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1 Coun Nonlinear Sci Nuer Siulat 14 (2009) Contents lists available at ScienceDirect Coun Nonlinear Sci Nuer Siulat journal hoepage: Short counication Notes on the hootopy analysis ethod: Soe definitions and theores Shijun Liao State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai , PR China article info abstract Article history: Received 18 April 2008 Accepted 18 April 2008 Available online 30 April 2008 PACS: a Nx Wz We describe, very briefly, the basic ideas and current developents of the hootopy analysis ethod, an analytic approach to get convergent series solutions of strongly nonlinear probles, which recently attracts interests of ore and ore researchers. Definitions of soe new concepts such as the hootopy-derivative, the convergence-control paraeter and so on, are given to redescribe the ethod ore rigorously. Soe leas and theores about the hootopy-derivative and the deforation equation are proved. Besides, a few open questions are discussed, and a hypothesis is put forward for future studies. Ó 2008 Elsevier B.V. All rights reserved. Keywords: Nonlinear equations Series solution Hootopy analysis ethod 1. Introduction Nonlinear equations are difficult to solve, especially analytically. Perturbation techniques [1 12] are widely applied in science and engineering, and do great contribution to help us understand any nonlinear phenoena. However, it is well known that perturbation ethods are strongly dependent upon sall/large physical paraeters, and therefore are valid in principle only for weakly nonlinear probles. The so-called non-perturbation techniques, such as the Lyapunov s artificial sall paraeter ethod [13], the d-expansion ethod [14,15], Adoian s decoposition ethod [16 19], and so on, are forally independent of sall/large physical paraeters. But, all of these traditional non-perturbation ethods can not ensure the convergence of solution series: they are in fact only valid for weakly nonlinear probles, too. The hootopy analysis ethod (HAM) [20 27] is a general analytic approach to get series solutions of various types of nonlinear equations, including algebraic equations, ordinary differential equations, partial differential equations, differential-integral equations, differential-difference equation, and coupled equations of the. Unlike perturbation ethods, the HAM is independent of sall/large physical paraeters, and thus is valid no atter whether a nonlinear proble contains sall/large physical paraeters or not. More iportantly, different fro all perturbation and traditional non-perturbation ethods, the HAM provides us a siple way to ensure the convergence of solution series, and therefore, the HAM is valid even for strongly nonlinear probles. Besides, different fro all perturbation and previous non-perturbation ethods, the HAM provides us with great freedo to choose proper base functions to approxiate a nonlinear proble [21,26]. Since Liao s book [21] for the hootopy analysis ethod was published in 2003, ore and ore researchers have been successfully applying this ethod to various nonlinear probles in science and engineering, such as the viscous flows of non-newtonian fluids [28 38], the KdV-type equations [39 43], nonlinear heat transfer [44 46], finance probles [47,48], Rieann probles related to nonlinear shallow water equations [49], projectile otion [50], Glauert-jet flow [51], nonlinear water waves [52], groundwater flows E-ail address: sjliao@sjtu.edu.cn /$ - see front atter Ó 2008 Elsevier B.V. All rights reserved. doi: /j.cnsns

2 984 S. Liao / Coun Nonlinear Sci Nuer Siulat 14 (2009) [53], Burgers Huxley equation [54], tie-dependent Eden Fowler type equations [55], differential-difference equation [56], Laplace equation with Dirichlet and Neuann boundary conditions [57], theral hydraulic networks [58], boundary layer flows over a stretching surface with suction and injection [59], and so on. Especially, soe new solutions of a few nonlinear equations are reported [60,61]: these new solutions have never been reported by all other previous analytic ethods and even by nuerical ethods. This shows the great potential of the HAM for strongly nonlinear probles in science and engineering. The HAM is based on hootopy, a fundaental concept in topology and differential geoetry [62], which can be traced back to Poincaré [63]. Briefly speaking, by eans of the HAM, one constructs a continuous apping of an initial guess approxiation to the exact solution of considered equations. An auxiliary linear operator is chosen to construct such kind of continuous apping, and an auxiliary paraeter is used to ensure the convergence of solution series. The ethod enjoys great freedo in choosing initial approxiations and auxiliary linear operators. By eans of this kind of freedo, a coplicated nonlinear proble can be transferred into an infinite nuber of sipler, linear sub-probles, as shown by Liao and Tan [26]. For exaple, let us consider a nonlinear algebraic equation f ðxþ ¼0: First of all, we construct such a hootopy: H½x; qš ¼ð1 qþ½f ðxþ fðx 0 ÞŠþqf ðxþ; where x 0 is an initial guess of x, and q 2½0; 1Š is called hootopy-paraeter. 1 Obviously, at q ¼ 0 and q ¼ 1, one has H½x; 0Š ¼f ðxþ f ðx 0 Þ; H½x; 1Š ¼f ðxþ; respectively. Thus, as q increases fro 0 to 1, H½x; qš varies continuously fro f ðxþ f ðx 0 Þ to f ðxþ. Such kind of continuous variation is called deforation in topology [62]. Now, enforcing H½x; qš ¼0, i.e. ð1 qþ½f ðxþ f ðx 0 ÞŠ þ qf ðxþ ¼0; we have now a faily of algebraic equations. Obviously, the solution of the above faily of algebraic equations is dependent upon the hootopy-paraeter q. So, the faily of equations can be rewritten as ð1 qþff ½/ðqÞŠ f ðx 0 Þg þ qf ½/ðqÞŠ ¼ 0: At q ¼ 0, it gives f ½/ðqÞŠ f ðx 0 Þ¼0; when q ¼ 0; whose solution is obviously /j q¼0 ¼ /ð0þ ¼x 0 : At q ¼ 1, one has f ½/ðqÞŠ ¼ 0; when q ¼ 1: It is exactly the sae as the original algebraic equation f ðxþ ¼0, thus /j q¼1 ¼ /ð1þ ¼x: Therefore, as the hootopy-paraeter q increases fro 0 to 1, /ðqþ varies (or defors) fro the initial guess x 0 to the solution x of f ðxþ ¼0. We call the faily of equations like (1) the zeroth-order deforation equation (the ore rigorous definition will be given in the following section). Because /ðqþ is now a function of the hootopy-paraeter q, we can expand it into Maclaurin series ð1þ /ðqþ ¼x 0 þ Xþ1 x k q k ; where /ð0þ ¼x 0 is eployed, and x k ¼ 1 o k /ðqþ ¼ D k! oq k k ð/þ: q¼0 ð2þ ð3þ Here, the series (2) is called hootopy-series, D k ð/þ is called the kth-order hootopy-derivative of / (ore rigorous definitions and soe related theores will be given in the following section). If the hootopy-series (2) is convergent at q ¼ 1, then using the relationship /ð1þ ¼x, one has the so-called hootopy-series solution x ¼ x 0 þ Xþ1 x k : Unfortunately, convergence radii of any Maclaurin series of functions are less than 1. So, here, we had to assue that the hootopy-series is convergent at q ¼ 1. This restriction can be overcoe by introducing an auxiliary paraeter, as shown later. ð4þ 1 In the theory of topology, q is called the ebedding paraeter.

