vs h, wherey(t) denotes the limit of the series (12). Because the limit of all convergent series solutions (12) is the same for a given a, there exists a horizontal line segment above the region h R h. So, by plotting the curve y(t) vs h at a high enough order approximation, one can find an approximation of the set R h. 4. Applications In this part, we introduce some applications on HAM to solve multi-order fractional differential equations. In these applications we select the most popular system of fractional differential equations D 0.5 Y = G(t, Y (t)). This system represents a classical problem in fluid dynamics and it has many applications in recent studies of scaling phenomena. 4.1 Example 1 Consider the following multi-order fractional differential equation y (t) D 1.5 y(t)+6 5 y (t)+d 0.5 y(t)+1 y(t) =f(t), (17) 5 with initial conditions y(0) = 0 and y (0) = 0, and choose f(t)sothattheexact solution is y(t) =t 2 + t 5/2. According to Theorem 1, this can be converted into an equivalent four dimensional system as follows, D 1/2 y 0 (t) = y 1 (t), D 1/2 y 1 (t) = y 2 (t), D 1/2 y 2 (t) = y 3 (t), (18) D 1/2 y 3 (t) = f(t)+y 3 (t) 6 5 y 2(t) y 1 (t) 1 5 y 0(t), with the initial conditions y 0 (t) =0,y 1 (t) =0,y 2 (t) =0andy 3 (t) =0. From (18), it is straightforward to use the set of base functions {t 0.5n : n 1, n N},

10 314 H. M. Jaradat, Fadi Awawdeh, and E. A. Rawashdeh to represent y i (t), y i (t) = b i,k t 0.5k, k=1 where b i,k is a coefficient to be determined later. From the initial conditions, we choose y 0,0 (t) =0,y 1,0 (t) =0,y 2,0 (t) =0,y 3,0 (t) =0, as our initial approximations of y i (t), i =0, 1, 2, 3. Besides that we select the auxiliary linear operators L i [f] =D (0.5) f, with the property L i [C i ]=0, in which C i, i =0, 1, 2, 3 are arbitrary constants that can be determined by the initial conditions. Furthermore, (18) suggest the that we define nonlinear operators as F i (ŷ i (t; q)) = D 1/2 ŷ i 1 (t; q) ŷ i (t; q), i =1, 2, 3 F 4 (ŷ i (t; q)) = D 1/2 ŷ 3 (t; q) ŷ 3 (t; q)+ 6 5ŷ2(t; q)+ŷ 1 (t; q)+ 1 5ŷ0(t; q) f(t), and by (16), we have R 0,m (t) = D 1/2 y 0,m 1 (t) y 1,m 1 (t), R 1,m (t) = D 1/2 y 1,m 1 (t) y 2,m 1 (t), R 2,m (t) = D 1/2 y 2,m 1 (t) y 3,m 1 (t), R 3,m (t) = D 1/2 y 3,m 1 (t) y 3,m 1 (t)+ 6 5 y 2,m 1(t)+ y 1,m 1 (t)+ 1 5 y 0,m 1(t) f(t)(1 χ m ). So, the mth order deformation equations are L i [y m (t) χ m y m 1 (t)] = h i R i,m ( y m 1 (t)), y i,m (0) = 0, i =0, 1, 2, 3, and we successfully obtain using a mathematical software the solutions y i (t) =y i,0 (t)+ y i,m (t) m=1

11 An Analytical Scheme for Multi-order Fractional Differential Equations 315 Figure 1: h-curve y(0.3) vs h for 10th order of approximation. The convergence of these series and rate of the approximation for HAM strongly depend upon the value of the auxiliary parameter h, as pointed out by Liao [20]. In order to find the range of admissible values of h, theh-curves are plotted in Fig. 1 for the 10th-order of approximation. We can see that the range for values of h for y(t) =y 0 (t) is 2 h 0. The 15th terms from the series solution expression by HAM are y(t) t 2 + t 5/2, which is in a good agreement with the exact solution as shown in Fig Example 2 Consider the Bagley-Torvik equation [AD 2 + BD3/2 + CD 0 ]y (t) =f (t), (19) which arises in the modelling of the motion of a rigid plate immersed in a Newtonian fluid. As a possible application we take A = B = C =1,y(0) = 0, y (0) = 0 and f(t) is chosen so that the exact solution is y(t) = t 2. The bagley-trovik equation, together with the initial conditions, can be written as a system of fractional differential equations of order 1/2oftheform D 1/2 y 0 (t) = y 1 (t), D 1/2 y 1 (t) = y 2 (t), D 1/2 y 2 (t) = y 3 (t), (20) D 1/2 y 3 (t) = f(t) y 3 (t) y 0 (t),

12 316 H. M. Jaradat, Fadi Awawdeh, and E. A. Rawashdeh Figure 2: Comparison of the numerical solution. Hollow dots: 15th-order HAM approximation; continued solid lines: exact solution. with initial conditions y 0 (0) = 0,y 1 (0) = 0,y 2 (0) = 0,y 3 (0) = 0. From (20), we use the set of base functions {t 0.5n : n N}, to represent y i (t), i =0, 1, 2, 3, y i (t) = b i,k t 0.5k, k=1 where b i,k is a coefficient to be determined later. We choose y 0,0 (t) =0,y 1,0 (t) =0,y 2,0 (t) =0,y 3,0 (t) =0, as our initial approximations of y i (t), i =0, 1, 2, 3, and the auxiliary linear operator L i [f] =D 0.5 f, with property L i [C i ]=0,

