NUMERICAL SOLUTION OF TIME-FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS USING SUMUDU DECOMPOSITION METHOD

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1 NUMERICAL SOLUTION OF TIME-FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS USING SUMUDU DECOMPOSITION METHOD KAMEL AL-KHALED 1,2 1 Department of Mathematics and Statistics, Sultan Qaboos University, P.O. Box 36, Al-Khod 123, Sultanet Oman; kamel@squ.edu.om 2 Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid 22110, P.O.Box (3030), Jordan; kamel@just.edu.jo Received April 5, 2014 In this paper, Sumudu decomposition method is developed to solve general form of fractional partial differential equation. The proposed method is based on the application of Sumudu transform to nonlinear fractional partial differential equations. The nonlinear term can easily be handled with the help of Adomian polynomials. The fractional derivatives are described in the Caputo sense. The Sumudu method is found to be fast and accurate. Illustrative examples are given to demonstrate the validity and applicability of the proposed technique. Key words: Fractional PDEs, Adomian polynomials, Sumudu transform. 1. INTRODUCTION Nonlinear partial differential equations appear in many branches of physics, engineering and applied mathematics. It has turned out that many phenomena in engineering, physics and other sciences can be described very successfully by models using mathematical tools from fractional calculus. For better understanding of a phenomenon described by a given nonlinear fractional partial differential equation, the solutions of differential equations of fractional order are much involved. Fractional derivatives provide more accurate models of real world problems than integer order derivatives. Because of their many applications in scientific fields, fractional partial differential equations are found to be an effective tool to describe certain physical phenomena, such as diffusion processes 13, electrical and rheological materials properties and viscoelasticity theories 15. It is important to solve time fractional partial differential equations. It was found that fractional time derivatives arise generally as infinitesimal generators of the time evolution when taken along time scaling limit. Hence, the importance of investigating fractional equations arises from the necessity to sharpen the concepts of equilibrium, stability states, and time evolution in the long time limit. In general, there exists no method that yields an exact solution for nonlinear fractional partial differential equations. Several different and powerful methods for solving fractional partial differential equations have been proposed in order to obtain the approximate solutions. In 5, the authors established the double Laplace formulas for partial fractional derivatives, and apply these formulas to RJP Rom. 60(Nos. Journ. Phys., 1-2), Vol , Nos. (2015) 1-2, P , (c) 2015 Bucharest, - v.1.3a*

2 100 Kamel Al-Khaled 2 solve a fractional heat equation with certain initial and boundary conditions. While in 29, they investigate the transport equations in fractal porous media by using the fractional complex transform method. A fractional sub-equation method is introduced in 16 to construct solutions of the Boussinesq and KdV-mKdV equations of fractional order. In this paper, we apply the Sumudu transform to solve general form of nonlinear fractional partial differential equations of the form D α t u(x,t) = Lu(x,t) + Nu(x,t) + g(x,t) (1) with n 1 < α n, and subject to the initial condition (r) u(x,0) t r = u (r) (x,0) = f r (x), r = 0,1,...,n 1. (2) where D α t u(x,t) is the Caputa fractional derivatives, g(x,t) is the source term, L is the linear operator and N is the general nonlinear operator. The Sumudu transform was first proposed by Watugala 26, 27. In 1, 17 some fundamental properties of the Sumudu transform were established in light of which they developed efficient and straightforward methodologies for treating differential equations. The Sumudu transform method is one of the most important transform methods, it is a powerful tool for solving many kinds of PDEs in various fields of science and engineering 1. In 14, the authors start from the definition of the Sumudu transform on general time scales to define the discrete Sumudu transform, and present its basic properties. Also various methods are combined with Sumudu transformation such as Homotopy Analysis Sumudu Method 6, 25, and Sumudu decomposition method 18. The fractional derivatives are considered in the Caputa sense 9. In recent years, there has been a growing interest in the field of fractional calculus. Oldham and Spanier 21, Miller and Ross 19, Momani 20, Dumitru 8 and Podlubny provide the history and a comprehensive treatment of this subject. Several fields of applications of fractional differentiation and fractional integration are already well established, some others just started. Many applications of fractional calculus can be found in turbulence and fluid dynamics, stochastic dynamical systems, plasma physics and controlled thermonuclear fusion, nonlinear control theory, image processing, nonlinear biological systems, for more see 21, 22, 24 and the references therein. Indeed, it provides several potentially useful tools for solving differential equations. The paper is structured in five sections: In section 2, we introduce some necessary definitions of the fractional calculus theory and Sumudu transform. In section 3, we describe the analysis of the Adomian Sumudu method. In section 4, we present two examples to show the efficiency of using Adomian decomposition method to solve fractional PDEs. Finally, relevant conclusions are drawn in the last section.

