3. Out Line :
Abstract
Keywords
Introduction
Homotopy perturbation method
Fredholm integral equation of the second kind
Volterra integral equations of the second kind
Conclusion
References
4. Abstract :
The aim of this paper is convergence study of
homotopy perturbation method (HPM) for solving
integral equations in general case. The homotopy
perturbation method is a powerful device for solving
a wide variety of problems. Using the homotopy
perturbation method, it is possible to find the exact
solution or an approximate solution of the problem.
Some illustrative examples are presented.
6. 1. Introduction:
Various kinds of analytical methods and numerical methods [1, 2,
3] were used to solve integral equations. In this paper, we apply
Homotopy perturbation method [4, 5, 6, 7, 8] to solve integral
equations. The method has been used by many authors to handle a
wide variety of scientific and engineering applications to solve
various functional equations. In this method, the solution is
considered as the sum of an infinite series, which converges rapidly
to accurate solutions. Using the homotopy technique in topology, a
homotopy is constructed with an embedding parameter p 2 [0, 1]
which is considered as a ”small parameter”.
7. This method was further developed and improved by He and
applied to nonlinear oscillators with discontinuities [7],
nonlinear wave equations [5], boundary value problems
[6], limit cycle and bifurcation of nonlinear problems [8]. It can
be said that He’s homotopy perturbation method is a universal
one, and is able to solve various kinds of nonlinear functional
equations. For example, it was applied to nonlinear Schrdinger
equations [9]. Other recent works in this field are found in
[8, 9, 10,]
8. 2. Homotopy perturbation method :
To convey an idea of the HPM, we consider a general equation of
the type:
L(u) = 0 (2.1)
where L is an integral or differential operator. We define a convex
homotopy H(u, p) by
H(u, p) = (1 − p)F(u) + pL(u) = 0, (2.2)
where F(u) is a functional operator with known solutions v0 which
can be obtained easily. It is clear that, for:
H(u, p) = 0 (2.3)
from which we have H(u, 0) = F(u) and H(u, 1) = L(u). This shows
that H(u, p) continuously traces an implicitly defined curve from a
starting point H(v0, 0) to a solution H(u, 1).
9. The embedding parameter p monotonously increases from zero
to a unit as the trivial problem F(u) = 0 continuously deforms to
original problem L(u) = 0. The embedding parameter p 2 [0, 1] can
be considered as an expanding parameter to obtain:
u = u0 + pu1 + u2 + ... (2.4)
When p → 1, Eq.(2.3) corresponds to Eq.(2.1) and becomes the
approximate solution of Eq.(2.1), i.e.,
The series (2.5) is convergent for most cases, and the rate of
convergence depends on L(u), [8].
u = u0 + u1 + u2 + ... (2.5)
10. 3. Fredholm integral equation of the
second kind :
Now we consider the Fredholm integral equation of the second
kind in general case, which reads
u(x) = f(x) + λ k(x, t)u(t)dt, (3.1)
where k(x, t) is the kernel of the integral equation. In view of
Eq.(2.2)
(1 − p)[u(x) − f(x)] + p[u(x) − f(x) − λ k(x, t)u(t)dt] = 0, (3.2)
or
u(x) = f(x) + pλ k(x, t)u(t)dt. (3.3)
11. Substituting Eq.(2.4) into Eq.(3.3), and equating the terms with
identical powers of p, we have
: = f(x),
: = λ k(x, t)(u0)dt,
: = λ k(x, t)(u1)dt,
: = λ k(x,t)(u2)dt,
⁞
therefor, we obtain iteration formula for Eq.(3.1) as follow:
12. According to Eq.(3.4) we define partial sum as follow :
(x) = f (x),
(x) = λ k(x,t) (t)dt, m>0 (3.4)
S0 (x) = f (x),
(3.5)
In view Eqs. (3.4) and (3.5) we have
S0 (X) = f (x),
(x) = f (x) + λ k(x,t) sn (t )dt.
13. 4. Volterra integral equations of
the second kind :
First, we consider the Volterra integral equations of the second
kind, which reads
u(x) = f(x) + λ k(x, t)u(t)dt, (4.1)
where K(x, t) is the kernel of the integral equation. As in the
case of the Fredholm integral equation we can use Homotopy
perturbation method to solve Volterra in-tegral equations of the
second kind.
However,there is one important difference: if K (x,t ) and f(x) are
real and continuous, then the series converges for all values of
λ.
14. In this work, we introduce the study of the problem of
convergence of the homotopy perturbation method.
The sufficient condition for convergence of the method has
been presented, and the examination of this condition for the
integral equations and integro-differential equation.
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