📂Banach Space

프레드홀 적분 방정식

Definition¹

The following integral equation is referred to as a Fredholm Integral Equation of the first kind.

$$g(s) = \int K(s, t) f(t) dt \tag{1}$$

Here, $K$ is called the kernel. The following form is referred to as the Fredholm integral equation of the second kind.

$$g(s) = f(s) + \int K(s, t) f(t) dt \tag{2}$$

Explanation

Solving the integral equation $(1), (2)$ typically means finding $f$ that satisfies $(1), (2)$ given $g$ and $K$. This is an inverse problem of finding the cause $f$ when the result $g$ is already known. Depending on the given function, the integral itself can be challenging, and solving it inversely is notoriously difficult. When the definite integral value $3$ is given, it is easy to see that there are infinitely many $f$ that satisfy the following condition.

$$3 = \int_{0}^{1} f(x) dx$$

Only when appropriate constraints are given, or when integral values over multiple domains are provided, can we find the exact $f$.