Application of He’s homotopy perturbation method to nonlinear integro-differential equations
Introduction
In recent years, the application of the homotopy perturbation method (HPM) [5], [14], [15] in nonlinear problems has been developed by scientists and engineers, because this method continuously deforms the difficult problem under study into a simple problem which is easy to solve. The homotopy perturbation method [8], proposed first by He in 1998 and was further developed and improved by He [6], [9], [10], [13]. The method yields a very rapid convergence of the solution series in the most cases. Usually, one iteration leads to high accuracy of the solution. Although goal of He’s homotopy perturbation method was to find a technique to unify linear and nonlinear, ordinary or partial differential equations for solving initial and boundary value problems. Most perturbation methods assume a small parameter exists, but most nonlinear problems have no small parameter at all. A review of recently developed nonlinear analysis methods can be found in [11]. Recently, the applications of homotopy perturbation theory among scientists were appeared [1], [2], [7], [14], which has become a powerful mathematical tool, when it is successfully coupled with the perturbation theory [9], [12], [13], [15]. Also recently, El-Shahed [3] applied He’s homotopy perturbation method to Volterra’s integro-differential equations.
In this paper, we propose HPM to solve the nonlinear integro-differential equations. Volterra integro-differential equation is given byand Fredholm type is given by
Section snippets
Basic idea of homotopy perturbation theory
To illustrate HPM consider the following nonlinear integral equation:with boundary conditionswhere A is a general integral operator, B is a boundary operator, f(r) is a known analytic function, and Γ is the boundary of the domain Ω.
The operator A can be generally divided into two parts L and N, where L is linear, whereas N is nonlinear. Therefore, Eq. (2.1) can be rewritten as follows:He [4] constructed a homotopy v : Ω × [0, 1] → R which satisfies
Numerical example
This section contains six examples of Volterra and Fredholm nonlinear integro-differential equations. Example 1 Case 1. Consider the following nonlinear integro-differential equation:for x ∈ [0, 1] with the exact solution . By HPM, let L(u) = u′(x) − g(x) = 0. Hence, we may choose a convex homotopy such thatand continuously trace an implicitly defined curve from a starting point H(v, 0) to a solution function H(v, 1). Substituting (2.7)
Conclusion
In this work, we applied an application of He’s homotopy perturbation method for solving the nonlinear integro-differential equations. The solution obtained by HPM is valid for not only weakly nonlinear equations, but also strong ones. Furthermore their first-order approximations of extreme accuracy for most cases.
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