Application of Homotopy Analysis Method for Linear Integro-Differential Equations

Transcription

1 International Mathematical Forum, 5, 21, no. 5, Application of Homotopy Analysis Method for Linear Integro-Differential Equations Zulkifly Abbas a, Saeed Vahdati a,1, Fudziah Ismail a,b and A. Karimi Dizicheh a a Laboratory of Computational Science and Informatics Institute for Mathematical Research University Putra Malaysia, Serdang 434, Selangor, Malaysia b Department of Mathematics Faculty of Science University Putra Malaysia, Serdang 434, Selangor, Malaysia Abstract This paper presents the application of the Homotopy Analysis Method (HAM) as a numerical solution to linear integro-differential equation. The HAM contains the auxiliary parameter, taht provides a powerful tool to analyze strongly linear and nonlinear problems. Examples are provided to demonstrate the advantages of HAM over the homotopy perturbation method (HPM), sine-cosine wavelets and CAS wavelets methods. Mathematics Subject Classification: 45J5, 65R2, 33F5, 74G1 Keywords: Homotopy analysis method; Homotopy perturbation method; Sine-Cosine wavelets; CAS Wavelets; Integro-differential equations 1 Introduction In 1992, Liao [1, 11] proposed a new analytical technique; namely, the Homotopy Analysis Method (HAM) based on homotopy of topology. However, in Liao s PhD dissertation [1], he did not introduce the auxiliary parameter, but simply followed the traditional concept of homotopy to construct the following one-parameter family of equations: 1 Corresponding author. (Saeed Vahdati) address: sdvahdati@gmail.com (1 p)l(u)+pn (u) =, (1)

2 238 Zulkifly Abbas et al where L is an auxiliary linear operator, N is a nonlinear operator related to the original nonlinear problem N (u) = and p is the embedding parameter. In [12], Liao expressed the above equation in a different form as (1 p)l(u) = pn (u). (2) However, Liao found that in some cases the series solution is divergent. To overcome this disadvantage, in 1997, Liao introduced the so-called auxiliary parameter in [13, 14] to construct the following two-parameter family of equations: (1 p)l(u u )= pn (u). (3) where u is an initial guess. Obviously, equation (1) is a special case of equation (3) when = 1. Liao [14]-[19] pointed out that the convergence of the solution series given by the HAM is determined by, thus one can always get a convergent series solution by means of choosing a proper value of. So, the auxiliary parameter provides us a simple way to ensure the convergence of HAM series solutions. Using the definition of Taylor series with respect to the embedding parameter p (which is a power series of p), Liao gave a general equations for high-order approximations. In 1998, He constructed the one-parameter family of equations [4, 5]: (1 p)l(u)+pn (u) =, (4) which is exactly the same as Liao s early one-parameter family equation (1), and is a special case of Liao s modified two-parameter equation (3) when = 1. In this article we consider the following integro-differential equation: 1 f (x) =λ k(t, x)f(t)dt + g(x) (5) f() = f where g L 2 [, 1), k L 2 ([, 1) [, 1)), and f is an unknown function. M. Tavvasoli et al. [6] proposed a direct method based on sine-cosine wavelets to solve this kind of equation.then, they [7] defined the (HPM) for this equation. Their comparison between sine-cosine wavelets and homotopy perturbation method showed that the homotopy perturbation method was more efficient and easier than the sine-cosine wavelets method. Han Danfu and Shang Xufeng [3] used CAS wavelets to solve this equation. They compared the result with the other numerical method which had been introduced in [9]. In this article, we will define the (HAM) to solve this equation. Using the auxiliary parameter, we will show that the homotopy analysis method is more efficient than sine-cosine wavelets, CAS wavelets and homotopy perturbation methods.

