SOLUTION OF FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS BY ADOMIAN DECOMPOSITION METHOD

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1 SOLUTION OF FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS BY ADOMIAN DECOMPOSITION METHOD R. C. Mittal 1 and Ruchi Nigam 2 1 Department of Mathematics, I.I.T. Roorkee, Roorkee, India rcmmmfma@iitr.ernet.in 2 Department of Mathematics, I.I.T. Roorkee, Roorkee, India reply2rn@gmail.com Received 26 November 27; accepted 2 March 28 ABSTRACT Fractional integro-differential equations arise in the mathematical modelling of various physical phenomena like heat conduction in materials with memory, diffusion processes etc. In this paper, we have taken the fractional integro-differential equation of type D q y(t) =a(t)y(t)+f(t)+ K(t, s)f (y(s))ds Adomian decomposition approach is found to be good enough to solve these types of equations. Numerical examples are presented to illustrate the procedure. Comparison with collocation method has also been pointed out. Keywords: Adomian decomposition method, Fractional integro-differential equation, Caputo fractional derivative. 1 INTRODUCTION Fractional integro-differential equations arise in modelling processes in applied sciences (physics, engineering, finance, biology...). Many problems in acoustics, electromagnetics, viscoelasticity, hydrology and other areas of application can be modeled by fractional differential equations. Consider the fractional order integro-differential equation of the type D α y(t) =a(t)y(t)+f(t)+ K(t, s)f (y(s))ds, t [, 1] (1) y() = α where D α is Caputo s fractional derivative and α is a parameter describing the order of the fractional derivative, and F (y(x)) is a nonlinear continuous function. Such kind of equations arise in the mathematical modeling of various physical phenomena, such as heat conduction in materials with memory. Moreover, these equations are encountered in combined conduction, convection and radiation problems (see for example (Caputo 1967; Olmstead and Handelsman 1976; Mainardi 1997)). Int. J. of Appl. Math. and Mech. 4(2): 87-94, 28.

2 88 R. C. Mittal and Ruchi Nigam In recent years, fractional differential equations have been investigated by many authors. Rawashdeh (Rawashdeh 25) used the collocation spline method to approximate the solution of fractional equations. Momani (Momani 2) obtained local and global existence and uniqueness solution of the integro-differential equation given by (1). In this paper Adomian decomposition method has been used to solve the fractional integro-differential equations. Adomian decomposition method has been widely used by many researchers to solve the problems in applied sciences (Adomian 1994; Adomian 1989; Kaya and El-Sayed 23). Decomposition method (Adomian 1988; Adomian 1985) provides an analytical approximation to linear and nonlinear problems. In this method the solution is considered as the sum of an infinite series, rapidly converging to an accurate solution. The Adomian decomposition method provides solutions without any need for linearization or discretizations. Essentially the method provides a systematic computational procedure for equations of physical significance. 2 FRACTIONAL CALCULUS Many definitions of fractional calculus have been proposed (Adam 24; Podulbny 1999). Most frequently occurring are Caputo and Riemann-Liouville (Caputo 1967). In this section, we discuss the definition and some of the basic properties of these two types of fractional derivatives. 2.1 Abel-Reimannn fractional integral and derivatives The Abel-Riemann (A-R) fractional integral of any order α>for a function ψ(t) with t R + is defined as J α ψ(t) = 1 (t τ) α 1 ψ(τ)dτ, t > α>. (2) Γ(α) J = I (Identity operator). The A-R integrals possess the semigroup property J α J β = J α+β, for all α,β. (3) The A-R fractional derivative (of order α>) is defined as the left inverse of the corresponding A-R fractional integral, i.e. D α J α = I For positive integer m such that m 1 <α m, (D m J m α )J α = D m (J m α J α )=D m J m = I, so that D α = D m J m α, i.e. { D α 1 d m t ψ(τ) dτ, m 1 <α<m ψ(t) = Γ(m α) dt m (t τ) α+1 m (5) d m ψ(t), α = m. dt m (4) Properties of the operators J α and D α can be found in following { J α t γ = Γ(γ+1) D α t γ = Γ(γ+1+α) tγ+α, Γ(γ+1) Γ(γ+1 α) tγ α, for t>, α, γ> 1. (Podulbny 1999), we mention the Int. J. of Appl. Math. and Mech. 4(2): 87-94, 28.

