Computers and Mathematics with Applications

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1 Computers and Mathematics with Applications 1 (211) Contents lists available at ScienceDirect Computers and Mathematics with Applications journal homepage: Variational iteration method and homotopy perturbation method for fourth-order fractional integro-differential equations Yasir Nawaz Donghua University, 1882 West Yan an Road, Shanghai 251, China a r t i c l e i n f o a b s t r a c t Keywords: Variational iteration method Homotopy perturbation method Boundary value problems Integro-differential equations Caputo fractional derivative In this article, linear and nonlinear boundary value problems for fourth-order fractional integro-differential equations are solved by variational iteration method and homotopy perturbation method. The fractional derivatives are described in the Caputo sense. The solutions of both problems are derived by infinite convergent series which are easily computable and then graphical representation shows that both methods are most effective and convenient one to solve linear and nonlinear boundary value problems for fourth-order fractional integro-differential equations. 21 Elsevier Ltd. All rights reserved. 1. Introduction In recent years various analytical and numerical methods have been applied for the approimate solutions of fractional differential equations (FDEs). Since eact solutions of most of the fractional differential equations do not eist, approimation and numerical methods are used for the solutions of the FDEs. He [1 4] was the first to propose the variational iteration method (VIM) and homotopy perturbation method (HPM) for finding the solutions of linear and nonlinear problems. VIM is based on Lagrange multiplier and HPM is a coupling of the traditional perturbation method and homotopy in topology. These methods have been successfully applied by many authors [3,5 1] for finding the analytical approimate solutions as well as numerical approimate solutions of functional equations which arise in scientific and engineering problems. The main feature for the use of VIM and HPM is that they can overcome the difficulties which arise in the Adomian decomposition method during computations of Adomian polynomials; see [8]. Many physical phenomena [11 15] can be modeled by fractional differential equations which have diverse applications in various physical processes such as acoustics, electromagnetism, control theory, robotics, viscoelastic materials, diffusion, edge detection, turbulence, signal processing, anomalous diffusion and fractured media. Momani and Aslam Noor [1] established the implementation of ADM to derive analytic approimate solutions of the linear and nonlinear boundary value problems for fourth-order fractional integro-differential equations. The purpose of this article is to etend the analysis of VIM and HPM to construct the approimate solutions of the following linear and nonlinear boundary value problems for fourth-order fractional integro-differential equations D α y() = f () + γ y() + subject to the following boundary conditions: y() = γ, y () = γ 2, y(b) = β, y (b) = β 2, [g(t)y(t) + h(t)f(y(t))]dt < < b, 3 < α 4 (1) (2) (3) address: yasir_maths@yahoo.com /$ see front matter 21 Elsevier Ltd. All rights reserved. doi:1.11/j.camwa

2 Y. Nawaz / Computers and Mathematics with Applications 1 (211) where D α is the fractional derivative in the Caputo sense and F(y()) is any nonlinear function, γ, γ, γ 2, β and β 2 are real constants and f, g and h are given and can be approimated by Taylor polynomials. 2. Basic definitions In this section, we give some basic definitions and properties of fractional calculus theory which are further used in this article. Definition 1. A real function f (), > is said to be in space C µ, µ R if there eists a real number p > µ, such that f (t) = t p f 1 (t), where f 1 (t) C(, ), and it is said to be in the space C n µ if and only if f n C µ, n N. Definition 2. The Riemann Liouville fractional integral operator of order α >, of a function f C µ, µ 1, is defined as J α f (t) = 1 Γ (α) J f (t) = f (t). t (t s) α 1 f (s)ds, α > (4) Some properties of the operator J α, which are needed here, are as follows: For f C µ, µ 1, α, β and γ 1: (1) J α J β f (t) = J α+β f (t) (2) J α J β f (t) = J β J α f (t) (3) J α t γ = Γ (γ +1) Γ (α+γ +1) tα+γ. Definition 3. The fractional derivative of f (t) in the Caputo sense is defined as D α 1 f (t) = Γ (m α) t for m 1 < α m, m N, t >, f C m 1. (t s) m α 1 f (m) (s)ds, (5) Lemma 1. If m 1 < α m, m N, f Cµ m, µ 1, then the following two properties hold: (1) D α J α f (t) = f (t) (2) m 1 (J α D α )f (t) = f (t) f (k) ( + ) tk k!. k= () 3. Analysis of VIM To illustrate the basic concepts of variational iteration method, consider the fractional differential equation (1) with boundary conditions (2) (3). According to the variational iteration method, we can construct the correction functional for Eq. (1) as: ] y k+1 () = y k () + J [λ β D α y k () f () γ ỹ k () [g(p)ỹ k (p) + h(p)f(ỹ k (p))]dp = y k () + 1 Γ (β) ( s) β 1 λ(s) D α y k (s) f (s) γ ỹ k (s) s [g(p)ỹ k (p) + h(p)f(ỹ k (p))]dp ds (7) where J β is the Riemann Liouville fractional integral operator of order β = α + 1 m, λ is a general Lagrange multiplier and ỹ k denotes restricted variation i.e. δỹ k =. We make some approimation for the identification of an approimate Lagrange multiplier, so the correctional functional (7) can be approimately epressed as: s y k+1 () = y k () + λ(s) D 4 y k (s) f (s) γ ỹ k (s) [g(p)ỹ k (p) + h(p)f(ỹ k (p))]dp ds. (8) Making the above correction functional stationary, we obtain the following stationary conditions: 1 λ (s) s= =, λ (s) s= =, λ (s) s= =, λ(s) s= =, λ (iv) (s) =. This gives the following Lagrange multiplier λ(s) = 1 (s )3. (9)

