Research Article Note on the Convergence Analysis of Homotopy Perturbation Method for Fractional Partial Differential Equations

Size: px

Start display at page:

Download "Research Article Note on the Convergence Analysis of Homotopy Perturbation Method for Fractional Partial Differential Equations"

Marylou Stone
5 years ago
Views:

1 Abstract and Applied Analysis, Article ID 8392, 8 pages Research Article Note on the Convergence Analysis of Homotopy Perturbation Method for Fractional Partial Differential Equations Asma Ali Elbeleze, 1,2 Adem KJlJçman, 3 and Bachok M Taib 1 1 Faculty of Science and Technology, Universiti Sains Islam Malaysia (USIM, 718 Nilai, Malaysia 2 Department of Mathematics, Zawia University, Zawia, Libya 3 Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia, 434 Serdang, Selangor, Malaysia Correspondence should be addressed to Adem Kılıçman; akilic@upmedumy Received 7 September 213; Revised 12 November 213; Accepted 3 December 213; Published 22 January 214 Academic Editor: Hossein Jafari Copyright 214 Asma Ali Elbeleze et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited We apply the homotopy perturbation method to obtain the solution of partial differential equations of fractional order This method is powerful tool to find exact and approximate solution of many linear and nonlinear partial differential equations of fractional order Convergence of the method is proved and the convergence analysis is reliable enough to estimate the maximum absolute truncated error of the series solution The fractional derivatives are described in the Caputo sense Some examples are presented to verify convergence hypothesis and simplicity of the method 1 Introduction Recently, the partial differential equations of fractional order have attracted much attention This is mostly due to their frequent appearance in many applications in fluid mechanics, viscoelastic, biology, engineering, and physics [1, 2] Most of partial differential equations of fractional order do not have exact analytical solution, so approximations and numerical techniques must be used Some of these methods are series solution methods which include Adomain decomposition method [3], homotopy analysis method [4, 5], variationalmiteration method [6], and homotopy perturbation method [7 9] The homotopy perturbation method [1] proposed by He in 1998 This method is useful tool for obtaining exact and approximate solution of linear and nonlinear partial differential equations of fractional order There is no need for a small parameter or linearization, the solution procedure is very simple, and only few iterations lead to high accurate solutions which are valid for the all solution domains The solution is expressed as the summation of an infinite series which is supposed to be convergent to the exact solution This method has been used to solve effectively, easily, and accurately many types of fractional equations of linear and nonlinear problems with approximations For example, [11] applied HPM to solve a class of initialboundary value problems of fractional partial differential equations over finite domain [12] usedhpmforsolvingtheklein-gordonpartial differential equations of fractional order Furthermore, many authors applied HPM for solving and investigating linear and nonlinear partial differential equations of fractional ordering; see [13, 14] For more details about homotopy perturbation method and its applications, we refer to [15, 16] Our aim in this study is to extend the applications of HPM to obtain approximate solution of some partial differential equations of fractional order such as Burgers equation of fractional order and fractional fourth-order parabolic partial differential equation and obtain the convergence of this method The paper is organized as follows In Section 2,somebasic definitions and properties of fractional calculus theory are given In Section 3, the basic idea of HPM is presented In Section 4, analysis of HPM is given Some examples are given in Section 5 Concluding remarks are listed in Section 6 2 Preliminaries In this section, we give some basic definitions and properties of fractional calculus theory which are used in this paper

