The Homotopy Analysis Method for a Fourth-Order Initial Value Problems

Where f is continuous function on [a, b] and for each i=0,3... the parameters αi are real constants. Such type of initial value problems arise in the mathematical modeling of physical and engineering sciences. Liao [1-5] presented the homotopy analysis method for solving different problems in several dimensions. In this method, produced solution is in infinite series form and this form is converges to the exact solution.


Introduction
In this section, we consider the general fourth-order initial value problems of the type: Where f is continuous function on [a, b] and for each i=0,3… the parameters α i are real constants. Such type of initial value problems arise in the mathematical modeling of physical and engineering sciences. Liao [1][2][3][4][5] presented the homotopy analysis method for solving different problems in several dimensions. In this method, produced solution is in infinite series form and this form is converges to the exact solution.
In this paper, we applies the homotopy analysis method to solve the linear and nonlinear fourth-order initial value problems by using the transformation of problem to a system of first-order differential eqn. (1). Several Numerical examples have been presented in Section 3, to illustrate the advantage and the effectiveness of the proposed method.

Homotopy Analysis Method
In this section, we will briefly describe the use of HAM for the differential equation: Where N is a linear or nonlinear operators, x denote the independent variable, u(x) is an unknown function and g(x) is a known analytic function.
By means of generalizing the traditional homotopy analysis method, Liao [1] constructs the so-called zeroth-order deformatin equation: Where L is an auxiliary linear operator with the property L[0]=0 N is the nonlinear operator related to the original eqn. (2), q∈[0,1] is the embedding parameter in topology (called the homotopy parameter), c¹0 is the so called "convergence-control parameter", and H(x) is an auxiliary function [6][7][8][9].
Note that in the frame of the homotopy, we have great freedom to choose the auxiliary linear operator L, the initial guess u 0 (x) the auxiliary function H(x) and the convergence-control-parameter c.
Obviously, when q=0 and q=1 one has: thus as q increases from 0 to 1, the solution φ(x,q) varies from the initial guesses u 0 (x) to the exact solution u (x) Expanding φ i (x,q)in Taylor series with respect to q one has: where: if the auxiliary linear operator, the initial guesses, the auxiliary parameters c, and the auxiliary function are so properly chosen, the series solutions eqn. (4) converge at q=1, and we have: wich must be of the solution of the original nonlinear equations, as proved by Liao [1].
According to eqn. (5), the governing equations can be deduced from the zeroth-order deformation eqn. (2). Define the vector: (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: Expanding φ i (x,q) in Taylor series with respect to q one has: where: if the auxiliary linear operator, the initial guesses, the auxiliary parameters c i and the auxiliary function are so properly chosen, the series solutions in eqn. (14) converge at q=1 and we have: wich must be of the solutions of the original nonlinear equations, as proved by Liao [1].
According to eqn. (15) the governing equations can be deduced from the zeroth-order deformation eqn. (12).Define the vector: Differentiating eqn. (12) mtimes 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: and It should be emphasized that ( ), 1 m u t m ≥ is governed by the linear eqn. (16) with the linear initial conditions that come from the original problem, which can be easily solved by symbolic computation softwares such as Maple and Mathematica.

Application
In this section, we shall apply HAM to several test examples to demonstrate its accuracy. To show the efficiency of the present method for our problem in comparison with the exact solution we report absolute error which is defined by: Consider the following non linear fourth-order problem: with the following boundary conditions: and It should be emphasized that ( ), 1 m u x m ≥ is governed by the linear eqn. (6) with the linear initial conditions that come from the original problem, which can be easily solved by symbolic computation softwares such as Maple and Mathematica [10].

The modified homotopy analysis method
In this part, we will briefly describe use of the HAM for problem eqn. (1) the modified HAM suggests that we reformulating the problem eqn. (1) as an equivalent system of a first-order differential equation, we use the transformation: we can rewrite the fourth-order initial value problem eqn. (1) as the system of ordinary differential equations: with the initial conditions: We will present a brief description of the standard HAM for a system of ODEs, we consider the differential equation: (12) where N i are nonlinear operators (and i-1…4) and x denote the independent variable, u i (x) are unknown functions and gi(x) are known analytic functions, for gi(x)=0 eqn. (11) reduces to the homogeneous equation [11][12][13].
By means of generalizing the traditional homotopy analysis method, Liao constructs the so-called zeroth-order deformatin equation [1]: Where L is an auxiliary linear operator with the property L[0]=0 N is the nonlinear operator related to the original eqn. (1) q∈[0,1] is the embedding parameter in topology (called the homotopy parameter), c i ¹0 is the so called "convergence-control parameter", and H(x) is an auxiliary function [14].
Note that in the frame of the homotopy, we have great freedom to choose the auxiliary linear operator L, the initial guess u i,0 (x), the auxiliary function H(x) and the convergence-control-parameter c i .
Obviously, when q=0 and q=1 one has: The analytic solution is: Using the transformation from eqn. (8) we can rewrite the fourthorder initial value problem in eqn. (19) as the system of ordinary differential equations: with the initial conditions: Where c i are constants of integrations. Furthermore, eqn. (21) suggest that we define a system of linear operators as: Now, the solution of the mth-order deformation eqn (22) for  Table 1 exhibits the exact solution and the series solution along with the errors obtained by using the homotopy analysis method [13][14][15]. It is obvious that the errors can be reduced further and higher accuracy can be obtained by evaluating more components of ( ) u x .

Example 3.2
Consider the following nonlinear initial fourth-order problem: The analytic solution is: Using the transformation eqn. (8) we can rewrite the fourth-order initial value problem as the system of ordinary differential equations: with the initial conditions: as shown in Table 2 the series solutions ( 10 th order) to get the analytic solution

Example 3.3
We consider the nonlinear fourth-order initial problem: The exact solution is: Using the transformation, we can rewrite the fourth-order initial value problem as the system of ordinary differential equations: with the initial conditions: as shown in Table 3 the series solutions (10 th order) to get the exact solution y(x)=sin(x) which are coinciding with the exact solution.

Example 3.4
Consider the linear system of fourth-order initial value problem: The analytic solution is: Using the transformation we can rewrite the system of ODEs from eqn. (23)as the system of ordinary differential equations: with the initial conditions: as shown in Table 4 exhibits the errors obtained by using the approximate solution of the homotopy analysis method.

Conclusion
In this paper, the homotopy analysis method HAM has been successfully used for finding the solution of special forth-order initial value problem under the the transformation of the fourth-order value problem to the system of the first order differential equation. Several examples showed the validity and potential of this modification of the homotopy analysis method for linear and nonlinear special fourthorder initial value problem in the science and engineering.