3 2332 Y. Nawaz / Computers and Mathematics with Applications 1 (211) We obtain the following iteration formula by substitution of (9) into functional (7), 1 y k+1 () = y k () + Γ (α 3) s [g(p)y k (p) + h(p)f(y k (p))]dp ( s) α 4 (s ) 3 D α y k (s) f (s) γ y k (s) = y k () ( s) D α 1 α y k (s) f (s) γ y k (s) Γ (α) s [g(p)y k (p) + h(p)f(y k (p))]dp ds. This yields the following iteration formula: y k+1 () = y k () J D α α y k () f () γ y k () The initial approimation y can be chosen by the following way which satisfies initial conditions (2): ds [g(p)y k (p) + h(p)f(y k (p))]dp. (1) y () = γ + γ 1 + γ γ 3 3 (11) where γ 1 = y () and γ 3 = y () are to be determined by applying suitable boundary conditions (3). We can obtain the following first-order approimation by substitution of (11) into (1): y 1 () = y () J D α α y () f () γ y () [g(p)y (p) + h(p)f(y (p))]dp. (12) Similarly, we can obtain the higher-order approimations. If Nth-order approimate is enough, then imposing boundary conditions (3) in Nth-order approimation yields the following system of equations: y N (b) = β, y N (b) = β 2. From Eqs. (13) (14), we can find the unknowns γ 1 = y () and γ 3 = y (). Substituting the constant values of γ 1 and γ 3 in Nth-order approimation results the approimate solution of (1) (3). (13) (14) 4. Analysis of HPM To illustrate the basic concepts of HPM for fractional integro-differential equations, consider the fractional differential equation (1) with boundary conditions (2) (3). In view of HPM [3,4], construct the following homotopy for Eq. (1): (1 p)d α y() + p D α y() f () γ y() [g(t)y(t) + h(t)f(y(t))]dt = (15) or D α y() = p f () + γ y() + [g(t)y(t) + h(t)f(y(t))]dt where p [, 1] is an embedding parameter. If p =, then Eq. (1) becomes a linear equation, D α y() =, and when p = 1, then Eq. (1) turns out to be the original Eq. (1). In view of basic assumption of homotopy perturbation method, solution of Eq. (1) can be epressed as a power series in p: y() = y () + py 1 () + p 2 y 2 () + p 3 y 3 () +. (18) Setting p = 1 in (18) results the approimate solution of Eq. (1): y() = y () + y 1 () + y 2 () + y 3 () +. (19) (1) (17)

