The numerical solution of a Fredholm integral equation of the second kind using the Galerkin method based on optimal interpolation

Abstract

In this paper, we study the Galerkin method for obtaining approximate solutions to linear Fredholm integral equations of the second kind. The finite element solution is represented as a linear combination of basis functions, and the construction of suitable basis functions plays a crucial role in the accuracy of the approximation. We propose an optimal interpolation formula that exactly reproduces the functions $e^x$ and $e^{-x}$, and derive basis functions from its coefficients. This interpolation formula is constructed within the Hilbert space $W_2^{(1,0)}$. To evaluate the effectiveness of the proposed approach, we solve several integral equations using the Galerkin method with two types of basis functions: the newly constructed exponential basis and classical piecewise linear basis functions. Numerical experiments are presented to compare the accuracy of these approaches. Graphs and tables illustrate the approximation errors, demonstrating that both basis functions achieve an error order of $O\left(h\right)$, with the optimal interpolation-based basis yielding superior accuracy in certain cases.