Homogeneous Fredholm integral Equation
Here we find the solution of Homogeneous Fredholm Integral Equation if second kind
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Integral equation playlist:- • INTEGRAL EQUATION
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What is Integral Equation:- • Introduction to Integral Equation & its types
What is Fredholm integral equation and its kinds:- • TYPE 1 Fredholm integral equation with exa...
Homogeneous and non homogeneous Integral equation:- • Linear and Homogeneous Criterion for Inte...
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Other method to Find to solutions
1) Adomian Decomposition method:- • Example 1 Adomian Decomposition method II ...
2) Modified Decomposition method:- • Modified Decomposition method Example 1
3) Noise term phenomenon:- • Noise Term Phenomenon example 1
4) Direct computation method:- • Direct method example 1
5) Successive approximation method:- • Successive approximation method Example 1
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In this section we will study the homogeneous Fredholm integral equation with
separable kernel given by
obtained from (1) by setting f(x) = 0. It is easily seen that the trivial solution u(x) = 0 is
a solution of the homogeneous Fredholm integral equation (208). In this study our goal
will be focused on finding nontrivial solutions to (208) if exist. We can achieve our
goal by introducing the technique that will enable us to determine the nontrivial
solutions to (208). Generally speaking, the homogeneous Fredholm integral equation
with separable kernel may have nontrivial solutions. Our approach in obtaining these
desired solutions will be based mainly on the direct computation method that was
employed effectively for nonhomogeneous Fredholm integral equations. We point out
that Adomian decomposition method is not applicable for the homogeneous Fredholm
integral equations. This may be related to the fact that the nonhomogeneous part f(x)
does not exist in this type of problems, and therefore the zeroth component u0(x) cannot
be defined.
We recall that the direct computation method reduces the equation to an algebraic
equation if the kernel consists of one term only, or to a system of algebraic equations if
the kernel contains many separable terms. Additional discussions will be required for
determining possible values of λ that will give rise to the nontrivial solutions as will be
discussed soon.
Without loss of generality we may assume a one term kernel given by
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