4 Y. Nawaz / Computers and Mathematics with Applications 1 (211) The convergence of series (19) has been proved in [17]. Substitution (18) into (1), then equating the terms with identical power of p, we obtain the following series of linear equations: p : D α y =, p 1 : D α y 1 = f () + γ y () + p 2 : D α y 2 = γ y 1 () + p 3 : D α y 3 = γ y 2 () +. where the functions F 1, F 2,... satisfy the following condition: [g(t)y (t) + h(t)f 1 (y (t))]dt, (21) [g(t)y 1 (t) + h(t)f 2 (y 1 (t))]dt, (22) [g(t)y 2 (t) + h(t)f 3 (y 2 (t))]dt, (23) F(y (t) + py 1 (t) + p 2 y 2 (t) + ) = F 1 (y (t)) + pf 2 (y 1 (t)) + p 2 F 3 (y 2 (t)) +. From Eq. (2), the initial approimation can be chosen in the following way: 3 j y = γ j j! j= where γ 1 = y () and γ 3 = y () are to be determined by applying suitable boundary conditions (3). Eqs. (2) (23) can be solved by applying the operator J α, which is the inverse of the operator D α and then by simple computation, we approimate the series solution of HPM by the following N-term truncated series: θ N () = y () + y 1 () + y 2 () + + y N 1 (). (24) Note that in epression (24), constants γ 1 and γ 3 are undetermined. By imposing boundary conditions (3) in (24), we get the following system of equations y (b) + y 1 (b) + y 2 (b) + + y N 1 (b) = β, (25) (b) + y 1 (b) + y (b) y (b) = β N 1 2. (2) From Eqs. (25) (2), we can find the unknowns γ 1 and γ 3. Substituting the constant values of γ 1 and γ 3 in epression (24) results the approimate solution of (1) (3). y 5. Applications In this section we have applied variational iteration method and homotopy perturbation method to fourth-order linear and nonlinear fractional integro-differential equations with a known eact solution at α = 4. Eample 1. Consider the following linear fourth-order fractional integro-differential equation: D α y() = (1 + e ) + 3e + y() subject to the following boundary conditions: y() = 1, y () = 2, y(1) = 1 + e, y (1) = 3e. For α = 4, the eact solution of problem (27) (29) is y() = 1 + e. (2) y(t)dt < < 1, 3 < α 4 (27) According to variational iteration method, the iteration formula (1) for Eq. (27) can be epressed in the following form: y k+1 () = y k () J D α α y k () (1 + e ) 3e y k () + y k (t)dt. (3) In order to avoid difficult fractional integration, we can take the truncated Taylor epansion for the eponential term in (3): e.g., e /2 + 3 / and assume that an initial approimation has the following form which satisfies the initial conditions (28): y () = 1 + A B 3 (31) where A = y () and B = y () are unknowns to be determined. (28) (29)

5 2334 Y. Nawaz / Computers and Mathematics with Applications 1 (211) Table 1 Values of A and B for different values of α using (32). α = 3.25 α = 3.5 α = 3.75 α = 4 A B E1 E3 E2.7 Absolute error functions X Fig. 1. Absolute error functions E 1 (), E 2 () and E 3 () obtained by VIM with different values of α. X Fig. 2. Comparison of the first-order approimate solution obtained by VIM with the eact solution at α = 4 and α = 3.4. Now, by iteration formula (3), first-order approimation takes the following form: y 1 () = y () J α D α y () y () + y (t)dt = 1 + A B α (4 + A) (A 7) (4 + B) (B 4) + +. (32) Γ (α + 1) Γ (α + 2) Γ (α + 3) Γ (α + 4) Γ (α + 5) By imposing boundary conditions (29) in y 1 (), we obtain Table 1 which shows the values of A and B for different values of α. In Fig. 1, we draw absolute error functions E 1 () = (1 + e ) y 1,3.75, E 2 () = (1 + e ) y 1,3.5 and E 3 () = (1 + e ) y 1,3.25 for different values of α, where 1 + e is an eact solution of (27) (29) and y 1,3.75, y 1,3.5 and y 1,3.25 represent the values of y 1 () at α = 3.75, α = 3.5 and α = 3.25, respectively (see Fig. 2).