3 3 Numerical solution of time-fractional PDEs using Sumudu decomposition method BASIC OF FRACTIONAL CALCULUS This section is devoted to a description of the operational properties of the purpose of acquainting with sufficient fractional calculus theory, to enable us to follow the solution of the fractional Burgers equation. Many definitions and studies of fractional calculus have been proposed in the last two centuries. These definitions include, Riemman-Liouville, Weyl, Reize, Campos, Caputa, and Nishimoto fractional operator. Mainly, in this paper, we will re-introduce section 2 of 4. The Riemann- Liouville definition of fractional derivative operator Ja α is defined as follows: Definition 1 Let α R +. The operator J α, defined on the usual Lebesque space L 1 a,b by Ja α f(x) = 1 x (x t) α 1 f(t)dt Γ(α) a J 0 af(x) = f(x) for a x b, is called the Riemann-Liouville fractional integral operator of order α. Properties of the operator J α can be found in 23, we mention the following: For f L 1 a,b,α,β 0 and γ > 1 1. J α a f(x) exists for almost every x a,b. 2. Ja α Ja β f(x) = Ja α+β f(x) 3. J α a J β a f(x) = J β a J α a f(x) 4. Ja α x γ = Γ(γ+1) Γ(α+γ+1) (x a)α+γ As mentioned in 20, the Riemann-Liouville derivative has certain disadvantages when trying to model real-world phenomena with fractional differential equations. Therefore, we shall introduce now a modified fractional differentiation operator D α proposed by Caputo in his work on the theory of viscoelasticity 9. Definition 2 The fractional derivative of f(x) in the Caputo sense is defined as D α f(x) = J m α D m f(x) = 1 Γ(m α) x 0 (x t) m α 1 f (m) (t)dt, (3) m 1 < α m,m N,x > 0 Also, we need here two of its basic properties. Lemma 1 If m 1 < α m, and f L 1 a,b, then D α a J α a f(x) = f(x), and m 1 Ja α Da α f(x) = f(x) f (k) (0 ) (x a)k, x > 0. k! The Caputo fractional derivative is considered in the Caputo sense. The reason for adopting the Caputo definition is as follows 20. To solve differential equations, we

4 102 Kamel Al-Khaled 4 need to specify additional conditions in order to produce a unique solution. For the case of Caputo fractional differential equations, these additional conditions are just the traditional conditions, which are taken to those of classical differential equations, and are therefore familiar to us. In contrast, for Riemann-Liouville fractional differential equations, these additional conditions constitute certain fractional derivatives of the unknown solution at the initial point x = 0, which are functions of x. The unknown function u = u(x,t) is assumed to be a causal function of time, i.e., vanishing for t < 0. Also, the initial conditions are not physical; furthermore, it is not clear how much quantities are to be measured from experiment, say, so that they can be appropriately assigned in an analysis. For more details on the geometric and physical interpretation for fractional derivatives of both Riemann-Liouville and Caputo types see 9, 20. Definition 3 For m to be the smallest integer that exceeds α, the Caputo fractional derivatives of order α > 0 is defined as 1 t D α u(x,t) = α u(x,t) Γ(m α) 0 (t τ)m α 1 m u(x,τ) τ dτ, m 1 < α < m m t α = m u(x,t) t, α = m N m For mathematical properties of fractional derivatives and integrals one can consult the mentioned references. In early 90 s, Watugala 26, 27 introduced a new integral transform, named the Sumudu transform and applied it to the solution of ordinary differential equation in control engineering problems. Definition 4 Sumudu transform over the following set of functions { } t t A = f(t) M, τ 1,τ 2 > 0, f(t) < Me j, if t ( 1) j 0, ) (4) is defined as, for u (τ 1,τ 2 ), we have Sf(t) = G(u) = 0 f(ut)e t dt = 0 1 u f(t)e t/u dt, (5) In Belgacem et al. 7, the Sumudu transform was shown to be the theoretical dual of the Laplace transform. Hence, one should be able to rival it to a great extent in problem solving. Many of special properties of the Sumudu transform are mentioned and tabulated in 7, 17. Some special properties of the Sumudu transform are as follows: 1. S1 = 1 2. S t n Γ(n+1) = u n, n > 0 3. Sf(x) g(x) = Sf(x) Sg(x)

