Homotopy Perturbation Method for Solving Partial Differential Equations of Fractional Order

Transcription

1 Int Journa of Math Anaysis, Vo 6, 2012, no 49, Homotopy Perturbation Method for Soving Partia Differentia Equations of Fractiona Order A A Hemeda Department of Mathematics, Facuty of Science Tanta University, Tanta, Egypt aahemeda@yahoocom Abstract In this artice, we present an efficient and reiabe treatment of the homotopy perturbation method (HPM) for inear / noninear partia differentia equations with fractiona order The fractiona derivatives are described in the Caputo sense The modified agorithm provides approximate soutions in the form of convergent series with easiy computabe components The obtained resuts show that the mentioned method is easy to impement and accurate when appied to partia differentia equations with fractiona order Keywords: Homotopy perturbation method; Partia differentia equations with fractiona order; Caputo fractiona derivative 1 Introduction Recenty, the differentia equations of fractiona order have been the focus of many studies due to their frequent appearance in various appications in fuid mechanics, viscoeasticity, bioogy physics and engineering consequenty, considerabe attention has been given to the soutions of fractiona differentia equations and integra equations of physica interest 1-10] Most fractiona differentia equations do not have exact anaytic soutions, so approximation and numerica techniques must be used 11-15] The homotopy perturbation method, proposed first by He 16-20], is reativey new approach to provide an anaytica approximation to inear / noninear probems, and it is particuary vauabe as too for scientists and appied mathematicians, because it provide immediate and visibe symboic terms of anaytic soutions, as we as numerica approximate soutions to both inear and noninear differentia equations without inearization or discretization

2 2432 A A Hemeda The motivation of this artice, is to extend the approximation of the homotopy perturbation method 16-20] to sove inear / noninear partia differentia equations with time fractiona derivative 2 Definitions For the concept of fractiona derivative we wi adopt Caputo s definition which is a modification of the Riemann-Liouvie definition and has the advantage of deaing propery with initia vaue probems in which the initia conditions are given in terms of the fied variabes and their integer order which is the case in most physica processes Definition 21 A rea function f(t), t>0 is said to be in the space C γ,γ R if there exists a rea number P (> γ), such that f(t) =t P f 1 (t), where f 1 (t) C0, ), and it is said to be in the space Cγ m iff f (m) C γ, m N Definition 22 The Riemann-Liouvi fractiona integra operator of order μ 0 of a function f C γ,γ 1, is defined as: I μ f(t) = 1 Γ(μ) t 0 (t τ) μ 1 f(τ)dτ, μ > 0, t>0, (1a) I 0 f(t) =f(t) (1b) Properties of the operator I μ can be found in 1,7,8], we mention ony the foowing: Definition 23 For f C γ,γ 1, μ,β 0 and ν> 1 : I μ I β f(t) =I μ+β f(t), (2a) I μ I β f(t) =I β I μ f(t), (2b) I μ x ν = Γ(ν +1) Γ(μ + ν +1) xμ+ν Definition 24 The Caputo fractiona derivative D μ t N, is defined as: (2c) of f, f C m 1,m

3 Homotopy perturbation method 2433 D μ t f(t) = dm f(t), μ= m, dtm = I m μ t D m t f(t) = 1 Γ(m μ) t 0 (t τ) m μ 1 d m f(τ)dτ, m 1 <μ m, t > 0, dτ (3a) m D μ t I μ t f(t) =f(t), m 1 <μ m, t > 0, (3b) I μ t D μ t f(t) =f(t) m 1 k=0 d k dt k f(t)tk,m 1 <μ m, t > 0 (3c) k! 3 Homotopy perturbation method At first, we wi present a review of the homotopy perturbation method 16,17] then we wi present the agorithm of the new modification of the homotopy perturbation method Consider the foowing noninear differentia equation: with the boundary conditions: L(u)+N(u) =f(r), r Ω, (4) B(u, u/ n) =0,r Γ, (5) where L is a inear operator, N is a noninear operator, B is a boundary operator, Γ is the boundary of the domain Ω and f(r) is a known anaytic function By the homotopy technique, He defines the homotopy v(r, p) :Ω Ω 0, 1] R which satisfies: or H(v, p) =(1 p)l(v) L(u 0 ]+p L(v)+N(v) f(r)] = 0, (6)

