Step 1: Understanding the Concept:
We need to convert the second-order boundary value problem (BVP) into an equivalent Fredholm integral equation. The kernel of this equation is the Green’s function for the operator under the given boundary conditions.
Step 2: Key Formula or Approach:
The BVP is:
Treating as a forcing term , we first solve . After integrating twice and applying boundary conditions, we express in terms of an integral with a kernel . Substituting back , we obtain the Fredholm equation with kernel .
Step 3: Detailed Explanation:
1. Solve : Using . 2. Apply : 3. Substitution gives: 4. Rearranging yields:
where
5. Substituting :
with
Step 4: Final Answer:
The integral equation corresponding to the given BVP is:
where
This matches the required form.