3 S. Liao / Coun Nonlinear Sci Nuer Siulat 14 (2009) According to the fundaental theore of calculus about Taylor series, the coefficient x k of the hootopy-series (2) is unique. Therefore, the governing equation of x k is unique, too, and can be deduced directly fro the zeroth-order deforation equation (1). Taking the 1st-order hootopy-derivative on both sides of the zeroth-order deforation equation (1) gives the so-called 1st-order deforation equation: x 1 f 0 ðx 0 Þþf ðx 0 Þ¼0; whose solution is x 1 ¼ f ðx 0Þ f 0 ðx 0 Þ : Taking the 2nd-order hootopy-derivative 2 on both sides of (1) gives the 2nd-order deforation equation: x 2 f 0 ðx 0 Þþ 1 2 x2 1 f 00 ðx 0 Þ¼0; whose solution is x 2 ¼ x2 1 f 00 ðx 0 Þ 2f 0 ðx 0 Þ ¼ f 2 ðx 0 Þf 00 ðx 0 Þ 2½f 0 ðx 0 ÞŠ 3 : In this way, one obtains x k one by one in the order k ¼ 1; 2; 3;... Here, we ephasize that all of these high-order deforation equations are linear, and therefore are easy to solve. Then, we have the 1st-order hootopy-series approxiation x ¼ x 0 þ x 1 ¼ x 0 f ðx 0Þ f 0 ðx 0 Þ and the 2nd-order hootopy-series approxiation x x 0 þ x 1 þ x 2 ¼ x 0 f ðx 0Þ f 0 ðx 0 Þ f 2 ðx 0 Þf 00 ðx 0 Þ 2½f 0 ðx 0 ÞŠ 3 : ð6þ Note that (5) is exactly the sae as the faous Newton s iteration forula, and thus (6) can be regarded as the 2nd-order Newton s iteration forula. In fact, one can give a faily of Newton s iteration forula in a siilar way. The above analytic approach is independent of any physical paraeters at all: no atter whether a nonlinear proble contains sall/large physical paraeters or not, one can always introduce the hootopy-paraeter q 2 ½0; 1Š to construct a zeroth-order deforation equation and then to get the hootopy-series solution. Unfortunately, the hootopy-series like (2) is not always convergent at q ¼ 1, therefore the corresponding hootopy-series solution (4) ight be divergent. For exaple, it is well-known that Newton s 1st-order iteration forula (5) often gives divergent results. This is ainly because the above approach is based on such a assuption that the hootopy-series like (2) is convergent at q ¼ 1, but this assuption does not hold in general, especially for nonlinear probles with strong nonlinearity. To overcoe this restriction of the early HAM, Liao [22] introduced an auxiliary paraeter h 0 to construct such a kind of zeroth-order deforation equation ð5þ ð1 qþff ½/ðqÞŠ f ðx 0 Þg ¼ qhf ½/ðqÞŠ: ð7þ Since h 0, the above equation at q ¼ 1 becoes hf ½/ðqÞŠ ¼ 0; when q ¼ 1; which is equivalent to the original equation f ðxþ ¼0, provided x ¼ /ð1þ. All other forulas like (2) and (4) are the sae, except the high-order deforation equation. Siilarly, taking the 1st-order hootopy-derivative on both sides of (7), we have the corresponding 1st-order deforation equation x 1 f 0 ðx 0 Þ hf ðx 0 Þ¼0; whose solution is x 1 ¼ h f ðx 0Þ f 0 ðx 0 Þ : Taking the 2nd-order hootopy-derivative on both sides of (7) gives the 2nd-order deforation equation: x 2 f 0 ðx 0 Þ ð1 þ hþx 1 f 0 ðx 0 Þþ 1 2 x2 1 f 00 ðx 0 Þ¼0; whose solution is x 2 ¼ð1 þ hþx 1 f 0 ðx 0 Þ x2 1 f 00 ðx 0 Þ 2f 0 ðx 0 Þ ¼ hð1 þ hþf ðx 0Þ f 2 ðx 0 Þf 00 ðx 0 Þ 2½f 0 ðx 0 ÞŠ 3 : 2 According to the definition (3), we here differentiates the zeroth-order deforation equation (1) twice with respect to q, then divides by 2! and finally sets q ¼ 0.