3 HAM for linear integro-differential equations Wavelets Wavelet constitute a family of functions constructed from dilation and translation of a single function called the mother wavelet. When the dilation parameter a and the translation parameter b vary continuously, we have the following family of continuous wavelets as [1] ( ) t b ψ a,b (t) = a 1 ψ, a,b R, a. a If we restrict the parameter a and b to discrete values as: a = a k,b= nb a k, where a > 1, b >oand n, and k are positive integers, we have the following of discrete wavelets: ( ) ψ n,k (t) = a k 2 ψ a k t nb, which from a wavelet basis for L 2 (R). In particular, when a = 2 and b =1 then ψ n,k (t) forms an orthonormal basis [1]. 2.1 Sine-Cosine wavelets Sine-Cosine wavelets ψ n,m (t) =ψ(n, k, m, t) have four arguments; n =, 1, 2,, 2 k 1,k =, 1, 2,, the values of m as given in equation (6) and t as the normalized time. They are defined on the interval [, 1) as { k f ψ n,m (t) = m (2 k n t n), t< n+1 2 k 2 k (6), otherwise with f m (t) = 1 2, m = cos(2mπt), m =1, 2,...,L sin(2(m L)πt), m = L +1,L+2,...,2L, where L is any positive integer. The set of SCW are an orthonormal set[8]. 2.2 CAS wavelets CAS wavelets have been itroduced by Yousefi and Banifatemi [2]. The CAS wavelets are defined as follows: { k 2 2 CAS ψ n,m (t) = m (2 k n t n), t< n+1 2 k 2 k (8), otherwise where (7) CAS m (t) = cos(2mπt) + sin(2mπt) (9) and n =, 1, 2,, 2 k 1, k is assumed to be any nonnegative integer, m is any integer and t is the normalized time.

4 24 Zulkifly Abbas et al 2.3 Function approximation and wavelets direct method where A function f(t) defined over [,1) may be expanded as f(t) = 2 k 1 m= n= c n,m ψ n,m (t), (1) c n,m =(f(t),ψ n,m (t)), (11) with (, ) denoting the inner product. If the infinite series in equation (1) is truncated, then it can be written as f(t) 2L 2 k 1 m= n= c n,m ψ n,m (t) =C T Ψ(t). (12) Consider equation (5), if we approximate g, f and k by (6)-(12) as follows: g(x) G T Ψ(x), then f (x) F T Ψ(x), f() F T Ψ(x), k(t, x) ΨT (t)kψ(x), f(x) = x x f (t)dt + f() F T Ψ(t)dt + F T Ψ(x) F T P Ψ(x)+F T Ψ(x) = (F T P + F T )Ψ(x). By substituting the above relation into (5) we have: Ψ T (x)f = λ 1 Ψ T (x)k T Ψ(t)Ψ T (t)(p T F + F )dt +Ψ T (x)g = Ψ T (x)f = λψ T (x)k T (P T F + F )+Ψ T (x)g = (I λk T P T )F = λk T F + G. By solving this linear system we can find the vector F,so F T = F T P + F T = f(x) F T Ψ(x). where the operation matrix P is defined as follows: s Ψ(t)Ψ T (t)dt = P Ψ(s) (13) For details of the operation matrix P in sine-cosine wavelets and CAS wavelets, refer to [8] and [3], respectively.

5 HAM for linear integro-differential equations Homotopy Analysis Method In this paper we apply the homotopy analysis method for integro-differential equation. To show the basic idea, let us consider the following equation: N [ϑ(t)] = where N is a nonlinear operator, t denotes the independent variable,ϑ(t) is an unknown function, respectively. Generalizing the traditional homotopy method, Liao[15] constructs the so-called zero-order deformation equation (1 p)l[φ(t; p) ϑ [] (t)] = p H(t)N [ϑ(t)] (14) Where p [, 1] is the embedding parameter, is a nonzero auxiliary parameter, H(t) is an auxiliary function, L is an auxiliary linear operator, ϑ [] (t) is an initial guess of ϑ(t), φ(t; p) is an unknown function. When p = and p =1, then φ(t;)=ϑ [] (t), φ(t;1)=ϑ(t), respectively. Thus, as p increases from to 1, the solution φ(t; p) varies from the initial guess ϑ [] (t) to the solution ϑ(t). Expanding φ(t; p) in Taylor series with respect to p, one has Where + φ(t; p) =ϑ [] (t)+ ϑ [k] (t)p k, (15) ϑ [k] (t) = 1 k! k=1 k φ(t; p) p k. (16) p= If the auxiliary linear operator, the initial guess, the auxiliary parameter, and the auxiliary function are properly chosen, the series (15) converges at p =1,thus + ϑ(t) =ϑ [] (t)+ k=1 ϑ [k] (t), which must be one of the solutions of the original nonlinear equation, as proved by Liao[15]. According to the equation (16), the governing equation can be deduced from the zero-order deformation equation (14). Define the vector { } ϑ m = ϑ [] (t),ϑ [1] (t),,ϑ [m] (t). Differentiating equation (14) k times with respect to the embedding parameter p and then setting p = and finally dividing them by k!, we have the so-called kth-order deformation equation L[ϑ [k] (t) χ k ϑ [k 1] (t)] = H(t)R k ( ϑ k 1 ), (17)