5 5 Numerical solution of time-fractional PDEs using Sumudu decomposition method 103 Theorem 2 7 Let G(u) be the Sumudu transform of f(t), such that 1. G(1/s)/s, is a meromorphic function, with singularities having Re(s) < γ, and 2. there exists a circular region Γ with radius R and positive constants, M and k with G(1/s) < MR k s then the function f(t) is given by f(t) = S 1 G(t) = 1 γ+ e st G( 1 2πi s )ds s = st G(1/s) residuse e s γ i To solve fractional differential equations, the following Lemma of Sumudu transform will be needed. Lemma 3 12 The Sumudu transform Sf(t) of the fractional derivative introduced by Caputo is given by SD α t f(t) = G(u) u α n 1 f (k) (0), where G(u) = Sf(t) uα k In this paper, we are dealing with the numerical solutions of fractional PDEs. So, we derive analogous result of the above Lemma. Lemma 4 The Sumudu transform Sf(x, t) of the fractional derivative introduced by Caputo is given by SD α t f(x,t) = Sf(x,t) u α n 1 f (k) (x,0) u α k, n 1 < α n (6) Proof: Taking the Sumudu transform of both sides for the partial derivative of Dt α f(x,t) as in Definition 3, we get SDt α 1 t f(x,t) = S (t τ) m α 1 m f(x,τ) Γ(m α) τ m dτ. By the convolution theorem of Sumudu transform 12, we obtain SDt α u f(x,t) = Γ(m α) Sf m (x,t) St α+m 1. By using Sumudu transform of multiple differentiation, we get the desired result ANALYSIS OF THE METHOD To illustrate the basic idea of the Sumudu Adomian decomposition method, we consider a general nonlinear fractional partial differential equation with the initial

6 104 Kamel Al-Khaled 6 conditions of the form D α t u(x,t) = Lu(x,t) + Nu(x,t) + g(x,t) (7) with n 1 < α n, and subject to the initial condition (r) u(x,0) t r = u (r) (x,0) = f r (x), r = 0,1,...,n 1. (8) where Dt α u(x,t) is the Caputa fractional derivatives, g(x,t) is the source term, L is the linear operator and N is the general nonlinear operator. According to Sumudu decomposition method, we apply the Sumudu transform on both sides of equation (7), S Dt α u(x,t) = S Lu(x,t) + Nu(x,t) + g(x,t) (9) Using the differentiation property of Sumudu transform (6), we get m 1 u α Su(x,t) u (α k) u (k) (x,0) = S Lu(x,t) + Nu(x,t) + g(x,t) Simplify, we get m 1 Su(x,t) = u k f k (x) + u α S Lu(x,t) + Nu(x,t) + g(x,t) Now, we present the solution as an infinite series given by 2, 3 u(x,t) = u n (x,t). (12) The nonlinear term operator is decomposed as Nu(x,t) = A n (u) (13) where A n (u) are the Adomian polynomials 11, 28 of u 0,u 1,...,u n,... that are given by A n (u) = 1 d n ( ) n! dλ n N λ i λ=0 u i, n = 0,1,2,... (14) i=0 The Adomian polynomials A n can be calculated for all form of nonlinearity according to specific algorithms constructed as in 28. The formula (14) is easy to compute by using Mathematica software, or by writing a computer code to get as many polynomials as we need in the explicit solution. Substituting equations (12), (13) into (10) (11)