4 2434 A A Hemeda H(v, p) =L(v) L(u 0 )+pl(u 0 )+p N(v) f(r)] = 0, (7) where r Ω, p 0, 1] is an impeding parameter and u 0 is an initia approximation which satisfies the boundary conditions Obviousy, from Eqs (6) and (7), we have: H(v, 0) = L(v) L(u 0 )=0, (8) H(v, 1) = L(v)+N(v) f(r) =0 (9) The changing process of p from zero to unity is just of v(r, p) from u 0 (r) to u(r)in topoogy, this is caed deformation and L(v) L(u 0 ) and L(v)+N(v) f(r) are homotopic The basic assumption is that the soution of Eqs (6) and (7) can be expressed as a power series in p : v = v 0 + pv 1 + p 2 v 2 + (10) Setting p = 1 resuts in the approximate soution of Eq (4): u = im p 1 v = v 0 + v 1 + v 2 + (11) The convergence of the series (11) has been proved in 16,17] 4 The modification homotopy perturbation method Consider the foowing noninear partia differentia equation with time derivative of fractiona order: D μ t u(x, t) =L(u, u x,u xx )+N(u, u x,u xx )+f(x, t), t>0, (12) where L is a inear operator, N is a noninear operator which aso might incude other fractiona derivatives of order ess than μ, f is a known anaytic function and D μ t,m 1 <μ m, is the Caputo fractiona derivative of order μ, subject to the initia conditions:

5 Homotopy perturbation method 2435 k t u(x, 0) = h k(x), k=0, 1, 2,, m 1 (13) k In view of the homotopy technique, we can define the foowing homotopy 21]: ] m u m t L(u, u u x,u m xx ) f(x, t) =p t + N(u, u x,u m xx ) D μ t u, (14) or ] m u m u f(x, t) =p tm t + L(u, u x,u m xx )+N(u, u x,u xx ) D μ t u, (15) where p 0, 1] The homotopy parameter p aways changes from zero to unity When p = 0, Eq (14) becomes the inearized equation: m u t m = L(u, u x,u xx )+f(x, t), (16) and Eq (15) becomes the inearized equation: m u = f(x, t), (17) tm and when p =1, Eq (14) or Eq (15) turns out to be the origina equation (12) The basic assumption is that the soution of Eq (14) or Eq (15) can be written as a power series in p : u = u 0 + pu 1 + p 2 u 2 + (18) The n-term approximate soution for Eq (12) is: n 1 u(x, t) = u i (x, t) =u 0 + u u n 1 (19) i=0 Finay, we approximate the soution: by the truncated series: u(x, t) = u n (x, t) n=0 Φ N (x, t) = N 1 n=0 u n (x, t) (20)

6 2436 A A Hemeda 5 Appication probems Probem 51 The first appication is the time-fractiona derivative heat equation in one-dimension defined as: D μ t u = αu xx, 0 <μ 1, t>0, x R, (21a) with the initia condition: u(x, 0) = c,c,b, R, (21b) where u(x, t) is the heat conduction (diffusion) in one-dimensiona isotropic medium and α is a therma diffusion In view of Eq (15), the homotopy for Eq (21a) is: ] u u t = p t + αu xx D μ t u (22) Substituting (18) and the initia condition (21b) into (22) and equating the terms with identica powers of p, we obtain the foowing set of inear partia differentia equations: ( ) u 0 bx t =0,u 0(x, 0) = c, u 1 t = u 0 t + α(u 0) xx D μ t u 0, u 1 (x, 0) = 0, u 2 t = u 1 t + α(u 1) xx D μ t u 1, u 2 (x, 0) = 0, u 3 t = u 2 t + α(u 2) xx D μ t u 2, u 3 (x, 0) = 0, Consequenty, the first few components of the homotopy perturbation soution for Prob (21) are derived in the form:

