TITLE:
Reproducing Kernel Hilbert Space Technique for Fourth-Order Singular Boundary Value Problems
AUTHORS:
Khaled K. Jaber
KEYWORDS:
Hilbert Space, Integro-Differential Equation, Gram-Schmidt Process, Fredholm-Volterra Integral Equations, Reproducing Kernel Function
JOURNAL NAME:
Applied Mathematics,
Vol.17 No.6,
June
24,
2026
ABSTRACT: Reproducing kernel Hilbert space method is utilized in this paper as an efficient approach to solve singular fourth order boundary value problems of mixed form Fredholm-Volterra integro-differential equations. In order to obtain the required nodal values, the algorithm developed using two smooth reproducing kernel functions. The solution philosophy is based on applying the Gram-Schmidt process to the kernel function obtained in the space
W
2
5
[
0,1 ]
to produce an orthogonal basis. From that point onward, the orthogonal basis was built for the purpose of formulating and utilizing numerical solutions in the same space. Some linear and nonlinear numerical issues were broken down to delineate the technique and affirm the execution of the proposed strategy. The numerical outcomes emphasize the method role in improving the initial approximation, handling boundary conditions, and refining the solution throughout the iterative process to shed light on such singular equations.