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IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com

IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com

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    • H. Saeedi, F. Samimi / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.052-056 He’s Homotopy Perturbation Method for Nonlinear Ferdholm Integro-Differential Equations Of Fractional Order H. Saeedi *, F. Samimi ** *(Department of Mathematics, Shahid Bahonar University of Kerman, Iran ** (Department of Mathematics, Islamic Azad University of Zahedan, IranABSTRACT In this paper, homotopy perturbation method is devoted to the search for better and more efficientapplied to solve non-linear Fredholm integro- solution methods for determining a solution,differential equations of fractional order. approximate or exact, analytical or numerical, toExamples are presented to illustrate the ability of nonlinear models. So the aim of this work is tothe method. The results reveal that the proposed present a numerical method (Homotopy-perturbationmethods very effective and simple and show that method) for approximating the solution of athis method can be applied to the non-fractional nonlinear fractional integro-differential equation ofcases. the second kind: 1 𝐷𝛼 𝑓 𝑥 − 𝜆 𝑘 𝑥, 𝑡 𝐹 𝑓 𝑡 𝑑𝑡Keywords - Fractional calculus, Fredholm 0integro-differential equations, Homotopy- = 𝑔(𝑥) (1)perturbation method Where 𝐹(𝑓 𝑡 ) = 𝑓 𝑡 𝑞 𝑞>1 With these supplementary conditions:I. INTRODUCTION 𝑓 (𝑖) 0 = 𝛿 𝑖 , 𝑖 = 0,1, … , 𝑟 − 1, The generalization of the concept of derivative 𝑟−1 < 𝛼 ≤ 𝑟 , 𝑟 𝐷 𝛼 𝑓(𝑥) to non-integer values of a goes to the ∈ 𝑁 (2)beginning of the theory of differential calculus. In Where,𝑔 ∈ 𝐿2 ([0, 1)), 𝑘 ∈ 𝐿2 ([0,1)2 ) are knownfact, Leibniz, in his correspondence with Bernoulli, functions, f(x) is the unknown function, 𝐷 𝛼 is theL’Hopital and Wallis (1695), had several notes about Caputo fractional differentiation operator and q is a 1the calculation of 𝐷 2 𝑓(𝑥). Nevertheless, the positive integer. There are several techniques fordevelopment of the theory of fractional derivatives solving such equations like Adomian decompositionand integrals is due to Euler, Liouville and Abel method [7, 9], collocation method [8], CAS wavelet(1823). However, during the last 10 years fractional method [2] and differential transform method [1].calculus starts to attract much more attention of Most of the methods