He s Homotopy Perturbation Method for Nonlinear Ferdholm Integro-Differential Equations Of Fractional Order

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1 H Saeedi, F Samimi / International Journal of Engineering Research and Applications (IJERA) ISSN: wwwijeracom Vol 2, Issue 5, September- October 22, pp52-56 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, Iran ABSTRACT In this paper, homotopy perturbation method is applied to solve non-linear Fredholm integrodifferential equations of fractional order Eamples are presented to illustrate the ability of the method The results reveal that the proposed methods very effective and simple and show that this method can be applied to the non-fractional cases Keywords - Fractional calculus, Fredholm integro-differential equations, Homotopyperturbation method devoted to the search for better and more efficient solution methods for determining a solution, approimate or eact, analytical or numerical, to nonlinear models So the aim of this work is to present a numerical method (Homotopy-perturbation method) for approimating the solution of a nonlinear fractional integro-differential equation of the second kind: D α f λ k, t F f t dt = g() () Where F(f t ) = f t q q > With these supplementary conditions: I INTRODUCTION The generalization of the concept of derivative D α f() to non-integer values of a goes to the beginning of the theory of differential calculus In fact, Leibniz, in his correspondence with Bernoulli, L Hopital and Wallis (695), had several notes about the calculation of D 2f() Nevertheless, the development of the theory of fractional derivatives and integrals is due to Euler, Liouville and Abel (823) However, during the last years fractional calculus starts to attract much more attention of physicists and mathematicians It was found that various; especially interdisciplinary applications can be elegantly modeled with the help of the fractional derivatives For eample, the nonlinear oscillation of earthquake can be modeled with fractional derivatives [3], and the fluid-dynamic traffic model with fractional derivatives caneliminate the deficiency arising from the assumption of continuum traffic flow [4] In the fields of physics and chemistry, fractional derivatives and integrals are presently associated with the application of fractals in the modeling of electro-chemical reactions, irreversibility and electro magnetism [], heat conduction in materials with memory and radiation problems Many mathematical formulations of mentioned phenomena contain nonlinear integrodifferential equations with fractional order Nonlinear phenomena are also of fundamental importance in various fields of science and engineering The nonlinear models of real-life problems are still difficult to be solved either numerically or theoretically There has recently been much attention f (i) = δ i, i =,,, r, r < α r, r N (2) Where,g L 2 ([, )), k L 2 ([,) 2 ) are known functions, f() is the unknown function, D α is the Caputo fractional differentiation operator and q is a positive integer There are several techniques for solving such equations like Adomian decomposition method [7, 9], collocation method [8], CAS wavelet method [2] and differential transform method [] Most of the methods have been utilized in linear problems and a few numbers of works have considered nonlinear problems In this paper, homotopy perturbation method is applied to solve non-linear Fredholm integro-differential equations of fractional order II FRACTIONAL CALCULUS There are several definitions of a fractional derivative of order α> The two most commonly used definitions are the Riemann Liouville and Caputo Definition2The Riemann-Liouville fractional integral operator of order α is defined as J α f = ( t) α f t dt, α >, Γ(α) >, 3 J f = f() IIt has the following properties: 52 P a g e

2 H Saeedi, F Samimi / International Journal of Engineering Research and Applications (IJERA) ISSN: wwwijeracom Vol 2, Issue 5, September- October 22, pp52-56 J α γ = Γ γ + Γ α + γ + α+γ, γ >, (4) Definition22The Caputo definition of fractal derivative operator is given by D α f = J m α D m f = Γ(m α) t m α f m (t)dt, (5) Where, m α m, m N, > It has the following two basic properties: D α J α f = f, J α D α f = f m f k ( + ) k k!, k= > (6) III HOMOTOPY PERTURBATION METHOD FOR SOLVING EQ () To illustrate the basic concept of homotopy perturbation method, consider the following nonlinear functional equation: A U = f r (7) With the following boundary conditions: U, U =, r Γ n Where A is a general functional operator, B is a boundary operator, f(r) is a known analytic function, and Γ is the boundary of the domain Ω Generally speaking the operator A can be decomposed into two parts L and N, where L is a linear and N is a nonlinear operator Eq (7), therefore, can be rewritten as the following: L U + N U f r = We construct a homotopy V r, p : Ω, R, which satisfies H V, p = p L V L U + p A V f r =, p,, r Ω (8) Or H V, p = L V L U + pl U + p N V f r = Where, u is an initial approimation for the solution of Eq (7)In this method, we use the homotopy parameter p to epand V = V + pv + p 2 V 2 + The solution, usually an approimation to the solution, will be obtained by taking the limit as p tends to, U = lim p V = V + V + V 2 + (9) To illustrate the homotopy perturbation method, for nonlinear Ferdholm integro-differential equations of fractional order, we consider p D α f i + p D α f i g i λ k, t [f i t ] q dt = Or D α f i = p g i + λ k, t [f i t ] q dt () P is an embedding parameter which changes from zero to unity Using the parameter p, we epand the solution of the Eq () in the following form: f i t = f + pf + p 2 f 2 + (2) Substituting (2) into () and collecting the terms with the same powers of p, we obtain a series of equations of the form: p : D α f = (3) p : D α f = g + λ k, t f t q dt (4) p 2 : D α f 2 = λ k, t f t q dt (5) It is obvious that these equations can be easily solved by applying the operator J α, the inverse of the operatord α, which is defined by (3) Hence, the components f i (i =,, 2, ) of the HPM solution can be determined That is, by setting p = In () we can entirely determine the HPM series solutions, f = f i k= IV APPLICATIONS In this section, in order to illustrate the method, we solve two eamples and then we will compare the obtained results with the eact solutions Eample 4 Consider the following fractional nonlinear integro-differential equation: D α f = t 2 f t 2 dt, <, < α (6) With this supplementary condition f() = For α =, we can get the eact solution f = + 2 For α =, According to () we construct the 4 following homotopy 53 P a g e