4 986 S. Liao / Coun Nonlinear Sci Nuer Siulat 14 (2009) Siilarly, one has the corresponding first-order hootopy-series approxiation x ¼ x 0 þ x 1 ¼ x 0 þ h f ðx 0Þ f 0 ðx 0 Þ ; and the corresponding 2nd-order hootopy-series approxiation x x 0 þ x 1 þ x 2 ¼ð1 þ h þ h 2 Þx 0 þ h f ðx 0Þ f 0 ðx 0 Þ f 2 ðx 0 Þf 00 ðx 0 Þ 2½f 0 ðx 0 ÞŠ 3 : ð9þ Obviously, (5) and (6) are special cases of (8) and (9) when h ¼ 1, respectively. The auxiliary-paraeter h in (5) can be regarded as a iteration factor that is widely used in nuerical coputations. It is well known that a properly chosen iteration factor can ensure the convergence of iteration. Siilarly, it is found that the convergence of the hootopy-series like (2) is dependent upon the value of h: one can ensure the convergence of the hootopy-series solution siply by eans of choosing a proper value of h, as shown by Liao [21 23,25,26,60] and others [28]-[59]. In fact, it is the auxiliary paraeter h that provides us, for the first tie, a siple way to ensure the convergence of series solution. Due to this reason, it sees reasonable to renae h the convergence-control paraeter, which was suggested by Dr. Pradeep Siddheshwar in our private discussions. It should be ephasized that, without the use of the convergence-paraeter h, one had to assue that the hootopyseries like (2) is convergent. However, with the use of the convergence-paraeter h, such an assuption is unnecessary, because it sees that one can always choose a proper value of h to obtain convergent hootopy-series solution. So, the use of the convergence-paraeter h in the zeroth-order deforation equation greatly odifies the early hootopy analysis ethod. Since then, the hootopy analysis ethod have been developing greatly, and ore generalized zeroth-order deforation equations are suggested by Liao [21,23,24,26]. Currently, soe developents [64 66] of the HAM are reported. At the current stage of the HAM, it is urgently necessary to redescribe this ethod in a ore rigorous way. So, in this paper, definitions of soe new concepts such as the hootopy-derivative, the convergence-control paraeter, the convergence-control vector, and so on, are given so as to redescribe the ethod ore rigorously. Besides, soe leas and theores about the hootopy-derivative and the deforation equation are proved. Furtherore, a few open questions are discussed, and a hypothesis is put forward for future studies. All of these can help the HAM users easily understand this ethod, and siplify the applications of the HAM for new, coplicated nonlinear probles in science and engineering. 2. Properties of hootopy-derivative As entioned in Section 1 the so-called hootopy-derivative is used to deduce the high-order deforation equation. Here, we first give rigorous definitions and then prove soe properties of the hootopy-derivative. These properties are useful to deduce the high-order deforation equations. ð8þ Definition 2.1. Let / be a function of the hootopy-paraeter q, then D ð/þ ¼ 1 o /! oq q¼0 is called the th-order hootopy-derivative of /, where P 0 is an integer. ð10þ Definition 2.2. Let N½uŠ ¼0 denote a nonlinear equation, / be a function of the hootopy-paraeter q 2½0; 1Š, whose Maclaurin series is u k q k : The faily of equations P½/; qš ¼0; q 2½0; 1Š is called the zeroth-order deforation equation of N½uŠ ¼0, if, at q ¼ 1, it is equivalent to the original equation N½uŠ ¼0so that u ¼ /j q¼1 ¼ Xþ1 u k ; and besides, its solution is obvious at q ¼ 0. The series (11) is called the hootopy-series, the series (12) is called hootopyseries solution of N½uŠ ¼0, and the equations governing u k are called the kth-order deforation equations. Molabahrai and Khani [54] proved the following theore: Molabahrai and Khani s Theore. For hootopy-series u i q i ; ð11þ ð12þ

5 S. Liao / Coun Nonlinear Sci Nuer Siulat 14 (2009) it holds D ð/ k Þ¼ X r 1 ¼0 u r1 X r 1 r 2 ¼0 u r1 r 2 X r 2 r 3 ¼0 u r2 r 3 Xr k 3 X r k 2 u rk 3 r k 2 r k 2 ¼0 r k 1 ¼0 u rk 2 r k 1 u rk 1 ; where P 0 and k P 1 are positive integers. For details, please refer to Molabahrai and Khani [54]. Here, we prove soe other properties of the hootopyderivative. Theore 2.1. Let f and g be functions independent of the hootopy-paraeter q. For hootopy-series u i q i ; w ¼ Xþ1 v j q j ; j¼0 it holds D ðf / þ gwþ ¼fD ð/þþgd ðwþ: Proof. Because f and g are independent of q, and besides D defined by (10) is a liner operator, it obviously holds D ðf / þ gwþ ¼D ðf /ÞþD ðgwþ ¼fD ð/þþgd ðwþ: Theore 2.2. For hootopy-series u i q i ; w ¼ Xþ1 v j q j ; j¼0 it holds ðaþ D ð/þ ¼u ; ðbþ D ðq k /Þ¼D k ð/þ; ðcþ D ð/wþ ¼ X D i ð/þd i ðwþ ¼ X D i ðwþd i ð/þ; ðdþ D ð/ n w l Þ¼ X D i ð/ n ÞD i ðw l Þ¼ X where P 0, n P 0, l P 0 and 0 6 k 6 are integers. D i ðw l ÞD i ð/ n Þ; Proof. (A) According to Taylor theore, the unique coefficient u of the Maclaurin series of / is given by u ¼ 1 o /! oq ; q¼0 which gives (a) by eans of the definition of D ð/þ. (B) It holds q k / ¼ q k Xþ1 u i q i ¼ Xþ1 u i q iþk ¼ Xþ1 u k q ; which gives by eans of (a) that D ðq k /Þ¼u k ¼ D k ð/þ: (C) According to Leibnitz s rule for derivatives of product, it holds o ð/wþ oq ¼ X ¼k! o i / o i w ¼ X i!