have been utilized in linearphysicists and mathematicians. It was found that problems and a few numbers of works havevarious; especially interdisciplinary applications can considered nonlinear problems. In this paper,be elegantly modeled with the help of the fractional homotopy perturbation method is applied to solvederivatives. For example, the nonlinear oscillation of non-linear Fredholm integro-differential equations ofearthquake can be modeled with fractional fractional order.derivatives [3], and the fluid-dynamic traffic modelwith fractional derivatives caneliminate the II. FRACTIONAL CALCULUSdeficiency arising from the assumption of continuum There are several definitions of a fractional derivativetraffic flow [4]. In the fields of physics and of order 𝛼> 0. The two most commonly usedchemistry, fractional derivatives and integrals are definitions are the Riemann–Liouville and Caputo.presently associated with the application of fractals in Definition1.2.The Riemann-Liouville fractionalthe modeling of electro-chemical reactions, integral operator of order α ≥ 0 is defined asirreversibility and electro magnetism [10], heatconduction in materials with memory and radiation 𝐽𝛼 𝑓 𝑥 𝑥problems. Many mathematical formulations of 1mentioned phenomena contain nonlinear integro- = (𝑥 − 𝑡) 𝛼 −1 𝑓 𝑡 𝑑𝑡, 𝛼 > 0, 𝑥 Γ(𝛼) 0differential equations with fractional order. Nonlinear > 0, 3phenomena are also of fundamental importance in 𝐽0 𝑓 𝑥 = 𝑓(𝑥)various fields of science and engineering. Thenonlinear models of real-life problems are still IIt has the following properties:difficult to be solved either numerically ortheoretically. There has recently been much attention 52 | P a g e
    • H. Saeedi, F. Samimi / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.052-056 Γ 𝛾+1 The solution, usually an approximation to the 𝐽𝛼 𝑥𝛾 = 𝑥 𝛼+𝛾 , solution, will be obtained by taking the limit as p Γ 𝛼+ 𝛾+1 𝛾 > −1, (4) tends to 1,Definition2.2.The Caputo definition of fractal 𝑈 = lim 𝑉 = 𝑉0 + 𝑉1 + 𝑉2 𝑝→1derivative operator is given by +⋯ (9) To illustrate the homotopy perturbation method, for𝐷 𝛼 𝑓 𝑥 = 𝐽 𝑚−𝛼 𝐷 𝑚 𝑓 𝑥 nonlinear Ferdholm integro-differential equations of 𝑥 1 fractional order, we consider= 𝑥 Γ(𝑚 − 𝛼) 0− 𝑡 𝑚 −𝛼−1 𝑓 𝑚 (𝑡)𝑑𝑡, (5) 1− 𝑝 𝐷 𝛼 𝑓𝑖 𝑥 1Where, m − 1 ≤ α ≤ m, m∈N, x >0. It has the + 𝑝 𝐷 𝛼 𝑓𝑖 𝑥 − 𝑔 𝑖 𝑥 − 𝜆 𝑘 𝑥, 𝑡 [𝑓𝑖 𝑡 ] 𝑞 𝑑𝑡 0following two basic properties: =0 10 Or 𝐷𝛼 𝐽𝛼 𝑓 𝑥 = 𝑓 𝑥 , 𝐽𝛼 𝐷𝛼 𝑓 𝑥 𝐷 𝛼 𝑓𝑖 𝑥 = 𝑝 𝑔𝑖 𝑥 = 𝑓 𝑥 1 𝑚 −1 𝑘 + 𝑥𝑘 + 𝜆 𝑘 𝑥, 𝑡 [𝑓𝑖 𝑡 ] 𝑞 𝑑𝑡 (11) − 𝑓 (0 ) , 0 𝑘! 