Successive approximation method Example 1

Example 2:-  Coming soon
In this method, we replace the unknown function under the integral sign of the Fredholm
integral equation of the second kind by any selective real valued function u0(x) , a ≤ x ≤ b. Accordingly, the first approximation u1(x) of the solution u(x),The second approximation of u2(x) of the solution u(x) can be obtained by replacing
u0(x).This process can be continued in the same manner to obtain the nth approximation. In
other words, the various approximations of the solution u(x) of (140) can be obtained in
a recursive scheme.Even though we can select any real valued function for the zeroth approximation u0(x),
the most commonly selected functions for u0(x) are 0, 1 or x. At the limit, the solution
u(x) is obtained by as mention in video
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Integral equation playlist:-  INTEGRAL EQUATION
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What is Integral Equation:- Introduction to Integral Equation & i...

What is Fredholm integral equation and its kinds:-  TYPE 1 Fredholm integral equation wit...

Homogeneous and non homogeneous Integral equation:-  Linear and Homogeneous Criterion  for...

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Other method to Find to solutions
1) Adomian Decomposition method:-  Example 1 Adomian Decomposition metho...

2) Modified Decomposition method:-  Modified Decomposition method Example 1

3) Noise term phenomenon:-  Noise Term Phenomenon example 1

4) Direct method:-   Direct method example 1

5) Successive approximation method:- Successive approximation method Examp...

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It is important to distinguish between the recursive schemes used in the Adomian
decomposition method and in the successive approximations method. In the
decomposition method, we apply the approach to determine several components of the
solution u(x)
so that the zeroth component u0(x) is defined by all terms that are out of the integral sign
or part of these terms if the modified version is used. However, in the successive
approximations method, we apply the above recursive scheme (143) to determine
various approximations of the solution u(x) itself, and not components of u(x). Further,
we should note here that the zeroth approximation u0(x) is not defined but rather given
by a selective function, and as a result the solution u(x) is given by the formula (144).
The successive approximations method will be illustrated by the following examples.

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