ð iþ! oq i oq i which gives by the definition (10): 0 D ð/wþ ¼ 1 o ð/wþ! oq ¼ Xþ1 1 o i q¼0 i! oq i Siilarly, it holds D ð/wþ ¼ Xþ1 D i ðwþd i ð/þ:! o i w o i / i!ð iþ! oq i oq ; i q¼0 10 A 1 o i ð iþ! oq i q¼0 1 A ¼ Xþ1 D i ð/þd i ðwþ:

6 988 S. Liao / Coun Nonlinear Sci Nuer Siulat 14 (2009) (D) Write U ¼ / n and W ¼ w l. According to (c), it holds D ð/ n w l Þ¼D ðuwþ ¼ Xþ1 D i ðuþd i ðwþ ¼ Xþ1 D i ð/ n ÞD i ðw l Þ: Siilarly, it holds D ð/ n w l Þ¼ Xþ1 D i ðw l ÞD i ð/ n Þ; which ends the proof. h Theore 2.3. Let L be a linear operator independent of the hootopy-paraeter q. For hootopy-series u k q k ; it holds D ðl/þ ¼L½D ð/þš: Proof. Since L is independent of q, it holds L ½Lðu k ÞŠq k : Taking th-order hootopy-derivative on both sides of the above expression and using Theore 2.1(a), one has D ðl/þ ¼Lðu Þ. On the other side, according to Theore 2.1(a), it holds obviously L½D ð/þš ¼ Lðu Þ. Thus, D ðl/þ ¼L½D ð/þš holds. h Theore 2.4. For hootopy-series u k q k ; it holds the recurrence forulas D 0 e / ¼ e u 0 ; 1 X D e / ¼ D k e / D k ð/þ; where P 1 is integer. Proof. According to the definition (10) of the operator D, it holds obviously D 0 ðe / Þ¼e u 0 : Besides, one has oe / o/ ¼ e/ oq oq : Thus, according to Leibnitz s rule for derivatives of product, it holds 1 o e /! oq ¼ 1! o 1 e / o/ ¼ 1 X 1 oq 1 oq 1 o k e / k!ð Þ! o k / oq k oq k ¼ X 1 Setting q ¼ 0 in above expression and using the definition (10), one has D ðe / Þ¼ X 1 D k e / D k ð/þ; where P 1 is an integer. h Theore 2.5. For hootopy-series u k q k ; ð kþ " # 1 o k e / k! oq k " # 1 o k / : ð kþ! oq k

7 S. Liao / Coun Nonlinear Sci Nuer Siulat 14 (2009) it holds the recurrence forulas D 0 ðsin /Þ ¼sinðu 0 Þ; D 0 ðcos /Þ ¼cosðu 0 Þ; D ðsin /Þ ¼ X 1 D k ðcos /ÞD k ð/þ; D ðcos /Þ ¼ X 1 D k ðsin /ÞD k ð/þ; where P 1 is an integer. Proof. According to the definition (10), it holds obviously and D 0 ðsin /Þ ¼sinðu 0 Þ; D 0 ðcos /Þ ¼cosðu 0 Þ: p Write i ¼ ffiffiffiffiffiffiffi 1. Using Euler forula and Theore 2.1, it holds for an integer P 1 that e i/ e i/ D ðsin /Þ ¼D ¼ 1 2i 2i D ðe i/ Þ D ðe i/ Þ e i/ þ e i/ D ðcos /Þ ¼D 2 ¼ 1 2 D ðe i/ ÞþD ðe i/ Þ : ð14þ Using Theore 2.4 and then Theore 2.1, we have D ðe i/ Þ¼ X 1 D k ðe i/ ÞD k ði/þ ¼i X 1 D k ðe i/ ÞD k ð/þ and siilarly, D ðe i/ Þ¼ i X 1 D k ðe i/ ÞD k ð/þ: Substituting the above two expressions into (13) and (14), then using Theore 2.1 and Euler forula, we have D ðsin /Þ ¼ 1 X 1 1 X D k ð/þ D k ðe i/ ÞþD k ðe i/ Þ ¼ e D i/ þ e i/ k ð/þd k 2 2 ¼ X 1 D k ð/þd k ðcos /Þ; and siilarly D ðcos /Þ ¼ i 2 X 1 ¼ X 1 This ends the proof. Theore 2.6. If the two hootopy-series 1 X D k ð/þ D k ðe i/ Þ D k ðe i/ Þ ¼ e D i/ e i/ k ð/þd k 2i D k ð/þd k ðsin /Þ: h ð13þ u i q i ; w ¼ Xþ1 v j q j ; j¼0 satisfy / ¼ w in a doain q 2½0; aþ, then D ð/þ ¼D ðwþ and u ¼ v for any integer P 0 and a real nuber a > 0. Proof. Since / ¼ w, it holds X þ1 ðu k v k Þq k ¼ 0: The above expression holds for all points q 2½0; aþ, if and only if u ¼ v ; P 0; which gives, due to Theore 2.2 (a), that D ð/þ ¼D ðwþ:

8 990 S. Liao / Coun Nonlinear Sci Nuer Siulat 14 (2009) Reark 2.1. According to Theore 2.5, taking the th-order hootopy-derivative on the two sides of the equation / ¼ w gives the sae results as equating the like-power of q of the equation / ¼ w. Note that, here, it is unnecessary to regard q as sall paraeter at all. Fro the above theore, it is clear that the so-called hootopy perturbation ethod [67,68] (proposed in 1998) is exactly a copy of the early hootopy analysis ethod (proposed in 1992) and is a special case of the late hootopy analysis ethod in case of h ¼ 1. For details, please refer to Abbasbandy [44], Hayat & Sajid [46,69]. 