13 An Analytical Scheme for Multi-order Fractional Differential Equations 317 Figure 3: Comparison of the numerical solution. (- -): 10th-order HAM approximation; (- -: 15th-order HAM approximation; (-o-): exact solution. in which C i is an integral constant. Using (20) and (16), we have that R 0,m (t) = D 1/2 y 0,m 1 (t) y 1,m 1 (t), R 1,m (t) = D 1/2 y 1,m 1 (t) y 2,m 1 (t), R 2,m (t) = D 1/2 y 2,m 1 (t) y 3,m 1 (t), R 3,m (t) = D 1/2 y 3,m 1 (t)+y 3,m 1 (t)+y 0,m 1 (t) f(t)(1 χ m ), and so, the mth order deformation equations are y i,m (t) = χ m y i,m 1 (t)+h i J 0.25 [R i,m ( y m 1 (t))], y i,m (0) = 0, i =0, 1, 2, 3. Now we successfully obtain y 1 (t),y 2 (t),...,y m (t). In order to find range of admissible values of h, the h-curve is plotted for 10th-order approximation. We can see that the range of values for h is between 1.5 h 0.5. Then, we conclude that we have achieved a good approximation with the numerical solution of the equation by using the first few terms only of the linear equations derived above. Moreover, the overall errors can be made smaller by adding new terms of the HAM series solution. Fig. 3 presents a Comparison of the numerical solution of 10th-order, 15th-order HAM approximation and the exact solution.

14 318 H. M. Jaradat, Fadi Awawdeh, and E. A. Rawashdeh 5. Conclusion This study has focused on developing a simple procedure to obtain an explicit analytical solution concerning the multi-order fractional differential equations. The method presented was applied to problems that exist in the literature. The results evaluated are in very good agreement with the already existing ones, besides that even much more accurate. A series solution is evaluated in a very fast convergence rate where the accuracy is improved by increasing the number of terms considered. The recent appearance of linear and nonlinear fractional differential equations as models in some fields such as the modeling of induction machines and diffusion phenomenon makes it necessary to investigate the method of solutions for such equations and we hope that this work is a step in this direction. References [1] S. Abbasbandy, Soliton solutions for the Fitzhugh Nagumo equation with the homotopy analysis method, Appl. Math. Model., 32(2008), [2] F. Awawdeh, H. M. Jaradat, and O. Alsayyed, Solving system of DAEs by homotopy analysis method, Chaos Solitons Fract, 42(2009), [3] R. L. Bagley and P. J. Torvik, On the appearance of the fractional derivative in the behaviour of real materials, J. Appl. Mech., 51(1984), [4] L. Blank, Numerical treatment of differential equations of fractional order, Numerical Analysis Report 287, Manchester Center of Computational Mathematics, [5] K. Diethelm, An algorithm for the numerical solution of differential equations of fractional order, Elec. Transact. Numer. Anal., 5(1997), 1-6. [6] K. Diethelm and N. J. Ford, Multi-order fractional differential equations and their numerical solution, Appl. Math. Comp., 154(2004), [7] K. Diethelm and N. J. Ford, Numerical solution of the Bagley Torvik equation, BIT, 42(2002),

15 An Analytical Scheme for Multi-order Fractional Differential Equations 319 [8] K. Diethelm and N. J. Ford, Multi-order fractional differential equations and their numerical solution, Appl. Math. Comp., 154(2004), [9] K. Diethelm, N. J. Ford, and A. D. Freed, Detailed error analysis for a fractional Adams method, Numer Algor., 36(2004), [10] K. Diethelm, N. J. Ford, A. D. Freed, and Y. Luchko, Algorithms for the fractional calculus: A selection of numerical methods, Comput. Methods Appl. Mech. Engrg., 194(2005), [11] N. J. Ford and J. A. Connolly, Comparison of numerical methods for fractional differential equations, Comm. Pure Appl. Anal., 5(2006), [12] N. J. Ford and J. A. Connolly, Systems-based decomposition schemes for the approximate solution of multi-term fractional differential equations, J. Comput. Appl. Math., 229(2009), [13] T. Hayat and M. Sajid, On analytic solution for thin film flow of a fourth grade fluid down a vertical cylinder, Phys. Lett. A., 361(2007), [14] M. Inc, Application of homotopy analysis method for fin efficiency of convective straight fins with temperature-dependent thermal conductivity, Math. Comput. Simul. (2008), doi: /j.matcom [15] M. Inc, On numerical solution of Burgers equation by homotopy analysis method, Phys. Lett. A., 372(4)(2008), [16] M. Inc, On exact solution of Laplace equation with Dirichlet and Neumann boundary conditions by the homotopy analysis method, Phys. Lett. A., 365(5-6) (2007), [17] M. Inc, Application of homotopy analysis method for fin efficiency of convective straight fins with temperature-dependent thermal conductivity, Math. Comput. Simulat., 79(2)(2008), [18] S. J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman & Hall/CRC Press, Boca Raton, [19] S. J. Liao, Notes on the homotopy analysis method: some definitions and theorems, Commun Nonlinear Sci Numer Simul, 14(2009),

16 320 H. M. Jaradat, Fadi Awawdeh, and E. A. Rawashdeh [20] S. J. Liao, On the homotopy analysis method for nonlinear problems, Appl. Math. Comput., 147(2004), [21] S. J. Liao and Y. Tan, A general approach to obtain series solutions of nonlinear differential equations, Stud. Appl. Math., 119(2007), [22] F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics, in Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi, eds., Wein, 1997, Springer, pp