7 Homotopy perturbation method 2437 u 2 (x, t) = cαb2 2 u 3 (x, t) = cαb2 2 + cα2 b 4 4 u 0 (x, t) =c u 1 (x, t) = cαb2 2, t, ( ) ] bx t 2 μ Γ(3 μ) t + cα2 b 4 4 t2 2, ( ) bx 2t 2 μ Γ(3 μ) t 4 μ 1 ] πt 3 2μ (3 2μ)Γ(15 μ)γ(2 μ) ( ] ( ) bx )t 2 2t3 μ cα3 b 6 bx t3 Γ(4 μ) 6 6, and so on, in the same manner the rest of components can be obtained ug the Mathematica Package The four-term approximate soution for Prob (21) is given by: u(x, t) =c + cαb2 2 4 μ 1 ] πt 3 2μ + cα2 b 4 (3 2μ)Γ(15 μ)γ(2 μ) 4 cα3 b 6 6 When μ = 1 Eq (23) takes the form: u(x, t) =c ( ) bx 1+ ( ) αb 2 t + 1 ( αb 2 t 2 2! 2 ( ) bx 3t 2 μ Γ(3 μ) 3t ( ) ] bx 3t 2 2 2t3 μ Γ(4 μ) t3 6 (23) ) ! ( ) ] αb 2 3 t, (24) 2

8 2438 A A Hemeda which is first four terms of the series of the exact soution u(x, t) =c ( bx ) exp( αb2 t 2 ) Tabe 1 shows the approximate soution for Prob (21) obtained for different vaues of μ at c = α = = b =1 It is cear that the approximations obtained by the method are in high agreement with those obtained ug the exact soution It is evident that the efficiency of the method can be enhanced by computing further terms of u(x, t) Tabe 1: Numerica vaues when μ = 025, 05, 075 and 10 for Eq (21) t x μ =025 μ =05 μ =075 μ =10 u Exact Probem 52 The second appication is the time fractiona derivative heat equation in two-dimension defined as: D μ t u = α(u xx + u yy ), 0 <μ 1, t>0, x R, (25a) with the initia condition: u(x, y, 0) = c + ( )] by,c,b, R (25b) In view of Eq (15), the homotopy for Eq (25a) is: ] u u t = p t + α(u xx + u yy ) D μ t u (26)

9 Homotopy perturbation method 2439 Substituting (18) and the initia condition (25b) into (26) and equating the terms with identica powers of p, we obtain the foowing set of inear partia differentia equations: u 0 t =0,u 0(x, y, 0) = c + ( )] by, u 1 t = u 0 t + α (u 0) xx +(u 0 ) yy ] D μ t u 0,u 1 (x, y, 0) = 0, u 2 t = u 1 t + α (u 1) xx +(u 1 ) yy ] D μ t u 1,u 2 (x, y, 0) = 0, u 3 t = u 2 t + α (u 2) xx +(u 2 ) yy ] D μ t u 2,u 3 (x, y, 0) = 0, Consequenty, the first few components of the homotopy perturbation soution for Prob (25) are derived in the form: u 0 (x, y, t) =c u 1 (x, y, t) = cαb2 2 u 2 (x, y, t) = cαb2 2 + cα2 b 4 4 u 3 (x, y, t) = cαb ( )] by ( )] by, ( )] by t2 2, ( )] by t, t 2 μ Γ(3 μ) t ( )] by 2t 2 μ Γ(3 μ) t ]