3. Deforation equations In this section, the properties of hootopy-derivatives proved in Section 2 are eployed to deduce the high-order deforation equations for various types of zeroth-order deforation equations. Lea 3.1. Let u ð~x; tþq ¼0 denote a hootopy-series, where q 2½0; 1Š is the hootopy-paraeter, u is a function of the spatial variable ~x and the teporal variable t, respectively. Let L denote an auxiliary linear operator with respect to ~x and t, and u 0 an guess solution. It holds D fð1 qþl½/ u 0 Šg ¼ L½u ð~x; tþ v u 1 ð~x; tþš; where the operator D 1 is defined by (10) and v is defined by 0; 6 1; v ¼ 1; > 1: ð15þ Proof. Since L is a linear operator independent of q, it holds ð1 qþl½/ u 0 Š¼L½/ q/ þ u 0 q u 0 Š: Using Theores 2.1, 2.2 and 2.3, we have D fð1 qþl½/ u 0 Šg ¼ D fl½/ q/ þ u 0 q u 0 Šg ¼ LfD ½/ q/ þ u 0 q u 0 Šg ¼ L½D ð/þ D ðq/þþu 0 D ðqþš ¼ L½u u 1 þ u 0 D ðqþš; which equals to L½u Š when ¼ 1, and L½u u 1 Š when > 1, respectively. Thus, using the definition (15) of v, it holds D fð1 qþl½/ u 0 Šg ¼ L½u v u 1 Š: Theore 3.1. Write u ð~x; tþq ; ¼0 where q 2½0; 1Š is the hootopy-paraeter. Let L denote an auxiliary linear operator, N a nonlinear operator, u 0 ð~x; tþ a guess solution, h the convergence-control paraeter independent of q, and Hð~x; tþ an auxiliary function independent of q, respectively. For the zeroth-order deforation equation defined by ð1 qþl½/ u 0 Š¼qhHð~x; tþn½/š; the corresponding th-order deforation equation ( P 1) reads ð16þ L½u ð~x; tþ v u 1 ð~x; tþš ¼ hhð~x; tþd 1 ðn½/šþ; ð17þ where the operator D 1 is defined by (10) and v is defined by (15). Proof. Using Theore 2.6, we have D fð1 qþl½/ u 0 Šg ¼ D ðqhhð~x; tþn½/šþ: According to Lea 3.1, it holds D fð1 qþl½/ u 0 Šg ¼ L½u v u 1 Š: ð18þ ð19þ According to Theores 2.1 and 2.2, one has D ðqhhð~x; tþn½/šþ ¼ hhð~x; tþd 1 ðn½/šþ: ð20þ Substituting (19) and (20) into (18), one has the th-order deforation equation L½u v u 1 Š¼hHð~x; tþd 1 ðn½/šþ:

9 S. Liao / Coun Nonlinear Sci Nuer Siulat 14 (2009) Theore 3.2. Write u ð~x; tþq ; ¼0 w ¼ Xþ1 a k q k ; where q 2½0; 1Š is the hootopy-paraeter and a k is a constant. Let L denote an auxiliary linear operator, N a nonlinear operator, u 0 ð~x; tþ a guess solution, and Hð~x; tþ an auxiliary function independent of q, respectively. For the zeroth-order deforation equation defined by ð1 qþl½/ u 0 Š¼Hð~x; tþ X þ1 a k q k!n½/š; ð21þ the corresponding th-order deforation equation ( P 1) reads L½u ð~x; tþ v u 1 ð~x; tþš ¼ Hð~x; tþ X a k D k ðn½/šþ; ð22þ where the operator D k is defined by (10) and v is defined by (15). Proof. Using Theore 2.6, we have D fð1 qþl½/ u 0 Šg ¼ D ðhð~x; tþwn½/šþ: ð23þ According to Lea 3.1, it holds D fð1 qþl½/ u 0 Šg ¼ L½u v u 1 Š: ð24þ Using Theores 2.1 and 2.2, we have D ðhð~x; tþwn½/šþ ¼ Hð~x; tþ X D k ðwþd k ðn½/šþ ¼ Hð~x; tþ X a k D k ðn½/šþ; which gives, since a 0 ¼ 0, D ðhð~x; tþwn½/šþ ¼ Hð~x; tþ X a k D k ðn½/šþ: ð25þ Substituting (24) and (25) into (23) ends the proof. h Reark 3.1. The zeroth-order deforation equation (16) is a special case of the zeroth-order deforation equation (21) in case of a 1 ¼ h and a k ¼ 0 for k > 1. Theore 3.3. Write u ð~x; tþq ; ¼0 w ¼ Xþ1 b k ð~x; tþq k ; where q 2½0; 1Š is the hootopy-paraeter, b k ð~x; tþ is either equal to zero or a non-zero function, but at least one of the is nonzero. Let L denote an auxiliary linear operator, N a nonlinear operator, and u 0 ð~x; tþ a guess solution, respectively. For the zerothorder deforation equation defined by ð1 qþl½/ u 0 Š¼ Xþ1 b k ð~x; tþq k!n½/š; ð26þ the corresponding th-order deforation equation ( P 1) reads L½u ð~x; tþ v u 1 ð~x; tþš ¼ X b k ð~x; tþd k ðn½/šþ; ð27þ where the operator D k is defined by (10) and v is defined by (15). Proof. Using Theore 2.6, we have D fð1 qþl½/ u 0 Šg ¼ D ðwn½/šþ: ð28þ According to Lea 3.1, it holds D fð1 qþl½/ u 0 Šg ¼ L½u v u 1 Š: ð29þ

10 992 S. Liao / Coun Nonlinear Sci Nuer Siulat 14 (2009) Using Theores 2.1 and 2.2, we have D ðwn½/šþ ¼ X D k ðwþd k ðn½/šþ ¼ X b k ð~x; tþd k ðn½/šþ; which gives, since b 0 ð~x; tþ ¼0, that D ðwn½/šþ ¼ X b k ð~x; tþd k ðn½/šþ: ð30þ Substituting (29) and (30) into (28) ends the proof. h Reark 3.2. The zeroth-order deforation equation (21) is a special case of the zeroth-order deforation equation (26) in case of b k ð~x; tþ ¼a k Hð~x; tþ for k P 1. Theore 3.4. Write u ð~x; tþq ; ¼0 w ¼ Xþ1 b k ð~x; tþq k ; where q 2½0; 1Š is the hootopy-paraeter, b k ð~x; tþ is either equal to zero or a non-zero function, but at least one of the is nonzero. Let L denote an auxiliary linear operator, N a nonlinear operator, and u 0 ð~x; tþ a guess solution, respectively. Besides, let A½/;~x; t; qš be a function of /;~x; t and q, which satisfies A½/;~x; t; qš ¼0; when q ¼ 0 and q ¼ 1: For the zeroth-order deforation equation defined by ð1 qþl½/ u 0 Š¼ Xþ1 b k