Studies in Nonlinear Sci., 3 (3): 94-101, 2012
Solving IVPs of Higher-orderordinary Differential Equations
by Modified Homotopy Perturbation Method
1H. Jafari and 2M. Mahmoudi
1Department of Mathematics, University of Mazandaran, Babolsar, Iran
2Payam Noor University, Babol, Iran
Abstract: In this paper, we use modified homotopy perturbation method to solving Initial Value Problems (IVP) of higher-order ordinary differential equations. The proposed method can be applied to linear and nonlinear problems.
The results prove that the modified HPM is a powerful tool for the solution of singular IVPs.
Key words: Initial value problems · homotopy perturbation method · ordinary differential equations
INTRODUCTION
The Homotopy Perturbation Method (HPM) is a new and ingenious method for solving linear and nonlinear differential and integral equations of various kinds. Homotopy perturbation method [11-12] is an analytical method which can be applied to the solution of linear, nonlinear deterministic and stochastic operator equations. The HPM deforms a difficult problem into an infinite set of problems which are easier to solve without any need to transform nonlinear terms. The applications of HPM in nonlinear problems have been demonstrated by many researchers. In recent years, much attention has been devoted to the application of the HPM, to the solutions of various scientific models. The purpose of this paper is to introduce a new reliable modification of HPM. For this reason, a new differential operator is defined which can be used for higher-order initial value problems.
HOMOTOPY PERTURBATION METHOD
To illustrate the basic ideas of this method, we consider the following nonlinear differential equation.
(1)
With the boundary conditions
B (u, (2)
where A is a general differential operator, is a boundary operator, f(r) is a known analytic function, G is the boundary of the domain W. The operator, A can, generally speaking, be divided in to two parts L and N, where L is linear and N is nonlinear, therefore Eq. (1) can be written as:
(3)
By using homotopytechnique,one can construct a homotopy v(r,p):
(4)
Or
H(v,p)=L(v)-L()+p
Where pÎ[0,1] is an embedding parameter and u0 is the initial approximation of Eq. (1)
Which satisfies the boundary conditions.clearly,we have
H(v,0) = L(v)-L(u0) = 0 (5)
H(v,1) = A(v)-f(r) = 0
The changing process of p from zero to unity is just that of v(r,p)changing from u0(r) to u(r). In topology, this is called homotopy. According to (HPM),we assume that the solutions of Eqs. (4)can be written as a power series in P:
(6)
Setting P=1,resultsin the approximate solution of Eq.(6)
U = = (7)
The convergence of the series(7) has been proved in [12].
MODIFIED HOMOTOPY PERTURBATION METHOD
Consider the initial value problem in the n-order differential equation in the form:
(8)
where N is a nonlinear differential operator of order less than n-1, p(x) and g(x) are given functions and are given constants.
We propose the new differential operator, as below
(9)
So, the problem(8) can be written as
(10)
The inverse operator L is therefore considered a n-fold integral operator, as below:
(11)
NUMERICAL EXAMPLES
In this section, few examples are presented to understand better the confusion HPM
Example 1: Consider the nonlinear singular IVP
(12)
We construct the following homotopy
We propose the new differential operator, as below
Equating the terms with the identical powers of P,
The exact solution is
Example 2: Consider the nonlinear singular IVP
(13)
We propose the new differential operator, as below
in an operator form, equation(13) yield
We construct the following homotopy
Equating the terms with the identical powers of P,
The exact solution is y(x) = x3
Example 3: Consider the nonlinear IVP
(14)
To use the Taylor series of g(x) with order V which by choosing V=6
We propose the new differential operator, as below
There fore
In an operator form, equation (14) yield
We construct the following homotopy
Equating the terms with the identical powers of P,
The exact solution is y(x) = x3ex
Example 4: Consider thelinear IVP
(15)
We propose the new differential operator, as below
Therefore
In an operator form, equation (8) yield
We construct the following homotopy
Equating the terms with the identical powers of P,
The exactsolution is y(x) = -cos x
Example 5: Consider the linear singular IVP
(16)
We propose the new differential operator, as below
We construct the following homotopy
Equating the terms with the identical powers of P,
The exact solution is y(x) = x4
Example 6: Consider the linear IVP
(17)
We construct the following homotopy
We propose the new differential operator, as below
Equating the terms with the identical powers of P,
The exactsolution is
DISCUSSION AND CONCLUSION
In this paper, we use modified homotopy perturbation method to solving initial value problems (IVP) of higher-order ordinary differential equations. The (MHPM) proposed in this investigation is simple and effective for solving higher order of IVP and can provide an accuracy approximate solution or exact solution.
Mathematica has been used for computations in this paper.
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