10 2440 A A Hemeda 4 μ 1 ] πt 3 2μ + cα2 b 4 (3 2μ)Γ(15 μ)γ(2 μ) 4 ] t 2 2t3 μ cα3 b 6 Γ(4 μ) ( )] by t3 6, ( )] by and so on, in the same manner the rest of components can be obtained ug the Mathematica Package The four-term approximate soution for Prob (25) is given by: u(x, y, t) =c + ( )] by + cαb t 2 μ Γ(3 μ) 3t 4 μ 1 ] πt 3 2μ (3 2μ)Γ(15 μ)γ(2 μ) + cα2 b 4 4 cα3 b When μ = 1 Eq (27) takes the form: u(x, y, t) =c + 1 2! + ( αb 2 t 2 ( )] by ( )] ] by 3t 2 2 2t3 μ Γ(4 μ) ( )] by t3 6 (27) ( )] by 1+ ) ! ( ) αb 2 t 2 ( ) ] αb 2 3 t, (28) which is the first four terms of the series of the exact soution u(x, y, t) = c ( bx ) + ( by )] exp( αb2 t 2 ) 2

11 Homotopy perturbation method 2441 Tabe 2 shows the approximate soution for Prob (25) obtained for different vaues of μ at c = α = = b =1 It is cear that the approximations obtained by the method are in high agreement with those obtained ug the exact soution Tabe 2: Numerica vaues when μ =025, 05, 075 and 10 for Eq (25) t x y μ =025 μ =05 μ =075 μ =10 u Exact o Probem 53 The third appication is the foowing time fractiona derivative in xyt-pane defined as: D μ t u = u, 1 <μ 2, t>0, x,y R, (29a) with the initia conditions: u(x, y, 0) = (x + y), u t (x, y, 0) = cos(x + y) (29b) In view of Eq (15), the homotopy for Eq (29a) is: 2 u t 2 2 = p u t + 1 ] u D μ t u (30) Substituting (18) and the initia conditions (29b) into (30) and equating the terms with identica powers of p, we obtain the foowing set of inear partia differentia equations:

12 2442 A A Hemeda 2 u 0 t 2 =0,u 0 (x, y, 0) = (x + y), (u 0 ) t = cos(x + y), 2 u 1 t 2 = 2 u 0 t u 0 D μ t u 0,u 1 (x, y, 0) = 0, (u 1 ) t (x, y, 0) = 0, 2 u 2 t 2 = 2 u 1 t u 1 D μ t u 1,u 2 (x, y, 0) = 0, (u 2 ) t (x, y, 0) = 0, 2 u 3 = 2 u 2 t 2 t u 2 D μ t u 2,u 3 (x, y, 0) = 0, (u 3 ) t (x, y, 0) = 0, Consequenty, the first few components of the homotopy perturbation soution for Prob (29) are derived in the form: u 0 (x, y, t) = (x + y) cos(x + y)t, u 1 (x, y, t) = (x + y) t2 2 + cos(x + y)t3 6, ] t22 t4 u 2 (x, y, t) = (x + y) t4 μ Γ(5 μ) ] t 3 + cos(x + y) 6 t5 120 t5 μ, Γ(6 μ) t22 t4 u 3 (x, y, t) = (x + y) + 12 t t4 μ Γ(5 μ) 2t6 μ Γ(7 μ) 4 μ 2 ] πt 6 2μ (6 2μ)(5 2μ)Γ(3 μ)γ(25 μ) + cos(x + y) t 3 6 t5 60

13 Homotopy perturbation method 2443 ] + t t5 μ Γ(6 μ) + 2t7 μ Γ(8 μ) + t7 2μ, Γ(8 2μ) and so on, in the same manner the rest of components can be obtained ug the Mathematica Package The four-term approximate soution for Prob (29) is given by: u(x, y, t) = (x + y) 1 3t2 2 + t4 8 t t4 μ Γ(5 μ) 2t6 μ Γ(7 μ) 4 μ 2 ] πt 6 2μ + cos(x + y) (6 2μ)(5 2μ)Γ(3 μ)γ(25 μ) t + ] t32 t t t5 μ Γ(6 μ) + 2t7 μ Γ(8 μ) + t7 2μ (31) Γ(8 2μ) When μ = 2 Eq (31) takes the form: ] ] u(x, y, t) = (x + y) 1 t2 2! + t4 4! t6 cos(x + y) t t3 6! 3! + t5 5! t7, 7! (32) which is the first four terms of the series of the exact soution u(x, y, t) = (x + y t) Tabe 3 shows the approximate soution for Prob (29) obtained for different vaues of μ It is cear that the approximations obtained by the method are neary agreement with those obtained ug the exact soution Tabe 3: Numerica vaues when μ = 125, 15, 175 and 20 for Eq (29)