6 Y. Nawaz / Computers and Mathematics with Applications 1 (211) Table 2 Values of A and B for different values of α using (43). α = 3.25 α = 3.5 α = 3.75 α = 4 A B According to HPM, we construct the following homotopy: D α y() = p (1 + e ) + 3e + y() y(t)dt. (33) Substitution of (18) into (33) and then equating the terms with same powers of p yield the following series of linear equations: p : D α y =, p 1 : D α y 1 = (1 + e ) + 3e + y () p 2 : D α y 2 = y 1 () p 3 : D α y 3 = y 2 (). y 1 (t)dt, y 2 (t)dt, (34) y (t)dt, (35) Applying the operator J α to the above series of linear equations and using initial conditions (28), we get; y () = 1, y 1 () = A B3 + J α (1 + e ) + 3e + y () y n () = J α y n 1 () y n 1 (t)dt (3) (37) (38) y (t)dt, (39), n = 2, 3, 4,... (4) where A = y () and B = y () are to be determined. In order to avoid difficult fractional integration, we can take the truncated Taylor epansions for the eponential term in (39) (4): e.g., e /2 + 3 /. Thus, by solving Eqs. (38) (4), we obtain y 1, y 2,... e.g.: y 1 () = A B3 α + 4 Γ (α + 1) + y 2 () = Aα+1 α+2 4 α+1 Γ (α + 2) + 5 α+2 Γ (α + 3) + α+3 + (2 A) + (B 2) Γ (α + 2) Γ (α + 3) Γ (α + 4) + 2α+3 Γ (2α + 4) 2 2α+4 Γ (2α + 5) Now, we can form the 2-term approimation α+3 Γ (α + 4) + B α+4 Γ (α + 5) + 4 α+4 Γ (α + 5) 2α 4 Γ (2α + 1) + 2α+2 Γ (2α + 3) (41) 4 2α+5 Γ (2α + ). (42) φ 2 () = 1 + A B3 α α+1 α+2 α (4 + A) + (7 A) + (4 + B) Γ (α + 1) Γ (α + 2) Γ (α + 3) Γ (α + 4) α+4 + (4 B) Γ (α + 5) + 2α 4 Γ (2α + 1) + 2α+2 Γ (2α + 3) + 2α+3 Γ (2α + 4) 2 2α+4 Γ (2α + 5) 4 2α+5 Γ (2α + ) where A and B can be determined by imposing boundary conditions (29) on φ 2. Table 2 shows the values of A and B for different values of α. In Fig. 3, we draw absolute error functions E 4 () = (1 + e ) φ 2,3.75, E 5 () = (1 + e ) φ 2,3.5 and E () = (1 + e ) φ 2,3.25 for different values of α, where 1 + e is an eact solution of (27) (29) and φ 2,3.75, φ 2,3.5 and φ 2,3.25 represent the values of φ 2 at α = 3.75, α = 3.5 and α = 3.25, respectively. In Figs. 4a 4c, we compare the approimate solutions obtained by VIM and HPM with an eact solution, and it is clear from Figs. 4a 4c that the approimate solutions are in good agreement with an eact solution of (27) (29) at α = 4, α = 3.8 and α = 3.2. Also it is to be noted that the accuracy can be improved by computing more terms of approimated solutions and/or by taking more terms in the Taylor epansion for the eponential term. (43)

7 233 Y. Nawaz / Computers and Mathematics with Applications 1 (211) Fig. 3. Absolute error functions E 4 (), E 5 () and E () obtained by 2-term HPM with α = 3.25, α = 3.5 and α = Fig. 4a. Comparison of approimate solutions obtained by 2-term HPM and first-order VIM with eact solution at α = 4. Fig. 4b. Comparison of approimate solutions obtained by 2-term HPM and first-order VIM with eact solution at α = 3.8.

8 Y. Nawaz / Computers and Mathematics with Applications 1 (211) Fig. 4c. Comparison of approimate solutions obtained by 2-term HPM and first-order VIM with eact solution at α = 3.2. Table 3 Values of A and B for different values of α using (49). α = 3.25 α = 3.5 α = 3.75 α = 4 A B Eample 2. Consider the following nonlinear fourth-order fractional integro-differential equation: D α y() = 1 + subject to the following boundary conditions: y() = 1, y () = 1, y(1) = e, y (1) = e. e t y 2 (t)dt, < < 1, 3 < α 4 (44) For α = 4, the eact solution of the above problem (44) (4) is y() = e. According to the variational iteration method, iteration formula (1) for Eq. (44) can be epressed in the following form: y k+1 () = y k () J D α α y k () 1 e t y 2 k (t)dt. (47) In order to avoid difficult fractional integration, we can take the truncated Taylor epansion for the eponential term in (47): e.g., e /2 3 / and assume that an initial approimation has the following form which satisfies the initial conditions (45): y () = 1 + A B 3 (48) where A = y () and B = y () are unknowns to be determined. Now, by iteration formula (47), the first-order approimation takes the following form: y 1 () = y () J D α α y k () 1 (1 t + t 2 /2 t 3 /)y 2 k (t)dt = 1 + A B α 3 1 Γ (α + 1) + 1 (1 2A) + Γ (α + 2) Γ (α + 3) B 2 9. (49) Γ (α + 11) By imposing boundary conditions (4) in y 1 (), we obtain Table 3 which shows the values of A and B for different values of α. (45) (4)