3 Solution of fractional integro-differential equations by Adomian decomposition method Caputo fractional derivatives In the late sixties an alternative definition of fractional derivative was introduced by Caputo. Caputo and Mirandi used this definition in their work on the theory of viscoelasticity. According to Caputo s definition D α = J m α D m for m 1 <α m, i.e. { D α 1 t ψ m (τ) ψ(t) = dτ, m 1 <α<m Γ(m α) (t τ) α+1 m (6) d m ψ(t), α = m. dt m One of the basic property of the Caputo fractional derivative is m 1 J α D α ψ(t) =ψ(t) k= ψ (k) ( + ) tk k! Caputo s fractional differentiation is a linear operation, similar to integer order differentiation. D α [λf(t)+μg(t)] = λd α f(t)+μd α g(t), where λ and μ are constants. (7) 3 NUMERICAL SCHEME Consider the equation (1) where D α is the operator defined as (6). Operating with J α on both sides of the equation (1) we get n 1 y(t) = y k ( + ) tk k! + J α ( a(t)y(t)+f(t)+ K(t, s)f (y(s))ds ) (8) k= Adomian s method defines the solution y(t) by the series y = n= y n and the nonlinear function F is decomposed as F = n= A n where A n are the Adomian polynomials given by A n = 1 n! [ dn dλ F ( λ i y n i )] λ=, n =, 1, 2... (11) i= The components y,y 1,y 2,... are determined recursively by n 1 y = y k ( + ) tk k! + J α f(t) (12) k= y k+1 = J α (a(t)y k )+J α ( (9) (1) (K(t, s)a k )ds) (13) Having defined the components y,y 1,y 2,..., the solution y in a series form defined by (9) follows immediately. It is important to note that the decomposition method suggests that the th component y be defined by the initial conditions and the function f(t) as described above. The other components namely y,y 1,y 2, etc are derived recurrently. Int. J. of Appl. Math. and Mech. 4(2): 87-94, 28.

4 9 R. C. Mittal and Ruchi Nigam 4 NUMERICAL EXAMPLES Example 1 Consider the following fractional integro-differential equation. y.75 (t) =( t2 e t y() = exact solution is y(t) =t 3 5 )y(t)+ 6t2.25 Γ(3.25) + According to Adomian decomposition, y(t) = y (t) = m 1 k= k= y() t k e t sy(s)ds, (14) t k k! + 6 Γ(3.25) J α t 9/4 1 5 J α (t 2 e t y(t)) + J α k y() t k t k k! + 6 Γ(3.25) J α t 9/4 6 Γ(9/4+1) = + Γ(3.25) Γ(9/4+3/4+1) t(9/4+3/4) = t 3 y 1 (t) = 1 5 J α (t 2 e t y(t)) + J α e t sy (s)ds y 1 (t) = 1 5 J α (t 2 e t y(t)) + J α e t s 4 ds = 1 5 J α (t 2 e t y(t)) J α (t 2 e t y(t)) = Hence we have got the exact solution y(t) =t 3. e t sy(s)ds Rawashdeh (Rawashdeh 25) has solved this equation by collocation method (for h =.1, t =.1,.5, 1, absolute error = , , respectively). Example 2 Consider the following fractional integro-differential equation y.5 (t) =(cost sin t)y(t)+f(t)+ with the initial condition y() =. t sin xy(x)dx, (15) Rawashdeh (Rawashdeh 25) chose f(t) such that the exact solution to the problem is y(t) = t 2 + t. Let f(t) = 2 Γ(2.5) t Γ(1.5) t.5 + t(2 3cost t sin t + t 2 cos t) According to Adomian decomposition, y (t) = y() + J α (f(t)) = t 2 + t + J α (2t 3t cos t t 2 sin t + t 3 cos t) Int. J. of Appl. Math. and Mech. 4(2): 87-94, 28.

5 Solution of fractional integro-differential equations by Adomian decomposition method 91 Table 1: Absolute errors for example 2 using standard ADM t E 3 E 4 E y n+1 (t) =J α ((cos t sin t)y n (t)) + J α ( t sin xy n (x)dx) (16) In order to avoid difficult fractional integral, we can simplify the integration by taking the truncated Taylor expansions for the trigonometric terms in (15) e.g. cos t 1 t 2 /2+t 4 /24 and sin t t t 3 /6+t 5 /12. Here our approach is similar to that employed in (Hashim 26). Using the recursive algorithm (16), we can obtain y(t) =y (t)+y 1 (t)+... We computed the absolute error at t =.1,.5, and t =1. The following notations will be used: E 3 = y(t) φ 3 (t), E 4 = y(t) φ 4 (t), E 5 = y(t) φ 5 (t) Table 1 shows the absolute error in (15) using standard Adomian decomposition method(adm). Further improvement on the accuracy level can be made by finding more terms in the Adomian series solution and by taking more terms in the Taylor expansions for the trigonometric function. Another approach could be the modified decomposition method (Wazwaz 1999). Using the modified recursive scheme, and by selecting y = t 2 + t, we obtain y = t 2 + t y 1 = J α (2t 3t cos t t 2 sin t + t 3 cos t)+j α ((t cos t sin t)y (t)) +J α ( =, y n+2 =, n t sin xy (x)dx) so that the solution in closed form y(t) =t 2 + t. Rawashdeh (Rawashdeh 25) has solved this equation by collocation method (for h =.1, t =.1,.5, 1, absolute error=.3 1 9,.5 1 9, respectively). Example 3 Consider the following nonlinear equation D α y(t) =1+ e x y 2 (x)dx (17) y() = 1 The Adomian algorithm is y (t) =y + J α (1) = 1 + t α Γ(1 + α) Int. J. of Appl. Math. and Mech. 4(2): 87-94, 28.