ð~x; tþq k!n½/šþa½/;~x; t; qš; ð31þ the corresponding th-order deforation equation ( P 1) reads L½u ð~x; tþ v u 1 ð~x; tþš ¼ X b k ð~x; tþd k ðn½/šþ þ D ða½/;~x; t; qšþ; where the operator D k is defined by (10) and v is defined by (15). Proof. Using Theores 2.6 and 2.1, we have D fð1 qþl½/ u 0 Šg ¼ D ðwn½/šþþ D ða½/;~x; t; qšþ: ð33þ According to Lea 3.1, it holds ð32þ D fð1 qþl½/ u 0 Šg ¼ L½u v u 1 Š: ð34þ Using Theores 2.1 and 2.2, we have D ðwn½/šþ ¼ X D k ðwþd k ðn½/šþ ¼ X which gives, since b 0 ð~x; tþ ¼0, that D ðwn½/šþ ¼ X b k ð~x; tþd k ðn½/šþ; b k ð~x; tþd k ðn½/šþ: ð35þ Substituting (39) and (35) into (38) ends the proof. h Reark 3.3. The zeroth-order deforation equation (26) is a special case of the zeroth-order deforation equation (31) in case of A½/;~x; t; qš ¼0. Theore 3.5. Write u ð~x; tþq ; ¼0 where q 2½0; 1Š is the hootopy-paraeter. Let L denote an auxiliary linear operator, N a nonlinear operator, and u 0 ð~x; tþ a guess solution, respectively. Besides, let B½/;~x; t; qš be a function of /;~x; t and q, which satisfies B½/;~x; t; qš ¼0; when q ¼ 0; B½/;~x; t; qš ¼cð~x; tþn½/š; when q ¼ 1;

11 S. Liao / Coun Nonlinear Sci Nuer Siulat 14 (2009) where cð~x; tþ 0 is a non-zero function. For the zeroth-order deforation equation defined by ð1 qþl½/ u 0 Š¼B½/;~x; t; qš; ð36þ the corresponding th-order deforation equation ( P 1) reads L½u ð~x; tþ v u 1 ð~x; tþš ¼ D ðb½/;~x; t; qšþ; ð37þ where the operator D is defined by (10) and v is defined by (15). Proof. Using Theore 2.6, we have D fð1 qþl½/ u 0 Šg ¼ D ðb½/;~x; t; qšþ: ð38þ According to Lea 3.1, it holds D fð1 qþl½/ u 0 Šg ¼ L½u v u 1 Š: ð39þ This ends the proof. h Reark 3.4. Obviously, the zeroth-order deforation equation (16), (21), (26) and (31) are special cases of the zeroth-order deforation equation (36). However, up to now, it is unknown which kind of zeroth-order deforation equation is better or best, as discussed in Liao s book [21]. In ost cases, the zeroth-order deforation equation (16) can give satisfied hootopyseries solution, if the auxiliary linear operator L, the auxiliary function Hð~x; tþ and convergence-control paraeter h are properly chosen, as pointed out by Liao [21 23,25,60,26] and others [28 59]. In ost cases, in order to get the high-order deforation equation related to a nonlinear equation N½uŠ ¼0, one just needs to calculate the ter D k ðn½/šþ. This can be easily done by eans of the properties of the hootopy-derivatives given in Section Exaples In this section, soe siple exaples are used to show how to apply the theores given in this article to deduce highorder deforation equations of nonlinear probles. Exaple 1. Consider the nonlinear heat transfer proble [44]: ð1 þ uþu 0 þ u ¼ 0; uð0þ ¼1: Choosing Lu ¼ u 0 þ u as the auxiliary linear operator, and defining the nonlinear operator N½/Š ¼ð1 þ /Þ/ 0 þ /; where u k ðtþq k is a hootopy-series, we construct such a zeroth-order deforation equation ð1 qþl½/ u 0 ðtþš ¼ qhn½/š; subject to the initial condition / ¼ 1; when t ¼ 0; where u 0 ðtþ is an initial guess satisfying the initial condition. According to Theore 3.1, the corresponding th-order deforation equation reads L½u ðtþ v u 1 ðtþš ¼ hd 1 ðn½/šþ; subject to the initial guess u ð0þ ¼0: For this exaple, using Theores 2.1, 2.2 and 2.3, one has D k ðn½/šþ ¼ D k ð/ 0 ÞþD k ð/þþd k ð// 0 Þ¼u 0 k þ u k þ Xk The corresponding hootopy-series solution is given by uðtþ ¼ Xþ1 u k ðtþ; n¼0 u k n u 0 n :

12 994 S. Liao / Coun Nonlinear Sci Nuer Siulat 14 (2009) which is convergent for any physical paraeter 0 6 <þ1 if one chooses the convergence-control paraeter h ¼ ð1 þ Þ 1. For details, please refer to Abbasbandy [44]. Exaple 2. Consider the nonlinear oscillation equation [26]: u 00 ðtþþkuðtþþu 3 ðtþ ¼0; uð0þ ¼1; u 0 ð0þ ¼0: Let x denote the unknown frequency of the solution. Write s ¼ xt, the above equation becoes x 2 u 00 ðsþþkuðsþþu 3 ðsþ ¼0; uð0þ ¼1; u 0 ð0þ ¼0: Define N½/; XŠ ¼X 2 / 00 þ k/ þ / 3 ; where the prie denotes the differentiation with respect to s, and u k ðtþq k ; X ¼ Xþ1 x k q k are two hootopy-series. Choosing the auxiliary linear operator Lu ¼ u 00 þ u; we construct the following zeroth-order deforation equation ð1 qþl½/ u 0 Š¼qhN½/; XŠ; subject to the initial conditions / ¼ 1; / 0 ¼ 0; at t ¼ 0; q 2½0; 1Š where u 0 ðtþ is an initial guess satisfying the initial conditions. According to Theore 3.1, the corresponding high-order deforation equation reads L½u ðsþ v u 1 ðsþš ¼ hd 1 ðn½/; XŠÞ; subject to the initial conditions u ð0þ ¼0; u 0 ð0þ ¼0: In this case, using Theores , and