2 2 Abstract and Applied Analysis Definition 1 Arealfunctionf(x, x>is said to be in space Cμ, μ Rif there exists a real number p>μ,suchthatf(x x p f 1 (x where f 1 (x C(,, and it is said to be in the space C n μ if fn C μ, n N Definition 2 The Riemann-Liouville fractional integral operator of order α of a function f Cμ, μ 1is defined as J α f (x 1 x Γ (α (x t α 1 f (t dt, α>, t> (1 in particular J f(x f(x For β and γ 1TheoperatorJ α has the following properties (1 J α J β f(x J αβ f(x, (2 J α J β f(x J β J α f(x, (3 J α x γ Γ(γ 1/Γ(α γ 1 xαγ Definition 3 The Caputo fractional derivative of f C m 1, m N, is defined as D α x 1 f (x Γ (m α (x t m α 1 f m (t dt, (2 m 1<αm Lemma 4 If m 1<αm, m N, f C m μ, μ> 1,then the following two properties hold (1 D α [J α f(x] f(x, (2 J α [D α f(x] f(x m 1 k1 fk ((x k /k! 3 Homotopy Perturbation Method To illustrate the basic idea of this method, we consider the following nonlinear differential equation: with boundary conditions A (u f(r, r Ω (3 B(u, u, r Γ, (4 n where A is a general differential operator, B is a boundary operator, f(r is a known analytic function, and Γ is the boundary of the domain Ω In general, the operator A can be divided into two parts L and N, wherel is linear, while N is nonlinear Equation (3 therefor can be rewritten as follows: L (u N(u f(r (5 By the homotopy technique [1, 17] we construct a homotopy V(r,p:Ω [,1] Rwhich satisfies H(V,p(1 p[l(v L(u ] p [A (V f(r] p [, 1], r Ω (6 or H(V,pL(V L(u pl(u p[n(v f(r], (7 where p [,1] is an embedding parameter and u is an initial approximation of (3 which satisfies the boundary conditions From (6and(7, we have H (V, L(V L(u, (8 H (V,1 A(V f(r The change in the process of p from zero to unity is just that of V(r, p from u (r to u(r In topology, this is called deformation and L(V L(u,andA(V f(rare called homotopic Now, assume that the solution of (6 and(7 canbe expressed as V V pv 1 p 2 V 2 (9 The approximate solution of (3 can be obtained by setting p1: u lim p 1 V V V 1 V 2 (1 4 Analysis on Convergence and Solution Consider the following fractional partial differential equations: c D α t u (t f(t,u(t,dn 1 u (t,d n 2 u (t,,d n q u (t, t [, T] (11 u k ( b k, u(x, t g(x, t, k,1,2,, (12 where D α t α / t α is Caputo fractional derivative of order α, m 1 α m, n i N for every i Consider that f : [,T] R R R R is a continuous mapping f(t, u 1,u 2,,u n exists with continuous and bounded derivatives f/ u i satisfing the Lipschitz condition f(t,u 1 (t,d n 1 u 1 (t,,d n q u 1 (t f (t, u 2 (t,d n 1 u 2 (t,,d n q u 2 (t L (u 1,D n 1 u 1,D n 2 u 1,,D n q u 1 (u 2,D n 1 u 2,D n 2 u 2,,D n q u 2, t, (13 where L is Lipschitz constant To illustrate the basic concepts of HPM for fractional partial differential equation (11 with the initial conditions (12, we construct the following homotopy for (11: (1 p c a Dα t u (x, t p( c a Dα t u (x, t f(t,u(t,dn 1 u, D n 2 u (t,,d n q u (t (14

