Direct Solution of nth-Order IVPs by Homotopy Analysis Method

Direct solution of a class of nth-order initial value problems IVPs is considered based on the homotopy analysis method HAM . The HAM solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of the series solutions. The HAM gives approximate analytical solutions which are of comparable accuracy to the sevenand eightorder Runge-Kutta method RK78 .


Introduction
Higher-order initial value problems IVPs arise in mathematical models for problems in physics and engineering. Generally, second-and higher-order IVPs are more difficult to solve than first-order IVPs. It is possible to integrate a special nth-order IVP by reducing it to a first-order system and applying one of the established methods available for such system. However, it seems more natural to provide direct numerical methods for solving the nthorder IVPs.
It is the purpose of the present paper to present an alternative approach for the direct solution of nth-order IVPs based on the homotopy analysis method HAM . The analytic homotopy analysis method HAM , initially proposed by Professor Liao in his Ph.D. thesis 1 , is a powerful method for solving both linear and nonlinear problems. The interested reader can refer to the much-cited book 2 for a systematic and clear exposition on this method. In recent years, this method has been successfully employed to solve many types of nonlinear problems in science and engineering 3-17 . All of these successful applications verified the validity, effectiveness and flexibility of the HAM. More recently, Bataineh  In this work, we consider a class of nth-order IVPs of the form subject to the initial conditions

Basic Ideas of HAM
To describe the basic ideas of the HAM, we consider the following differential equation: where N is a nonlinear operator, t denotes the independent variable, x t is an unknown function. By means of generalizing the traditional homotopy method, Liao 2 constructs the so-called zero-order deformation equation where q ∈ 0, 1 is an embedding parameter, is a nonzero auxiliary function, L is an auxiliary linear operator, x 0 t is an initial guess of x t and φ t; q is an unknown function. It is important to note that one has great freedom to choose auxiliary objects such as and L in HAM. Obviously, when q 0 and q 1, both Differential Equations and Nonlinear Mechanics 3 hold. Thus as q increases from 0 to 1, the solution φ t; q varies from the initial guess x 0 t to the solution x t . Expanding φ t; q in Taylor series with respect to q, one has If the auxiliary linear operator, the initial guess, the auxiliary parameter , and the auxiliary function are so properly chosen, then the series 2.4 converges at q 1 and one has which must be one of the solutions of the original nonlinear equation, as proved by Liao 2 .
which is used mostly in the HPM 32 . According to 2.5 , the governing equations can be deduced from the zero-order deformation equations 2.2 . We define the vectors Differentiating 2.2 m times with respect to the embedding parameter q and then setting q 0 and finally dividing them by m!, we have the so-called mth-order deformation equation

2.10
It should be emphasized that x m t m ≥ 1 are governed by the linear equation 2.9 with the linear boundary conditions that come from the original problem, which can be easily solved by symbolic computation softwares such as Maple and Mathematica.

D i fferential Equations and Nonlinear Mechanics
A new approach in the HAM was proposed by Yabushita et al. 26 . We will call this method NHAM. Yabushita et al. 26 considered the following projectile problem: The standard HAM applied to this problem yields a divergent solution on some part of the solution domain. In NHAM, the zeroth-order deformation equations were constructed for not only 2.11 , but also for 2.12 . This slight modification in the NHAM gives a more accurate solution.

Numerical Experiments
To illustrate the effectiveness of the HAM we will consider four examples of nth-order IVPs 1.1 -1.2 .

Example 1
We first consider the nonlinear second-order IVP subject to the initial conditions The exact solution is To solve 3.1 -3.2 by means of HAM, we choose the initial approximation and the linear operator where c i i 1, 2 are constants of integration. Furthermore, 3.1 suggests that we define the nonlinear operator as Using the above definition, we construct the zeroth-order deformation equation as in 2.2 and the mth-order deformation equation for m ≥ 1 is as in 2.9 with the initial conditions now the solution of the mth-order deformation for m ≥ 1 is We now successively obtain . . .

3.11
Then the series solution expression can be written in the form and so forth. Hence, the series solution when −1 is which converges to the closed-form solution 3.3 .

Example 2
Consider the linear fourth-order IVP, 3.14 subject to the initial conditions The exact solution is To solve 3.14 -3.15 by means of HAM, we choose the initial approximation the solution of the mth-order deformation for m ≥ 1 is the same as 3.10 .

Differential Equations and Nonlinear Mechanics 7
We now successively obtain . . .

3.22
Then the series solution expression can be written in the form and so forth. Hence, the series solution when −1 is

Example 3
Now consider the nonlinear fourth-order IVP, subject to the initial conditions The exact solution is x t e t − 1.

3.28
According to the HAM, the initial approximation is the solution of the mth-order deformation for m ≥ 1 is the same as 3.10 . We now successively obtain

3.31
Then the series solution expression can be written in the form x t x 0 t x 1 t x 2 t · · · , 3.32 Differential Equations and Nonlinear Mechanics 9 and so forth. Hence, the series solution when −1 is

Example 4
Finally we consider the nonlinear Genesio equation 33 subject to the initial conditions where a, b, c are positive constants satisfying ab < c. First we solve 3.35 by means of HAM. According to the HAM, the initial approximation is and the linear operator is with the property L c 1 tc 2 t 2 c 3 0, 3.40 where c i i 1, 2, 3 are constants of integration. According to the zeroth-order deformation equation 2.2 and the mth-order deformation equation 2.9 with the initial conditions 3.42 the solution of the mth-order deformation for m ≥ 1 is the same as 3.10 . We now successively obtain when a 1.2, b 2.92 and c 6, In this technique, we construct the zeroth-order deformation equations for not only 3.35 but also for 3.36 as follows:

3.44
Differential Equations and Nonlinear Mechanics 11 and the mth-order deformation equation

3.45
with the initial conditions

3.47
Again, we successively obtain when a 1.2, b 2.92 and c 6, and so forth. Then the series solution expression can be written in the form 3.50 The series solutions 3.12 , 3.23 , 3.32 , 3.49 and 3.50 contain the auxiliary parameter . The validity of the method is based on such an assumption that the series 2.4 converges at q 1. It is the auxiliary parameter which ensures that this assumption can be satisfied. In general, by means of the so-called -curve, it is straightforward to choose a proper value of which ensures that the solution series is convergent. Figure 1 show the -curves obtained from the fifth -order HAM approximation solutions of 3.1 , 3.14 and 3.26 . From this figure, the valid regions of correspond to the line segments nearly parallel to the horizontal axis. Substituting the special choice −1 into the series solutions 3.12 , 3.23 and 3.32 yields the exact solution 3.3 , 3.16 and 3.28 . Also Figures 2 and 3 show the -curves obtained from the eleventh -order HAM and NHAM approximation solutions of 3.35 and 3.36 . In Figure 4 we obtain numerical solution of the Genesio equation using the eleventh -order HAM and NHAM approximation. It is demonstrated that the HAM and NHAM solutions agree very well with the solutions obtained by the seven-and eight-order Runge-Kutta method RK78 . Moreover we conclude that the proposed algorithm given by NHAM is more stable than the classical HAM. by using HAM, so according to Figure 4 we conclude that the behavior of numerical solution 3.35 is more stable than the numerical solution obtained by 19 using the classical HAM.

Conclusions
In this paper, the homotopy analysis method HAM was applied to solve a class of linear and nonlinear nth-order IVPs and the Genesio equation. HAM provides us with a convenient way of controlling the convergence of approximation series, which is a fundamental qualitative difference in analysis between HAM and other methods. The illustrative examples suggest that HAM is a powerful method for nonlinear problems in science and engineering.