6 242 Zulkifly Abbas et al where R k ( ϑ k 1 )= 1 (k 1)! k 1 N [φ(t; p)] p k 1, (18) p= and χ k = {, k 1, 1, k > 1. (19) It should be emphasized that ϑ [k] (t) for k 1 is governed by the linear equation (16) with the linear boundary conditions that comes from the original problem, which can be easily solved by symbolic computation software such as Matlab, Maple or Mathematica. In this paper all calculations were accomplished using Matlab software where the long format and the double precision have been used for high accuracy results. 3.1 Analysis of the method by the HAM Consider the following integro-differential equation u (t) =g(t)+λ 1 k(x, t)u(x)dx (2) For solving this equation by HAM, we construct the zeroth-order deformation as [ ] [ U(t, p, ) U(t, p, ) 1 ] (1 p) g(t) = p g(t) k(x, t)u(x, p, )dx t t (21) For p = and p = 1, we can write U(t,, ) = g(t)dt, U(t, 1, ) =u(t). (22) Considering Maclaurin series of U(t, p, ) corresponding to p, one has U(t, p, ) =U(t,, )+ + u [k] k=1 (t, ) p k (23) k! which (t, ) = k U(t, p, ) p k. p= u [k]

7 HAM for linear integro-differential equations 243 If p = 1, using equation (23), then u(t) = g(t)dt + + k=1 u [k] Thus we obtain the kth-order deformation equation L[u [k] (t, ) χ k u [k 1] (t, )] = R k ( u k 1 ), (t, ). (24) k! Now the solution of the kth-order deformation equation for k 1 becomes and u [k] (t, ) k! = u[k 1] u [1] (t, ) = ( 1 ( k(x, t) ) ) g(x)dx dx dt (25) ( ( ) ) (t, ) + u[k 1] (t, ) 1 k(x, t) u[k 1] (t, ) dx dt (k 1)! (k 1)! (k 1)! (26) As a note, the solution of the problem is similar to the Homotopy Perturbation Method if = 1 [7]. 4 Numerical examples Three examples below are provided to show the advantages of HAM over the other methods. The first example has been solved by SCW (with k = 4 and L = 1) [7] and HPM (with twelve terms). The other two examples are solved by CASW (with k = 2 and L = 1) [3] and HPM (with twelve terms). The sign u ũ expresses the absolute value of difference between the exact solution with the numerical solution. Example 1. Consider the integro-differential equation obtained from [2]. u (x) =3e 3x 1 3 (2e3 +1)x + 1 3xtu(t)dt (27) with the exact solution f(x) =e 3x. By HAM, we may construct the zeroth-order deformation (21). using equations (22)-(26) we obtain u [] (t, ) = g(t)dt u [] (t, ) =e 3t 1 ( 2e 3 +1 ) t 2, 6 Then, by

8 244 Zulkifly Abbas et al u [1] (t, ) = ( 1 u [2] (t, ) 2! = u [1] t t 2 e 3. ( k(x, t)u [] ) ) dx dt = ( 5 ( 1 (t, ) u[1] (t, ) ( 5 48 t2 + 5 ) 24 t2 e 3 24 e )t2, ) ) (t, ) dx dt = ), ( k(x, t)u [1] 2 ( 5 48 t t2 e 3 Therefore, the approximate solution of Example 1 can be readily obtained by u(t) = g(t)dt + + k=1 u [k] (t, ) = e 3t 1 k! 6 (2e3 +1)t 2 ( 5 24 e )t t ( t 2 e 3 48 t2 + 5 ) ( 24 t2 e t ) t2 e 3 + (28)