3 Abstract and Applied Analysis 3 or c a Dα t u (x, t (15 p(f(t,u(t,d n 1 u (t,d n 2 u (t,,d n q u (t Substituting (9into(15 and equating the terms with having identical power of p, we obtain the following series of equations: p 1 : c a Dα t u 1 (x, t p : c a Dα t u (x, t f(x, t, f(t,u,d n 1 u (t,d n 2 u (t,,d n q u (t, p 2 : c a Dα t u 2 (x, t f(t,u 1,D n 1 u 1 (t,d n 2 u 1 (t,,d n q u 1 (t, p n : c a Dα t u n (x, t f(t,u n 1,D n 1 u n 1 (t,d n 2 u n 1 (t,,d n q u n 1 (t (16 Operating with Riemann-Liouville fractional operator J α, which is the inverse operator of Caputo derivative C D α a in both sides of (16thesolution u 1 (x, t u (x, t n 1 k b k t k k! J α (f (x, t, J α (f (t, u,d n 1 u (t,d n 2 u (t,,d n q u (t u 2 (x, t J α (f (t, u 1,D n 1 u 1 (t,d n 2 u 1 (t,,d n q u 1 (t, u n (x, t J α (f (t, u n 1,D n 1 u n 1 (t,d n 2 u n 1, 41 Existence and Uniqueness of Solutions Theorem 5 Let f satisfy the Lipschitz condition (13 and then the problem (11 has unique solution u(x, t,whenever<γ< 1 Proof Let y and z be two different solutions of (11, and for All t [,T] and τ [,t] is bounded Let M max τt,tt (t τ α 1 ;then, y zj α (f (t,y(t,d n 1 y (t,d n 2 y (t,,d n q y (t y z J α (f (t,z(t,d n 1 z (t,d n 2 z (t,,d n q z (t, t 1 Γ (α (t τ α 1 f(τ,y(t,d n 1 y (t,d n 2 y, 1 t Γ (α (t τ α 1,D n q y (tdτ f(τ,z(t,d n 1 z (t,d n 2 z (t,,d n q z (tdτ, t y z 1 Γ (α (t τ α 1 f(τ,y(t,d n 1 y (t,d n 2 y,,d n q y (tdτ 1 t Γ (α (t τ α 1 f(τ,z(t,d n 1 z, D n 2 z (t,,d n q z (tdτ L t Γ (α (t τα 1 y z dτ, max y z L t Γ (α max (t τα 1 y z dτ, n y z [LMT Γ (α ] 1 n! y z γ y z, (1 γ y z (2,D n q u n 1 (t (17 The solution of (11 in series form is given by u (x, t u 1 (x, t u 2 (x, t u 3 (x, t (18 Define that (C[, T], is the Banach space, the space of all continuous functions on [, T] with the norm f (t max f (t (19 t [,T] Since 1 γ,then y z ; therefore, yz,andthis completes the proof 42 Proof of the Convergence Theorem 6 Let u n (x, t and u(x, t be defined in Banach space (C[, T], Then the series solution {u n (x, t} n1 defined by (18 converges to the solution of (11,if<γ<1

4 4 Abstract and Applied Analysis Proof Suppose that {s n } is the sequence of partial sums of the series (18 and we need to show that s n (t is a Cauchy sequence in Banach space (C[, T, ]Forthis,weconsider s n1 (t s n (t u n1 (t γ u n (t γ 2 u n 1 (t γ n1 u (t (21 Now, for every n, m N, n m, there are two arbitrary partial sums s n and s m ;byusing(21 andtriangleinequality successively, we have s n s m (s n (t s n 1 (t (s n 1 (t s n 2 (t (s m1 (t s m (t s n s n 1 s n 1 s n 2 s m1 s m [γ n γ n 1 γ n 2 γ m1 ] u (t γ m1 [γ n m 1 γ n m 2 γ1] u (t γ m1 ( 1 γn m 1 γ u (t (22 Since <γ<1,wehave(1 γ n m <1;then, s n s m Since u is bounded, γm1 (1 γ max u (t t [,T] (23 lim n,m s n (t s m (t (24 Therefore, s n (t is a Cauchy sequence in C[, T],sotheseries converges and the proof complete Since <γ<1,wehave1 γ n m <1;thus,wegettheformula (25 u (x, t m i This completes the proof 5 Applications γm1 u i (x, t (1 γ u (t (27 Example 8 Consider the following fractional partial differential equations with initial condition: α u u u tα x 2 u x 2, (x, t R [,1 2 (28 u (x, 2x (29 with the exact solution at special case α1 u (x, t 2x 12t (3 To solve (28 with initial condition (29, according to the homotopy perturbation technique, we construct the following homotopy: or (1 p ( α u t α α u u t α p( 2 x 2 u u x α u (31 tα α u t α α u u t α p( 2 x 2 u u x α u (32 tα Substituting of (9 in(32 and then equating the terms with same powers of p, we get the series p : α u, tα 43 Error Estimate Theorem 7 The maximum absolute truncation error of the series solution (18 of the problem (11 is estimated to be u (x, t m i γm1 u i (x, t (1 γ u (t (25 Proof From Theorem 6 and inequality (22, we have p 1 : α u 1 t α 2 u x 2 u u x α u t α, p 2 : α u 2 t α 2 u 1 x 2 u u 1 x u u 1 x, p 3 : α u 3 t α 2 u 2 x 2 u u 2 x u u 1 1 x u u 2 x, (33 u (t s m γm1 ( 1 γn m 1 γ u (t (26 p i : α u i t α 2 u i 1 x 2 i 1 j u i j 1 u j x