14 2444 A A Hemeda t x y μ =125 μ =15 μ =175 μ =20 u Exact Probem 54 Finay, the forth appication is the foowing noninear partia differentia equation with time fractiona derivative: D μ t u + u 2 xt + u xx =2(x +2t 2 ), 1 <μ 2, t>0, x R, (33a) with the initia conditions: u(x, 0) = 0, u t (x, 0) = a (33b) In view of Eq (15), the homotopy for Eq (33a) is: ] 2 u 2 t 2x u 2 4t2 = p t 2 u2 xt u xx D μ t u (34) Substituting (18) and the initia conditions (33b) into (34) and equating the terms with identica powers of p, we obtain the foowing set of inear partia differentia equations: 2 u 0 t 2 2x 4t2 =0,u 0 (x, 0) = 0, (u 0 ) t (x, 0) = a, 2 u 1 t 2 = 2 u 0 t 2 (u 0) 2 xt (u 0) xx D μ t u 0,u 1 (x, 0) = 0, (u 1 ) t (x, 0) = 0,

15 Homotopy perturbation method u 2 = 2 u 1 t 2 t 2(u 0) 2 xt (u 1 ) xt (u 1 ) xx D μ t u 1, u 2 (x, 0) = 0, (u 2 ) t (x, 0) = 0, Consequenty, the first few components of the homotopy perturbation soution for Prob (33) are derived in the form: u 0 (x, t) =at + xt 2 + t4 3, ] t 2 u 1 (x, t) =2x 2 t4 μ 8t6 μ Γ(5 μ) Γ(7 μ), u 2 (x, t) =x t 2 4t4 μ Γ(5 μ) + 2(4) μ 2 ] πt 6 2μ (6 2μ)(5 2μ)Γ(3 μ)γ(25 μ) 8t4 12 8t6 μ Γ(7 μ) + 8t 6 μ (5 μ)(6 μ)γ(4 μ) + 8t8 2μ Γ(9 2μ), and so on, in the same manner the rest of components can be obtained ug the Mathematica Package the three-term approximate soution for Prob (33) is given by: u(x, t) =at t4 3 16t6 μ Γ(7 μ) + 8t 6 μ (6 μ)(5 μ)γ(4 μ) + 8t8 2μ Γ(9 2μ) + x 3t 2 6t4 μ Γ(5 μ) + 2(4) μ 2 ] πt 6 2μ (35) (6 2μ)(5 2μ)Γ(3 μ)γ(25 μ) When μ = 2 Eq (35) takes the form: u(x, t) =at + xt 2, (36)

16 2446 A A Hemeda which is the exact soution for Prob (33) Tabe 4 shows the approximate soution for Prob (33) obtained for different vaues of μ at a = 1 It is cear that the approximations obtained by the method are the same with those obtained ug the exact soution u(x, t) = at + xt 2 when μ =2 Tabe 4: Numerica vaues when μ =125, 15, 175 and 20 for Eq (33) t x μ =125 μ =1, 5 μ =175 μ =10 u Exact Concusion From the obtained resuts it is cear that the modified homotopy perturbation method suggested in this artice provide the soutions in terms of convergent series with easiy computabe components, so it is an efficient and appropriate method for soving inear and noninear partia differentia equations of fractiona order References: 1] KK Odham, J Spanier, The Fractiona Cacuus, Academic Press, New York, ] M Caputo, Linear modes of dissipation whose Q is amost frequency independent, Part II, J Roy Astr Soc 13 (1967) 529 3] W Schneider, W Wyss, Fractiona diffusion equation and wave equation, J Math Phys 30 (1989) 134 4] KS Mier, B Ross, An Introduction to the Fractiona Cacuus and Fractiona Differentia Equations, John Wiey and Sons, Inc, New York, 1993

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18 2448 A A Hemeda Received: March, 2012