3 H Saeedi, F Samimi / International Journal of Engineering Research and Applications (IJERA) ISSN: wwwijeracom Vol 2, Issue 5, September- October 22, pp52-56 D 4f = p(g + Substituting (2) into (7) 2 t 2 f t 2 dt) (7) p D 4f = p D 4f = g + 2 t 2 [f t ] 2 dt p 2 : D 4f 2 = 2 t 2 [2f t f t ]dt p 3 D 4f 3 = 2 t 2 [2f t f 2 t + f 2 t ]dt p 4 D 4f 4 = 2 t 2 [2f t f 3 t + 2f 2 t f t ]dt f = f = f 2 () = f 3 = f 4 = Therefore the approimations to the solutions of Eample 4 For α = will be determined as 4 f = i= f i = f + f + f 2 + f 3 + f 4 + = The approimations to the solutions for α = is 2 f = Hence for α = 3, 4 f = Fig shows the numerical results for α = The comparisons show that as, the approimate solutions tend to f() = + 2, which is the eact solution of the equation in the case of α = Eample 42 Consider the following nonlinear Fredholm integro-differential equation, of order α = 2 D 2f = g + t f t 2 dt < (8) Where, g = Γ( 3 7 with these 2 ) 2 supplementary conditionsf =, with the eact solution, f = + According to () we construct the following homotopy: D 2f = p(g + t f t 2 dt) Substituting (2) into (8) p D 2f = p D 2f = g + t[f t ] 2 dt p 2 D 2f 2 = t[2f t f t ]dt (9) p 3 D 2f 3 = t[2f t f 2 t + f 2 t ]dt p 4 D 2f 4 = t[2f t f 3 t + 2f 2 t f t ]dt f = f = f 2 () = P a g e

4 H Saeedi, F Samimi / International Journal of Engineering Research and Applications (IJERA) ISSN: wwwijeracom Vol 2, Issue 5, September- October 22, pp52-56 f 3 = f 4 = Therefore the approimations to the solutions of Eample 42 Will be determined as f = i= f i = f + f + f 2 + f 3 + f 4 + = + And hence,f = +, which are the eact solution Eample 43 Consider the following nonlinear Fredholm integro-differential equation, of order α = 3 4 D 3 4f = 8 5Γ t f t 3 dt < (2) With these supplementary Condition f =, with the eact solution, f = + 2 According to () we construct the following homotopy: D 3 4f = p(g + t f t 3 dt) Substituting (2) into (2) p D 3 4f = p D 3 4f = g + t[f t ] 3 dt p 2 D 3 4f 2 = t[3f t 2 f t ]dt p 3 D 3 4f 3 = t[3f t 2 f 2 t + 3f t f t 2 ]dt (2) p 4 D 3 4f 4 = t[3f t 2 f 3 t + 6f t + f t + f 2 t + f 2 3 (t)]dt f = f = f 2 () = f 3 = f 4 = Therefore the approimations to the solutions of Eample 43 Will be determined as f = i= f i = f + f + f 2 + f 3 + f 4 + = + 2 And hence f = + 2, which are the eact solution V CONCLUSION This paper presents the use of the He s homotopy perturbation method, for non-linear ferdholm integro-differential equations of fractional order We usually derive very good approimations to the solutions It can be concluded that the He s homotopy perturbation method is a powerful and efficient technique infinding very good solutions for this kind of equations ACKNOWLEDGEMENTS The authors wish to thank the reviewers for their useful Comments on this paper REFERENCES [] Erturk VS, Momani S, Odibat Z Application of generalized differential transform method to multi-order fractional differential equations Commun Nonlinear Sci Numer Simulat,(28), 3: [2] H Saeedi, M M Moghadam, N Mollahasani, and G N Chuev, A CAS wavelet method for solving nonlinear Fredholm Integro-differential equations of fractional order, Communications in Nonliner Science and Numerical Simulation, vol 6, no 3, (2) pp [3] He JH Nonlinear oscillation with fractional derivative and its applications In: International conference on vibrating engineering 98 China: Dalian: (998), p [4] He JH "Some applications of nonlinear fractional differential equations and their approimations" Bull Sci Techno;5(2), (999), 86 9 [5] JH He, ComputeMethods Appl Mech Eng (999), P a g e

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