5 Abstract and Applied Analysis 5 Operating with Riemann-Liouville fractional operator J α, which is the inverse operator of Caputo derivative c D a a in both sides of (28, the solution reads u (x, t 2x, And so on, finally we get s n s m s n s n 1 s n 1 s n 2 s m1 s m u 3 (x, t u 1 (x, t u 2 (x, t 4xtα Γ (α1, 16xt2α Γ (2α 1, 16xt3α Γ (2α 1 (4 Γ (23α 1 Γ 2 (α1, (34 n j n 1 u j iu i m1 n 1 u j j u j i j iu u n u n 1 u m m n 2 iu i (36 2 2n2 ( γ 2 n1 u γn u Since t γ/2, < γ < 1 and α 1, and according to Theorem 6,we have s 1 s 4xt α Γ (α1 (2x ( 2tα Γ (α1 4( γ 2 2x γ u, s 2 s s 2 s 1 s 1 s 16xt 2α 4xt α Γ (2α 1 Γ (α1 8t 2α (2x ( Γ (2α (2x ( 2tα Γ (α1 2x (8( γ 2 2 2( γ 2 16( γ 2 2 2x γ 2 u, s 3 s 1 s 3 s 2 s 2 s 1 16xt 3α Γ (2α 1 (4 Γ (3α 1 Γ 2 (α1 16xt 2α Γ (2α 1 3α 8t (2x ( 2α 8t (2x ( Γ (3α Γ (3α (4 (8( γ 2 3 8( γ 2 2 Γ (2α Γ 2 (α1 Therefore, lim n,m s n s m lim n γ n u ;thatis, s n is, a Cauchy sequence Then the approximate solution in a series form is u (x, t u (x, t u 1 (x, t u 2 (x, t u 3 (x, t 2x (4 4xtα Γ (α1 16xt2α Γ (2α 1 16xt3α Γ (3α 1 Γ (2α 1 Γ 2 (α1 (37 Example 9 Consider the fourth-order parabolic partial differential equation of fractional order α u yz u zx u α ( t 2 cos x 1 4 ( x4 2 cos y 1 4 y 4 ( xy 2 cos z 1 4 u z 4, <x,y,z<π 3, subject to the initial conditions t [, 1], u(x,y,z, u t (x,y,z, 1<α2 xyz (cos xcos ycos z The exact solution for special case α2is (38 (39 32( γ 2 3 2x γ 3 u (35 u(x,y,z,t(xyz cos x cos y cos z e t (4