7 7 Numerical solution of time-fractional PDEs using Sumudu decomposition method 105 (11), we get S m 1 u n (x,t) = u k f k (x) + u α S L ( ) u n (x,t) + ( Matching both sides of (15) yields the following iterative algorithm m 1 Su 0 (x,t) = u k f k (x) Su 1 (x,t) = u α S Lu 0 (x,t) + A 0 (u(x,t)) + g(x,t) Su n+1 (x,t) = u α S Lu n (x,t) + A n (u(x,t)), n 1. ) A n (u) + g(x,t) Operating with the Sumudu inverse on both sides of the above equations yields u 0 (x,t) = S 1( m 1 ) u k f k (x) u 1 (x,t) = S 1( ) u α S Lu 0 (x,t) + A 0 (u(x,t)) + g(x,t) u n+1 (x,t) = S 1( ) u α S Lu n (x,t) + A n (u(x,t)), n 1. In this manner the rest of components of the decomposition solution can be obtained. The remaining components of the series (12) can be determined in a way that each component is determined by using the preceding components, i.e., each term of the series (12) is given by the following recursive relation u n+1 (x,t) = S 1( ) u α S Lu n (x,t) + A n (u(x,t)), n 1. Finally, the solution u n (x,t;α) can be approximated by the truncated series (15) n 1 u n (x,t) = u j (x,t) (16) j=0 such that lim u n(x,t) = u(x,t) (17) n In computing u(x, t), choosing large values for n, increasing the number of terms in the expression of A n and this causes propagation of round off errors. The ADM reduces significantly the massive computation which may arise if discretization methods are used. However, in many cases the exact solution in a closed may be obtained. Moreover, the decomposition series solutions generally converge very rapidly. The convergence series was investigated by several authors 10, and they obtained some results about the speed of convergence of this method.

8 106 Kamel Al-Khaled 8 4. NUMERICAL EXPERIMENTS The efficiency and accuracy of the Sumudu Adomian Decomposition method by applying it to two test problems. As mentioned in 20 these examples are somewhat artificial in the sense that the exact solution, for the special cases α = 1 or α = 2, is known in advance and the initial condition are directly taken from the exact solution. All the results are calculated by using the symbolic calculus soft-ware Mathematica. Example 1 Consider the linear fractional PDEs 15 α u t α + x u x + 2 u x 2 = 2(tα + x 2 + 1), 0 t 1, 0 x 1, 0 < α 1 (18) We solve equation (18) subject to the initial condition u(x,0) = x 2. The exact solution of the given problem is given by u(x,t) = x 2 +2t 2α Γ(α+1) Γ(2α+1). Taking the Sumudu transform on both sides of equation (18), and using the property of Sumudu transform together with the initial condition, we arrive at Su(x,t) = x 2 + u α S 2(t α + x 2 + 1) x u x 2 u x 2. Assuming the solution of (18) is in the form u(x,t) = u n (x,t), so get S u n (x,t) = x 2 + u α S 2(t α + x 2 + 1) x x u n (x,t) 2 x 2 u n (x,t). Matching both sides of the above equation, yields the following recursive relation Su 0 (x,t) = x 2 + u α S 2(t α + x 2 + 1) = x 2 + 2u 2α Γ(α + 1) + x 2 u α + u α (19) and, Su 1 (x,t) = u α S x x 2 u 0(x,t) x u 0(x,t) 2 In general, the n-th term has the form Su n+1 (x,t) = u α S x x u n(x,t) 2 x 2 u n(x,t) Operating with Sumudu inverse on both sides of equation (19), we get u 0 (x,t) = x 2 2α Γ(α + 1) + 2t Γ(2α + 1) + 2t α (x2 + 1) Γ(α + 1). Therefore, equation (20) reduces to (20) (21) Su 1 (x,t) = 2(x 2 + 1)u α 4(x 2 + 1)u 2α = (x 2 + 1)4u 2α + 2u α (22)

9 9 Numerical solution of time-fractional PDEs using Sumudu decomposition method 107 Table 1 Numerical values for the solution of (18) when α = 0.75, 0.95 and α = 1 using our approximation, and for α = 0.75, 1.0 using the exact solution. t x α = 0.75 α = 0.95 α = 1 Exact,α = 0.75 Exact,α = Also, taking the Sumudu inverse of both sides of equation (22), we obtain u 1 (x,t) = (x 2 4t 2α + 1) Γ(2α + 1) + 2tα Γ(α + 1) Similarly, u 2 (x,t) = (x 2 + 1) 8t 3α Γ(3α + 1) + 4t2α Γ(2α + 1). 2 u n+1 (x,t) = ( 1) n+1 (x 2 n+2 t (n+2)α + 1) Γ((n + 2)α + 1) + 2n+1 t (n+1)α Γ((n + 1)α + 1) The approximate solution is given by Taking n, we obtain φ n (x,t) = u 0 (x,t) + u 1 (x,t) u n (x,t) u(x,t) = lim φ n(x,t) = x 2 2α Γ(α + 1) + 2t n Γ(2α + 1). In Table 1, we list some values for the solution using our approach with α = 0.75,0.95, 1.0 and n = 8 for various values of x and t. It can be easily seen from the Table that the numerical solutions are in good agreement with the exact one. Example 2 Consider the nonlinear time-fractional hyperbolic PDEs 20 α u t α = ( u(x,t) u ), t > 0, x R, 1 < α 2 (23) x x with initial conditions u(x,0) = x 2, u t (x,0) = 2x 2. The values of α = 2 is the only case for which we know the exact solution u(x,t) = ( x t+1 )2. Taking the Sumudu