9 2338 Y. Nawaz / Computers and Mathematics with Applications 1 (211) Fig. 5. Absolute error functions E 7 (), E 8 () and E 9 () obtained by first-order VIM with α = 3.25, α = 3.5 and α = Fig.. Comparison of first-order approimate solution obtained by VIM with the eact solution at α = 4 and α = 3.4. In Fig. 5, we draw absolute error functions E 7 () = e y 1,3.75, E 8 () = e y 1,3.5 and E 9 () = e y 1,3.25 for different values of α, where e is an eact solution of (44) (4) and y 1,3.75, y 1,3.5 and y 1,3.25 represent the values of y 1 () at α = 3.75, α = 3.5 and α = 3.25, respectively (see Fig. ). Now, we solve Eqs. (44) (4) by homotopy perturbation method. According to HPM, we construct the following homotopy: D α y() = p 1 + e t y 2 (t)dt. (5) Substitution of (18) into (5) and then equating the terms with same powers of p yield the following series of linear equations: p : D α y =, p 1 : D α y 1 = 1 + p 2 : D α y 2 = 2 p 3 : D α y 3 =. e t y 2 (t)dt, e t y (t)y 1 (t)dt, e t (2y (t)y 2 (t) + y 2 1 (t))dt, (54) (51) (52) (53)

10 Y. Nawaz / Computers and Mathematics with Applications 1 (211) Table 4 Values of A and B for different values of α using (1). α = 3.25 α = 3.5 α = 3.75 α = 4 A B Fig. 7. Absolute error functions E 1 (), E 11 () and E 12 () obtained by 2-term HPM with α = 3.25, α = 3.5 and α = Applying the operator J α to the above series of linear equations and using initial conditions (45), we get; y () = 1, (55) y 1 () = A B3 + J α 1 + e t y 2 (t)dt, (5) y 2n () = J α e t (2y (t)y 2n (t) + 2y 1 (t)y 2n 1 (t) + + 2y n 1 (t)y 2n (n 1) (t) + y 2 n (t))dt, n = 1, 2, 3, 4,... (57) y 2n+1 () = J α 2e t (y (t)y 2n+1 (t) + y 1 (t)y (2n+1) 1 (t) + + y n (t)y 2n+1 n (t))dt, n = 1, 2, 3, 4,... (58) where A = y () and B = y () are to be determined. In order to avoid difficult fractional integration, we can take the truncated Taylor epansions for the eponential term in (5) (58): e.g., e /2 3 /. Thus, by solving Eqs. (55) (58), we obtain y 1, y 2,... e.g.: y 1 () = A B 3 + Γ (α + 1) + α α+1 Γ (α + 2) y 2 () = 2Aα+2 Γ (α + 3) + 2 α+3 Γ (α + 4) 4A α+3 Γ (α + 4) 2α+ Γ (α + 7) + 2B α+ Γ (α + 7) 4B α+7 Γ (α + 8) + Now, we can form the 2-term approimation α α+2 Γ (α + 3) + α+4 Γ (α + 5) + α+3 Γ (α + 4) 2B α+4 Γ (α + 5) + α+4 Γ (α + 5) 12 α+5 Γ (α + ) 8A α+5 Γ (α + ) (59) B α+5 Γ (α + ) 2t 2α+1 Γ (2α + 2) +. () φ 2 () = 1 + A B 3 + Γ (α + 1) + α+1 α+2 + (2A 1) Γ (α + 2) Γ (α + 3) + 3 α+3 α+4 (2B + 7) Γ (α + 4) Γ (α + 5) + 12α+5 Γ (α + ) 8A α+5 Γ (α + ) B α+5 Γ (α + ) 2 α+ α+ α+7 Γ (α + 7) + 2B Γ (α + 7) 4B Γ (α + 8) + 2t 2α+1 Γ (2α + 2) + (1) where A and B can be determined by imposing boundary conditions (4) on φ 2. Table 4 shows the values of A and B for different values of α. In Fig. 7, we draw absolute error functions, E 1 () = e φ 2,3.75, E 11 () = e φ 2,3.5 and