6 6 Abstract and Applied Analysis For solving (38, according the homotopy perturbation method, we have α u, tα α u 1 t α α u 2 t α yz u ( 2 cos x 1 4 x 4 ( xy 2 cos z 1 4 u z 4 α u α t, yz u ( 1 2 cos x 1 4 x 4 zx u ( 2 cos y 1 4 y 4 ( zx u 1 2 cos y 1 4 y 4 (xy u 1 2 cos z 1 4 z 4 By applying J α in both sides of (41, we got u (x,y,z,t xyz (cos xcos ycos z (, u 1 (x,y,z,t xyz (cos xcos ycos z t α ( Γ (α1 t α1 Γ (α2, u 1 (x,y,z,t xyz (cos xcos ycos z t 2α ( Γ (2α 1 t 2α1 Γ (2α 2, u n (x,y,z,txyz (cos xcos ycos z t nα ( Γ (nα 1 t nα1 Γ (nα 2 According to Theorem 6,we have s 2 s 1 s 2 s 1 s 1 s (41 (42 t 2α Γ (2α 1 t 2α1 t α Γ (2α 2 Γ (α1 tα1 Γ (α 2α /Γ (2α 1 t (2α1 /Γ (2α 2 ((t α /Γ (α1 t (α1 /Γ (α2 ((t (43 But for all t [,1]and α2,wehave t 2α /Γ (2α 1 t (2α1 /Γ (2α 2 t a /Γ (α1 t (α1 /Γ (α2 Thus, s 2 s 1 γ 2 u s 3 s 1 s 3 s 2 s 2 s 1 γ 47 < 1 γ21<1 (44 t 3α Γ (3α 1 t 3α1 t 2α Γ (3α 2 Γ (2α 1 t 2α1 Γ (2α 2 t (( 3α /Γ (3α 1 t (3α1 /Γ (3α 2 2α /Γ (2α 1 t (2α1 /Γ (2α 2 ((t t 3α /Γ (3α 1 t (3α1 /Γ (3α 2 ( t 2α /Γ (2α 1 t (2α1 /Γ (2α 2 Since, for all t [,1], <γ<1,andα2,wehave Thus t 3α /Γ (3α 1 t (3α1 /Γ (3α 2 t 2α /Γ (2α 1 t (2α1 /Γ (2α 2 where γ 3 ((1 γ 2 /(1 γγ 2 And so on, finally we get (45 γ 4 < 1, γ 47 < 1 (46 s 3 s 1 γ3 u, (47 s n s m s n s n 1 s n 1 s n 2 s m1 s m t nα Γ (nα 1 t nα1 Γ (nα 2 t (n 1α Γ ((n 1 α1 t (n 1α1 Γ ((n 1 α2 t (m1α Γ ((m1 α1 t (m1α1 Γ ((m1 α2

7 Abstract and Applied Analysis 7 (( (tnα /Γ (nα 1 (t nα1 /Γ (nα 2 (( (t(n 1α /Γ ((n 1 α1 (t(n 1α1 /Γ ((n 1 α2 (( (t(m1α /Γ ((m1 α1 (t(m1α1 /Γ ((m1 α2 (t nα /Γ (nα 1 (t nα1 /Γ (nα 2 ( (t (n 1α /Γ ((n 1 α1 (t(n 1α1 /Γ ((n 1 α2 (t (m1α /Γ ((m1 α1 (t(m1α1 /Γ ((m1 α2 (48 Since, for all t [,1], <γ<1,andα2,wehave (t nα /Γ (nα 1 (t nα1 /Γ (nα 2 γ n <1, (t (n 1α /Γ ((n 1 α1 (t (n 1α1 /Γ ((n 1 α2 γ n 1 <1, t (m1α Γ ((m1 α1 t (m1α1 γ m1 <1 Γ ((m1 α2 (49 Thus, s n s m γn u, (5 where γ n ((1 γ n m /(1 γγ m1 Therefore, lim s n s m γ n u (51 n,m That is {s n } is a Cauchy sequence The series of the solution is u(x,y,z,t (xyz cos x cos y cos z 6 Conclusion ( t α Γ (α1 t α1 Γ (α2 t 2α Γ (2α 1 t 2α1 Γ (2α 2 (52 In this paper, we applied the HPM for fractional partial differential equations and obtained highly approximate solutions with few iterations Further, we introduce the study problem of convergence of HPM The sufficient condition for convergence of this method has been presented In this paper, we studied the convergence analysis of homotopy perturbation method for fractional partial differential equations, Further, we consider the convergence analysis of homotopy perturbation method for fractional integro-differential equations as future work The convergence analysis is reliable enough to estimate the maximum absolute truncation error of the series solution Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper Acknowledgments The authors are very grateful to the referees for their valuable suggestions and comments that improved the paper References [1] W R Schneider and W Wyss, Fractional diffusion and wave equations, Mathematical Physics, vol3,no1,pp , 1989 [2] S Momani and Z Odibat, Numerical comparison of methods for solving linear differential equations of fractional order, Chaos, Solitons & Fractals,vol31,no5,pp ,27 [3] Z Odibat and S Momani, Numerical methods for nonlinear partial differential equations of fractional order, Applied Mathematical Modelling,vol32,no1,pp28 39,28 [4] H Jafari and S Seifi, Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation, Communications in Nonlinear Science and Numerical Simulation,vol 14, no 5, pp , 29 [5] H Jafari and S Seifi, Solving a system of nonlinear fractional partial differential equations using homotopy analysis method, Communications in Nonlinear Science and Numerical Simulation,vol14,no5,pp ,29 [6] S Momani and Z Odibat, Comparison between the homotopy perturbation method and the variational iteration method for linear fractional partial differential equations, Computers & Mathematics with Applications, vol54,no7-8,pp91 919, 27