𝑘 =0 P is an embedding parameter which changes from 𝑥>0 (6) zero to unity. Using the parameter p, we expand the solution of the Eq. (1) in the following form: 𝑓𝑖 𝑡 = 𝑓0 + 𝑝𝑓1 + 𝑝2 𝑓2 + ⋯. (12)III. HOMOTOPY PERTURBATION METHOD Substituting (12) into (11) and collecting the termsFOR SOLVING EQ. (1) with the same powers of p, we obtain a series ofTo illustrate the basic concept of homotopy equations of the form:perturbation method, consider the following non- 𝑝0 : 𝐷 𝛼 𝑓0 = 0 (13)linear functional equation: 1 𝑝1 : 𝐷𝛼 𝑓 = 𝑔 𝑥 + 𝜆 1 𝑘 𝑥, 𝑡 𝑓0 𝑡 𝑞 𝑑𝑡 (14) 0𝐴 𝑈 = 𝑓 𝑟 (7) 1With the following boundary conditions: 𝑝2 : 𝐷 𝛼 𝑓2 = 𝜆 𝑘 𝑥, 𝑡 𝑓 𝑡 1 𝑞 𝑑𝑡 (15) 0 𝜕𝑈 . 𝑈, = 0, 𝑟∈Γ . 𝜕𝑛Where A is a general functional operator, B is a It is obvious that these equations can be easily solvedboundary operator, f(r) is a known analytic function, by applying the operator 𝐽 𝛼 , the inverse of theand Γ is the boundary of the domain Ω. Generally operator𝐷 𝛼 , which is defined by (3). Hence, thespeaking the operator A can be decomposed into two components 𝑓𝑖 (i = 0, 1, 2 , ...) of the HPM solutionparts L and N, where L is a linear and N is a non- can be determined. That is, by setting p = 1. In (11)linear operator. Eq. (7), therefore, can be rewritten as we can entirely determine the HPM seriesthe following: solutions, 𝑓 𝑥 = ∞ 𝑓𝑖 𝑥 . 𝑘=0𝐿 𝑈 + 𝑁 𝑈 − 𝑓 𝑟 =0 IV. APPLICATIONS In this section, in order to illustrate the method, weWe construct a homotopy 𝑉 𝑟, 𝑝 : Ω × 0,1 → 𝑅, solve two examples and then we will compare thewhich satisfies obtained results with the exact solutions. 𝐻 𝑉, 𝑝 = 1 − 𝑝 𝐿 𝑉 − 𝐿 𝑈0 Example 4.1. Consider the following fractional + 𝑝 𝐴 𝑉 − 𝑓 𝑟 = 0, nonlinear integro-differential equation: 1 𝑝 ∈ 0,1 , 𝑟∈Ω (8) 32 2 𝐷𝛼 𝑓 𝑥 = 2− 𝑥 + 𝑥 2 𝑡 2 𝑓 𝑡 2 𝑑𝑡,Or 15 0 𝐻 𝑉, 𝑝 = 𝐿 𝑉 − 𝐿 𝑈0 + 𝑝𝐿 𝑈0 0 ≤ 𝑥 < 1, + 𝑝 𝑁 𝑉 − 𝑓 𝑟 =0 0< 𝛼≤1 (16)Where, 𝑢0 is an initial approximation for the solution With this supplementary condition 𝑓(0) = 1.of Eq. (7).In this method, we use the homotopy For 𝛼 = 1, we can get the exact solutionparameter p to expand 𝑓 𝑥 = 1 + 2𝑥. 1 For 𝛼 = , According to (11) we construct the𝑉 = 𝑉0 + 𝑝𝑉1 + 𝑝2 𝑉2 + ⋯ 4 following homotopy 53 | P a g e