10 108 Kamel Al-Khaled 10 transform on both sides of equation (23), and using the property of Sumudu transform together with the initial condition, we arrive at ( Su(x,t) = x 2 2x 2 u + u α S u(x,t) u ). x x The inverse of Sumudu transform implies that u n (x,t) = x 2 (1 2t) + S 1 u α S A n (u) x where A n (u) are Adomian polynomials (14) that represent the nonlinear term uu x. The first few components of Adomian polynomials, are given by A 0 (u) = u 0 u 0x, A 1 (u) = u 0 u 1x + u 1 u 0x, A 2 (u) = u 2 u 0x + u 1 u 1x + u 0 u 2x, and so on the rest of A n (u) s. The recursive relation is given as (24) u 0 (x,t) = x 2 (1 2t) u 1 (x,t) = S 1 u α S x A 0(u) u 2 (x,t) = S 1 u α S x A 1(u). u n (x,t) = S 1 u α S x A n 1(u) Upon passing simple calculations, we arrive at and, t α u 1 (x,t) = 6x 2( Γ(α + 1) 4tα+1 Γ(α + 2) + 8tα+2 ) Γ(α + 3) (25) t 2α u 2 (x,t) = 72x 2( Γ(2α + 1) 4t2α+1 Γ(2α + 2) + + 8t2α+2 Γ(2α + 3) 2Γ(α + 2)t2α+1 Γ(α + 1)Γ(2α + 2) 8Γ(α + 3)t2α+2 16Γ(α + 4)t2α+1 Γ(α + 2)Γ(2α + 3) Γ(α + 3)Γ(2α + 4) and so on, in this manner the rest of components of the decomposition series can be obtained. Table 2 shows some numerical values for the approximate solution of (23) for different values of α. In our approximation, only three terms of the decomposition series were used in evaluating the approximate solutions for Table 2. The accuracy can be improved by computing more terms of the approximate solution. Table 2 shows some numerical values for the solution of equation (23) for different values of α. )

11 11 Numerical solution of time-fractional PDEs using Sumudu decomposition method 109 Table 2 Numerical values for the solution of (23) when α = 1.5, 1.75 and α = 2 using our approximation, and for α = 2 using the exact solution. t x α = 1.5 α = 1.75 α = 2 Exact,α = From the numerical solutions in Tables 1,2, it can be seen that the exact solution (α = 1, α = 2), respectively, is quite close to the approximate solution when α = 0.95,α = Also, it is observed that the values of the approximate solution at different α s has the same behavior as those obtained using the exact solution, for which α = 1,α = 2. This shows the approximate solution is efficiency. In the theory of fractional calculus, it is obvious that when the fractional derivative α(m 1 < α m) tends to positive integer m, then the approximate solution continuously tends to the exact solution of the problem with derivative m = 1,m = 2. A closer look at the values in Tables 1 and 2, we observe that our approach do have this characteristic. 5. CONCLUSIONS The fundamental goal of this paper is to propose an efficient algorithm for the solution of nonlinear fractional PDEs. The Adomian decomposition Sumudu method is introduced for solving nonlinear fractional PDEs. To, show the applicability and efficiency of the proposed method, the method is applied to obtain the solutions of several examples. The results obtained by using the proposed method agree well with the results obtained by other methods 20. It is worth mentioning that the proposed technique is capable of reducing the volume of the computational work as compared to the classical methods. Finally, we conclude that the Adomian decomposition Sumudu method is very powerful and efficient in finding analytical as well as numerical solutions for wide classes of nonlinear fractional partial differential equations. The authors would like to thank the reviewers for their comments to im- Acknowledgements. prove the paper.

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