11 234 Y. Nawaz / Computers and Mathematics with Applications 1 (211) Fig. 8a. Comparison of approimate solutions obtained by 2-term HPM and first-order VIM with eact solution at α = 4. Fig. 8b. Comparison of approimate solutions obtained by 2-term HPM and first-order VIM with eact solution at α = 3.8. Fig. 8c. Comparison of approimate solutions obtained by 2-term HPM and first-order VIM with eact solution at α = 3.2. E 12 () = e φ 2,3.25 for different values of α, where e is an eact solution of (44) (4) and φ 2,3.75, φ 2,3.5 and φ 2,3.25 represent the values of φ 2 at α = 3.75, α = 3.5 and α = 3.25, respectively. In Figs. 8a 8c, we compare the approimate

12 Y. Nawaz / Computers and Mathematics with Applications 1 (211) solutions obtained by VIM and HPM with an eact solution, and it is clear from Figs. 8a 8c that the approimate solutions are in good agreement with an eact solution of (44) (4) at α = 4, α = 3.8 and α = 3.2. Also it is to be noted that the accuracy can be improved by computing more terms of approimated solutions and/or by taking more terms in the Taylor epansion for the eponential term.. Conclusion In this article, variational iteration method (VIM) and homotopy perturbation method (HPM) have been successfully applied to linear and nonlinear boundary value problems for fourth-order fractional integro-differential equations. Two eamples are presented to illustrate the accuracy of the present schemes of VIM and HPM. Comparisons of VIM and HPM with eact solution have been shown by graphs and absolute error functions are plotted which show the efficiency of the methods. References [1] J.H. He, Variational iteration method a kind of non-linear analytical technique: some eamples, International Journal of Nonlinear Mechanics 34 (1999) [2] J.H. He, Homotopy perturbation method: a new nonlinear analytic technique, Applied Mathematics and Computation 135 (23) [3] J.H. He, Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering 178 (1999) [4] J.H. He, A coupling method of homotopy technique and perturbation technique for nonlinear problems, International Journal of Nonlinear Mechanics 35 (1) (2) [5] Ahmet Yildirim, Solution of BVPs for fourth-order integro-differentials by using homotopy perturbation method, Computers and Mathematics and Applications 5 (28) [] N.H. Sweilam, Fourth order integro-differential equations using variational iteration method, Computers and Mathematics and Applications 54 (27) [7] S. Momani, Z. Odibat, Application of homotopy-perturbation method to fractional IVPs, Journal of Computational and Applied Mathematics 27 (1) (27) 9. [8] Z. Odibat, S. Momani, The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics, Computers and Mathematics with Applications 58 (29) [9] S. Abbasbandy, An approimation solution of a nonlinear equation with Riemann Liouville s fractional derivatives by He s variational iteration method, Journal of Computational and Applied Mathematics 27 (27) [1] V. Daftardar-Gejji, Hossein Jafari, Solving a multi-order fractional differential equation using adomian decomposition, Applied Mathematics and Computation 189 (27) [11] I. Podlubny, Fractional Differential Equations, Academic Press, New York, [12] I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fractional Calculus & Applied Analysis 5 (22) [13] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, [14] D.A. Benson, S.W. Wheatcraft, M.M. Meerschaert, Application of a fractional advection dispersion equation, Water Resource Research 3 () (2) 143. [15] D.A. Benson, S.W. Wheatcraft, M.M. Meerschaert, Water Resource Research 3 () (2) [1] S. Momani, M. Aslam Noor, Numerical methods for fourth order fractional integro-differential equations, Applied Mathematics and Computations 182 (2) [17] J.H. He, Non-Perturbative Methods for Strongly Nonlinear Problems, Dissertation, de-verlag im Internet GmbH, Berlin, 2.