    • H. Saeedi, F. Samimi / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.052-056 1 1 𝐷 4 𝑓 𝑥 = 𝑝(𝑔 𝑥 + 𝑥2 𝑡 2 𝑓 𝑡 2 𝑑𝑡) (17) 0Substituting (12) into (17) 1 𝑝0 ∶ 𝐷 4 𝑓0 𝑥 = 0 1 1 𝑝1 ∶ 𝐷 4 𝑓 𝑥 = 𝑔 𝑥 + 1 𝑥 2 𝑡 2 [𝑓 𝑡 ]2 𝑑𝑡 0 0 1 1𝑝2 : 𝐷 4 𝑓 𝑥 = 2 𝑥 2 𝑡 2 [2𝑓0 𝑡 𝑓1 𝑡 ]𝑑𝑡 0 13 1𝑝 ∶ 𝐷 4 𝑓3 𝑥 = 𝑥 2 𝑡 2 [2𝑓0 𝑡 𝑓2 𝑡 + 𝑓12 𝑡 ]𝑑𝑡 0 1 1 𝑝4 ∶ 𝐷 4 𝑓 𝑥 = 4 𝑥 2 𝑡 2 [2𝑓0 𝑡 𝑓3 𝑡 0 + 2𝑓2 𝑡 𝑓 𝑡 ]𝑑𝑡 1 . . .Consequently, by applying the operators 𝐽 𝛼 to theabove sets Example 4.2. Consider the following nonlinear Fredholm integro-differential equation, of order 1 𝑓0 𝑥 = 1 𝛼=2 1 1 2 1154 1 764 9 𝐷 2 𝑓 𝑥 = 𝑔 𝑥 + 𝑥𝑡 𝑓 𝑡 𝑑𝑡𝑓 = 1 𝑥 4− 𝑥 4 0 523 541 0≤ 𝑥<1 (18) 893 9 𝑓2 (𝑥) = 1386 𝑥 4 1 17 Where, 𝑔 𝑥 = 3 𝑥 − 12 𝑥 with these 718 9 Γ( ) 2 𝑓3 𝑥 = 𝑥 4 supplementary conditions𝑓 0 = 1, with the exact 1169 317 9 solution, 𝑓 𝑥 = 1 + 𝑥 𝑓4 𝑥 = 1250 𝑥 4 According to (11) we construct the following . homotopy: . 1 1 2 . 𝐷 2 𝑓 𝑥 = 𝑝(𝑔 𝑥 + 𝑥𝑡 𝑓 𝑡 𝑑𝑡) (19)Therefore the approximations to the solutions of 0 1 Substituting (12) into (18)Example 4.1. For 𝛼 = 4 will be determined as 1 𝑝0 ∶ 𝐷 2 𝑓0 𝑥 = 0 1 ∞ 1 𝑓 𝑥 = 𝑓𝑖 𝑥 = 𝑓0 + 𝑓 + 𝑓2 + 𝑓3 + 𝑓4 + ⋯ 1 𝑝1 ∶ 𝐷 2 𝑓 𝑥 = 𝑔 𝑥 + 1 𝑥𝑡[𝑓0 𝑡 ]2 𝑑𝑡 𝑖=0 0 1 1154 4 1 𝑥 = 1+ 𝑝2 ∶ 𝐷 2 𝑓2 𝑥 = 𝑥𝑡[2𝑓0 𝑡 𝑓 𝑡 ]𝑑𝑡 1 523 0 1 1The approximations to the solutions for 𝛼 = is 1 2821 2 2 𝑝3 ∶ 𝐷 2 𝑓3 𝑥 = 𝑥𝑡[2𝑓0 𝑡 𝑓2 𝑡 + 𝑓12 𝑡 ]𝑑𝑡 𝑓 𝑥 = 1+ 𝑥. 0 1250 1 3 1 Hence for 𝛼 = 4 , 𝑝4 ∶ 𝐷 2 𝑓4 𝑥 = 𝑥𝑡[2𝑓0 𝑡 𝑓3 𝑡 346 4 0 𝑓 𝑥 = 1+ 𝑥3. + 2𝑓2 𝑡 𝑓 𝑡 ]𝑑𝑡 159 1 1 1 3Fig. 1 shows the numerical results for 𝛼 = . . . . 4 2 4 .The comparisons show that as   1 , the .approximate solutions tend to 𝑓(𝑥) = 1 + 2𝑥, Consequently, by applying the operators 𝐽 𝛼 to thewhich is the exact solution of the equation in the case above setsof 𝛼 = 1. 𝑓0 𝑥 = 1 1379 3 𝑓 = 𝑥− 1 𝑥 2 2000 41 3 𝑓2 (𝑥) = 𝑥 2 200 54 | P a g e
    • H. Saeedi, F. Samimi / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.052-056 617 3 232 7 𝑓3 𝑥 = 𝑥 2 𝑓3 𝑥 = 𝑥 4 5269 921 471 3 779 7 𝑓4 𝑥 = 𝑥 2 𝑓4 𝑥 = 5000 𝑥 4 6173 . . . . . .Therefore the approximations to the solutions ofExample 4.2. Will be determined as Therefore the approximations to the solutions of ∞ Example 4.3. Will be determined as 𝑓 𝑥 = 𝑓𝑖 𝑥 = 𝑓0 + 𝑓 + 𝑓2 + 𝑓3 + 𝑓4 + ⋯ 1 ∞ 𝑖=0 𝑓 𝑥 = 𝑓𝑖 𝑥 = 𝑓0 + 𝑓1 + 𝑓2 + 𝑓3 + 𝑓4 + ⋯ =1+ 𝑥 𝑖=0And hence,𝑓 𝑥 = 1 + 𝑥, which are the exact = 1 + 𝑥2solution. And hence 𝑓 𝑥 = 1 + 𝑥 2 , which are the exactExample 4.3. Consider the following nonlinear solution.Fredholm integro-differential equation, of order 3 𝛼=4 V. CONCLUSION This paper presents the use of the He’s homotopy 3 8 5 15 perturbation method, for non-linear ferdholm 𝐷 4 𝑓 𝑥 = 5 𝑥 4− 𝑥 integro-differential equations of fractional order. We 5Γ 8 4 usually derive very good approximations