6 92 R. C. Mittal and Ruchi Nigam y n+1 (t) =J α ( where e x A n (x)dx) A = y 2, A 1 =2y y 1, A 2 =2y y 2 + y are the Adomian polynomials for the nonlinear term F (y) =y 2 Following the same procedure as the previous example, we take the truncated Taylor expansions for the exponential term e t 1 x + x2 x Figure 1 shows the approximate solution of (17) for some values of α obtained after 3 iterations. The value of α =1is the only case for which we know the exact solution and our approximate solution is in good agreement with the exact values. To determine the accuracy of the approximations against the exact solution, let e n = e x φ n 1 denote the error function which are shown in Figure 2. We can see that even though we take a small number of terms in the expansions for the exponential, good accuracy is achieved with a very minimum amount of computation. Note that as we calculate more terms of the approximate solution, error reduces further showing convergence. y t Α.25 Α.5 Α.75 Α t Figure 1: Approximate solution for example 3 5 CONCLUSION In this paper, we have applied the Adomian decomposition method for solving the fractional integro-differential equations. The reliability of the method and reduction in the size of the computational work give this method a wider applicability. In the paper we have taken three examples. The first two examples are the same as solved by (Rawashdeh 25) using the collocation spline method. He obtained the solution having absolute error of the order of 1 4 and 1 8 while we have got the analytic solution of the same by Adomian method with reduced effort. The third example is a nonlinear and has been taken just to confirm the scheme. Although Int. J. of Appl. Math. and Mech. 4(2): 87-94, 28.

7 Solution of fractional integro-differential equations by Adomian decomposition method 93 Error e2 e3 e t Figure 2: Error functions e2,e3, and e4 other methods are also available, the Adomian decomposition method justifies its applicability and efficiency in solving the proposed equations. REFERENCES Adam Loverro (24). Fractional Calculus: History, Definitions and Applications for the Engineer. Adomian G (1985). On the solution of algebraic equations by the decomposition method. Journal of Mathematical Analysis and Applications 15, pp Adomian G (1988). A review of the decomposition method in applied mathematics. Journal of Mathematical Analysis and Applications 135, pp Adomian G (1989). Nonlinear Stochastic Systems and Application to Physics. Kluwer Academic Publisher. Adomian G (1994). Solving Frontier Problems of Physics: The Decomposition Method. Kluwer Academic Publisher. Caputo M (1967). Linear models of dissipation whose Q is almost frequency independent-ii. Geophysical Journal of the Royal Astronomical Society 13, pp Hashim I (26). Adomian decomposition method for solving BVPs for fourth-order integrodifferential equations. Journal of Computational and Applied Mathematics 193, pp Kaya D and El-Sayed SM (23). On a generalized fifth order kdv equations. Physics Letters 31(1), pp Mainardi F (1997). Fractional calculus: Some basic problems in continuum and statistical mechanics. A Carpinteri and F Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, pp New York: Springer-Verlag. Int. J. of Appl. Math. and Mech. 4(2): 87-94, 28.

8 94 R. C. Mittal and Ruchi Nigam Momani SM (2). Local and global existence theorems on fractional integro-differential equations. Journal of Fractional Calculus 18, pp Olmstead WE and Handelsman RA (1976). Diffusion in a semi-infinite region with nonlinear surface dissipation. SIAM Rev. 18, pp Podulbny I (1999). Fractional Differential Equations. Academic Press. Rawashdeh EA (25). Numerical solution of fractional integro-differential equations by collocation method. Applied Mathematics and Computation 176(1), pp Wazwaz Abdul-Majid (1999). A reliable modification of adomian decomposition method. Applied Mathematics and Computation 12(1), pp Int. J. of Appl. Math. and Mech. 4(2): 87-94, 28.