8 8 Abstract and Applied Analysis [7] S Momani and Z Odibat, Homotopy perturbation method for nonlinear partial differential equations of fractional order, Physics Letters A,vol365,no5-6,pp345 35,27 [8] O Abdulaziz, I Hashim, and S Momani, Application of homotopy-perturbation method to fractional IVPs, Computational and Applied Mathematics,vol216,no2,pp , 28 [9] J H He, Homotopy perturbation method: a new nonlinear analytical technique, Applied Mathematics and Computation, vol135,no1,pp73 79,23 [1] J-H He, Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering,vol178,no3-4,pp , 1999 [11] A M A El-Sayed, A Elsaid, I L El-Kalla, and D Hammad, A homotopy perturbation technique for solving partial differential equations of fractional order in finite domains, Applied Mathematics and Computation, vol218,no17,pp , 212 [12] A M A El-Sayed, A Elsaid, and D Hammad, A reliable treatment of homotopy perturbation method for solving the nonlinear Klein-Gordon equation of arbitrary (fractional orders, Applied Mathematics, vol 212, Article ID , 13 pages, 212 [13] A Elsaid and D Hammad, A reliable treatment of homotopy perturbation method for the Sine-Gordon equation of arbitrary (fractional order, Fractional Calculus and Applications,vol2,no1,pp1 8,212 [14] A K Golmankhaneh, A K Golmankhaneh, and D Baleanu, On nonlinear fractional KleinGordon equation, Signal Processing, vol 91, no 3, pp , 211 [15] J-H He, Some asymptotic methods for strongly nonlinear equations, International Modern Physics B, vol2, no1,pp ,26 [16] J-H He, Asymptotic methods for solitary solutions and compactons, Abstract and Applied Analysis, vol212,articleid , 13 pages, 212 [17] J-H He, A coupling method of a homotopy technique and a perturbation technique for non-linear problems, International Non-Linear Mechanics,vol35,no1,pp37 43,2

9 Advances in Operations Research Advances in Decision Sciences Applied Mathematics Algebra Probability and Statistics The Scientific World Journal International Differential Equations Submit your manuscripts at International Advances in Combinatorics Mathematical Physics Complex Analysis International Mathematics and Mathematical Sciences Mathematical Problems in Engineering Mathematics Discrete Mathematics Discrete Dynamics in Nature and Society Function Spaces Abstract and Applied Analysis International Stochastic Analysis Optimization

Research Article Application of Homotopy Perturbation and Variational Iteration Methods for Fredholm Integrodifferential Equation of Fractional Order

Abstract and Applied Analysis Volume 212, Article ID 763139, 14 pages doi:1.1155/212/763139 Research Article Application of Homotopy Perturbation and Variational Iteration Methods for Fredholm Integrodifferential