9 HAM for linear integro-differential equations 245 Table 1: Numerical results of Example 1 x SCW HAM u ũ u ũ HPM: = 1 = 1.5 = 1.6 = e e e-16.e e e e e-15.e e e e e-14.e e e e e e e e e e-13.e e e e e-13.e e e e e-13.e e e e e-13.e e-13 Example 2. Consider the integro-differential equation u (x) =xe x + e x x + 1 xu(t)dt The exact solution for this problem is u(x) =xe x [3]. By HAM, we may construct the zeroth-order deformation (21). using equations (22)-(26) we have Then, by u [] (t, ) = u [1] (t, ) = u [2] (t, ) 2! g(t)dt u [] (t, ) =tet 1 2 t2, ( 1 ( ) ) k(x, t)u [] dx dt = 5 12 t2, ( 1 = u [1] (t, ) u [1] (t, ) k=1 ( k(x, t)u [1] (t, ) ) ) dx dt = 5 12 t t 2, u [3] ( ( ) ) (t, ) = u[2] (t, ) u[2] (t, ) 1 k(x, t) u[2] (t, ) dx dt = 3! 2! 2! 2! 5 12 t t 2 + ( 5 12 t t 2 ) 1 6 t2 ( ),. Therefore, the approximate solution of Example 2 can be readily obtained by + u [k] (t, ) u(t) = g(t)dt + = te t 1 k! 2 t t t t t t 2 + ( 5 12 t t 2 ) 1 6 t2 ( )+

10 246 Zulkifly Abbas et al Table 2: Numerical results of Example 2 x CASW HAM u ũ u ũ HPM: = 1 = 1.1 = 1.2 = e e e e e e e e-14.e e e e e-14.e e e e e-13.e e e e e-13.e e e e e e e e e e e e e e e-13.e e e e e e e-13 Example 3. Consider the integro-differential equation: u (x) = x + xtu(t)dt with the exact solution u(x) = x[3]. By HAM, we may construct the zeroth-order deformation (21). Then, by using equations (22)-(26) we obtain u [] (t, ) = u [1] (t, ) = u [2] (t, ) 2! u [3] g(t)dt u [] (t, ) =t 1 6 t2, ( 1 ( ) ) k(x, t)u [] dx = 7 48 t t 2, dt = 7 48 t2, (t, ) 3!. Therefore, the approximate solution of Example 3 can be readily obtained by = 7 48 t t 2 + ( 7 48 t t 2 ) 1 8 t2 ( ), u(t) =t 1 6 t t t t t t 2 + ( 7 48 t t 2 ) 1 8 t2 ( )+

11 HAM for linear integro-differential equations 247 Table 3: Numerical results of Example 3 x CASW HAM u ũ u ũ HPM: = 1 = 1.1 = 1.2 = e e-13.e e e e e-13.e e e e e-12.e e e e e-12.e e e e e-12.e e e e e-12.e e e e e-12.e e e e e-11.e e e e e-11.e e e-11 5 Conclusion Homotopy analysis method is known to be a powerful device for solving many functional equations such as ordinary, partial differential equations, integral equations and so many other equations. In this article, we used homotopy analysis method for solving the linear integro-differential equations. Numerical results showed the advantage of the HAM over the HPM, SCW and CASW methods. (29) References [1] A. Boggess, F. J. Narcowich, A First Course in Wavelets with Fourier Analysis, Prentice-Hall, 21. [2] A. M. Wazwaz, A First Course in Integral Equations, New Jersey, [3] H. Danfu, S. Xufeng, Numerical solution of integro-differential equations by using CAS wavelet operational matrix of integration, Applied mathematics and computation, 194(27), [4] J. H. He, Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering, 178(1999), [5] J. H. He, A coupling method of homotopy technique and perturbation technique for nonlinear problems, International Journal of Non-Linear Mechanics, 35-1(2),

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13 HAM for linear integro-differential equations 249 [19] S.J. Liao, Series solutions of unsteady boundary layer flows over a stretching flat plate, Studies in Applied Mathematics, 117(26), [2] S. Yousefi, A. Banifatemi, Numerical solution of Fredholm integral equations by using CAS wavelets, Applied mathematics and computation, 183(26), Received: April, 29