Molabahrai and Khani s Theore, we have D k ðn½/; XŠÞ ¼ D k ðx 2 / 00 ÞþkD k ð/þþd k ð/ 3 Þ¼ X k X i u00 k i j¼0 The corresponding hootopy-series solutions are given by uðtþ ¼ Xþ1 u k ðxtþ; x ¼ Xþ1 x k ; x i j x j þ ku k þ Xk which are convergent if the convergence-control paraeter h is chosen properly. For exaple, when k ¼ 0, the hootopyseries solutions are convergent for any a physical paraeter 0 6 <þ1 by using h ¼ ð1 þ Þ 1. For details, please refer to Liao and Tan [26]. u k i X i j¼0 u i j u j : 5. Discussions The introduction of the convergence-control paraeter h greatly iproves the early hootopy analysis ethod. It is the convergence-control paraeter h that provides us, for the first tie, a siple way to ensure the convergence of series solution of nonlinear probles. Different fro all previous analytic ethods, one can ensure the convergence of series solution of strongly nonlinear probles by eans of choosing a proper value of the convergence-control paraeter h. This is an obvious advantage of the HAM. Besides, unlike all perturbation and previous non-perturbation ethods, the HAM provides us with great freedo to choose proper base functions so as to give better approxiations of nonlinear probles. Note that (16) can be rewritten as ð1 qþ ^L½/ u 0 Š¼qN½/Š; ð40þ where ^L ¼ L hhð~x; tþ

13 S. Liao / Coun Nonlinear Sci Nuer Siulat 14 (2009) can be regarded as a auxiliary linear operator, too. The above expression opens out that, in the frae of the late HAM, the auxiliary linear operator is chosen in several steps: a basic linear operator is first chosen, then an auxiliary function is deterined due to the rule of solution expression suggested by Liao [21], and finally the convergence-control paraeter h is deterined to ensure the convergence of the hootopy-series solution. In fact, the zeroth-order deforation equations (21), (26), (31) and (36) are obtained by generalizing the concept of convergence-control paraeter h. Define the vector ~a ¼ fa 1 ; a 2 ; a 3 ;... g: Obviously, the vector ~a in the zeroth-order deforation equation (21) is a kind of generalization of the convergence-control paraeter h in Eq. (16), and thus is called the convergence-control vector. Siilarly, the vector ~ b ¼fb1 ð~x; tþ; b 2 ð~x; tþ; b 3 ð~x; tþ;...g in Eqs. (26) and (31) is a further generalization of the so-called convergence-control vector ~a in Eq. (21). Siilarly, by properly choosing the convergence-control vectors ~a or ~ b, one can ensure the convergence of the hootopy-series solutions of Eqs. (26), (31), and (36). For the zeroth-order deforation equation (16), Liao suggested to choose a proper value of h by plotting the so-called h-curves. Let dð~x; tþ denote the residual error of the th-order hootopy-series approxiation, and D ¼ RR d 2 ð~x; tþdv dt denote the integral of the residual error. Plotting the curves of D h, it is straightforward to find a region of h in which D decreases to zero as the order of approxiation increases. Then, a convergent hootopy-series solution is obtained by choosing a value in this region. For the zeroth-order deforation equation (21), Marinca [65] currently proposed an interesting approach to deterine the convergence-control vector ~a by iniizing the residual error. Obviously, soe rigorous atheatical theores are urgently needed to find the best convergence-control paraeter h for (16), the best convergence-control vector ~a for (21) and the best convergence-control vector ~ b for (26) and (31), respectively. Obviously, the various types of the zeroth-order deforation equations such as (16), (21), (26), (31) and (36) provide us great freedo and flexibility to apply the HAM. Besides, for each type of these zeroth-order deforation equations, one has great freedo and flexibility to choose the auxiliary linear operator L: even the order of L can be different fro original nonlinear probles, as shown by Liao and Tan [26], who illustrated that a 2nd-order nonlinear PDE can be replaced by an infinite nuber of 4th or 6th-order linear PDEs. Such kind of freedo and flexibility greatly siplifies the resolving of coplicated nonlinear equations. By eans of the HAM, a nonlinear ODE is often replaced by an infinite nuber of linear ODEs, and a nonlinear PDE can be transferred into an infinite nuber of linear ODEs. Besides, a nonlinear differential equation with variable coefficients can be replaced by an infinite nuber of linear differential equations with constant coefficients. Certainly, this kind of freedo and flexibility increases the possibility of finding satisfactory