The decomposition method and the modified decomposition method introduce the solution of Fredholm integral equation in the form of a rapidly convergent power series with elegantly computable terms. However, if f(x) consists of more than one term, the modified decomposition method minimizes the volume of the computational work.
Example 1:- • Modified Decomposition...
Previous Video:- Adomian decomposition Method : • What is Adomian Decomp...
What is Integral Equation:- • Introduction to Integr...
What is Fredholm integral equation and its kinds:- • TYPE 1 Fredholm integr...
Homogeneous and non homogeneous Integral equation:- • Linear and Homogeneous...
There are so many ways to find the solutions of Non Homogeneous Fredholm integral equation.
The following ways are
1) Adomian Decomposition method:- • Example 1 Adomian Deco...
2) Modified Decomposition method:- • Modified Decomposition...
3) Noise term phenomenon:- • Noise Term Phenomenon ...
4) Direct method:- • Direct method example 1
5) Successive approximation method:- • Successive approximati...
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The Modified Decomposition Method
It is worth noting that the Adomian decomposition method may be sometimes
implemented in a different but easier manner in order to facilitate the computational
work. It is recommended to apply the modified decomposition method, developed by
Wazwaz [ 59], for cases where the nonhomogeneous part f(x) in (1) consists of a
combination of many terms. This modified technique, as will be seen later, will
minimize the volume of calculations and reduce the several integral evaluations that
result in applying the standard Adomian decomposition method.
It is also of interest, before giving a clear discussion of this method, to note that this
modified technique will be carried out with promising results in Volterra integral
equations and nonlinear integral equations in forthcoming chapters. The technique
avoids the cumbersome integrations of other methods.
In the modified method, we split the given function f(x) into two parts defined by
where f0(x) consists of number of terms of f(x), and f1(x) includes the remaining terms of
f(x). We note that a necessary condition is required to apply this approach in that f(x)
should consist of more than one term as shown by (38). In view of (38), the integral
equation (1) becomes
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