to the 1 3 solutions. It can be concluded that the He’s homotopy + 𝑥𝑡 𝑓 𝑡 𝑑𝑡 perturbation method is a powerful and efficient 0 0≤ 𝑥<1 (20) technique infinding very good solutions for this kind of equations.With these supplementary Condition 𝑓 0 = 1, withthe exact solution, 𝑓 𝑥 = 1 + 𝑥 2According to (11) we construct the following ACKNOWLEDGEMENTShomotopy: The authors wish to thank the reviewers for their 3 useful Comments on this paper. 𝐷 4𝑓 𝑥 1 3= 𝑝(𝑔 𝑥 + 𝑥𝑡 𝑓 𝑡 𝑑𝑡) (21) 0Substituting (12) into (21) REFERENCES 3 [1] Erturk VS, Momani S, Odibat Z. 𝑝0 ∶ 𝐷 4 𝑓0 𝑥 = 0 Application of generalized differential 1 3 transform method to multi-order fractional𝑝1 ∶ 𝐷 4 𝑓 𝑥 = 𝑔 𝑥 + 1 𝑥𝑡[𝑓0 𝑡 ]3 𝑑𝑡 0 differential equations. Commun Nonlinear 1 3 Sci Numer Simulat,(2008), 13:1642–54.𝑝2 ∶ 𝐷 4 𝑓2 𝑥 = 𝑥𝑡[3𝑓0 𝑡 2 𝑓 𝑡 ]𝑑𝑡 1 [2] H. Saeedi, M. M. Moghadam, N. 0 3 1 Mollahasani, and G. N. Chuev, A CAS𝑝3 ∶ 𝐷 4 𝑓3 𝑥 = 𝑥𝑡[3𝑓0 𝑡 2 𝑓2 𝑡 wavelet method for solving nonlinear 0 Fredholm Integro-differential equations of + 3𝑓0 𝑡 𝑓 𝑡 2 ]𝑑𝑡 1 fractional order, Communications in 1 3 Nonliner Science and Numerical Simulation,𝑝4 ∶ 𝐷 4 𝑓4 𝑥 = 𝑥𝑡[3𝑓0 𝑡 2 𝑓3 𝑡 + 6𝑓0 𝑡 0 vol. 16, no. 3, (2011) pp. 1154–1163.. + 𝑓1 𝑡 + 𝑓2 𝑡 + 𝑓32 (𝑡)]𝑑𝑡 [3] He JH. Nonlinear oscillation with fractional . derivative and its applications . In: . International conference on vibrating . engineering’98. China: Dalian: (1998), p.Consequently, by applying the operators 𝐽 𝛼 to the 288–91.above sets [4] He JH. "Some applications of nonlinear 𝑓0 𝑥 = 1 fractional differential equations and their approximations". Bull Sci Techno;15(2), 2137 7 (1999), 86–90.𝑓 = 𝑥2 −1 𝑥 4 [5] J.H. He, Compute.Methods Appl. Mech. 2500 Eng. (1999), 178.257. 203 7 𝑓2 (𝑥) = 479 𝑥 4 55 | P a g e
    • H. Saeedi, F. Samimi / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.052-056[6] J.H. He, Int. J. Nonlinear Mech. (2000), 35.37.[7] Momani S, Noor M. Numerical methods for fourth order fractional integro-differential equations. Appl Math Comput, 182: (2006), 754–60.[8] Rawashdeh E. Numerical solution of fractional integro-differential equations by collocation method. Appl Math Comput,176:16, 2006 . [9] Ray SS. Analyticalsolution for the space fractional diffusion equation by two-step adomian decomposition method. Commun Nonlinear SciNumerSimulat, 14: (2009), 1295–306.[10] Tenreiro Machado JA. Analysis and design of fractional-order digital control systems. Sys Anal Modell Simul;107–22.1997. 56 | P a g e