series solution of a given nonlinear proble. However, it also enhances the difficulties in applying and learning it. Up to now, it is even unknown whether or not there exists the best auxiliary linear operator and the best zeroth-order deforation equation for a nonlinear equation which has at least one solution. To siplify the applications of the HAM, Liao [21] suggested soe rules, i.e. the rule of solution expression, the rule of solution existence, and the rule of ergodicity for coefficient of hootopy-series solution. Obviously, soe rigorous atheatical theores are urgently needed to choose the auxiliary linear operators. The freedo and flexibility in the choose of the auxiliary linear operator L in the zeroth-order deforation equation can be used to develop soe new nuerical techniques for strongly nonlinear probles. For exaple, the so-called general boundary eleent ethod [70 73], which is based on the HAM, gives accurate convergent results of the viscous driven flows (governed by the exact Navier Stokes equations) in a square cavity with the high Reynolds nuber Re ¼ 7500, as shown by Zhao and Liao [74]. Currently, Wu and Cheung [49] applied the HAM to give an explicit nuerical approach for Rieann probles related to nonlinear shallow water equations. All of these illustrate the great potential of the HAM cobined with traditional nuerical techniques. The zeroth-order deforation equations (21), (26), (31) and (36) are rather general. Using the, Liao [21] proved that the HAM logically contains other previous non-perturbation ethods, such as Lyapunov s artificial sall paraeter ethod [13], the d-expansion ethod [14,15] and Adoian s decoposition ethod [16 19]. Thus, the HAM unifies the previous non-perturbation ethods. Besides, the so-called hootopy perturbation ethod [67,68] (proposed in 1998) is exactly the sae as the early hootopy analysis ethod (proposed in 1992) and is a special case of the late hootopy analysis ethod in case of h ¼ 1, as illustrated by Abbasbandy [44] and proved by Sajid et al. [46,69] in general. Indeed, Dr. He [67,68] siply copied Liao s idea of the early HAM, and his so-called hootopy perturbation ethod proposed in 1998 (6 year later) has nothing new except its nae, as pointed out by Hayat and Sajid [46,69]. Frankly speaking, the HAM is a ethod for the tie of coputer: without high-perforance coputer and sybolic coputation software such as Matheatica, aple and so on, it is ipossible to solve high-order deforation equations quickly so as to get approxiations at high enough order. Without coputer and sybolic coputation software, it is also ipossible to choose a proper value of the convergence-control paraeter h by eans of analyzing the high-order approxiations. It is true that expressions given by the HAM are often lengthy and thus can be hardly expressed on only one page. However, by eans of coputer and sybolic coputation software, it often needs only a few seconds to calculate these lengthy results! Note that, one needs uch ore tie to calculate a traditional analytic expression in a length of half page by eans of a traditional coputational tool such as a slide rule. So, if we regard keyboard of coputer as a pen, hard disk as papers, and CPU as a slide rule, we can calculate lengthy analytic expressions given by the HAM in a few seconds by eans of a coputer! So, it sees that the traditional concept analytic solution, which was fored hundreds year ago in the tie of slide rule, should be odified in the tie of coputer that has changed our life copletely.

14 996 S. Liao / Coun Nonlinear Sci Nuer Siulat 14 (2009) Finally, according to our experience, it sees that, as long as a nonlinear equation has at least one solution, then one can always construct a kind of zeroth-order deforation equation to get convergent hootopy-series solution. However, hundreds of successful exaples is not better than a rigorous atheatical proof. So, to end this article, we give here such a hypothesis: Hypothesis 1. If a nonlinear equation has at least one solution, then there exists at least one zeroth-order deforation equation such that its hootopy-series solution converges to the solution of the original nonlinear equation. Acknowledgeents Thanks to Prof. Pradeep Siddheshwar, Departent of Matheatics, Bangalore University, India, for his valuable suggestion of renaing h as the convergence-control paraeter. This work is supported by National Natural Science Foundation of China (Approve No ). References [1] Krylov N, Bogoliubov NN. 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