An efficient technique for solving nonlinear Fredholm integral equations over Large intervals

Source abstract

Solving nonlinear Fredholm integral equations over large intervals represents a significant computational challenge in scientific computing due to the extensive integration domain, which often leads to numerical instability and prohibitive computational costs in conventional methods. Existing approaches predominantly rely on domain truncation, introducing substantial approximation errors and limiting practical applicability. To bridge this research gap, this paper proposes a novel numerical technique that combines an exponential mapping with a Laguerre collocation method. The key innovation of our approach lies in its ability to accurately and efficiently solve nonlinear Fredholm integral equations on large or semi-infinite intervals without resorting to domain truncation. The exponential mapping transforms the original problem into an equivalent formulation on a semi-infinite domain, where Laguerre polynomials provide a natural and efficient approximation framework. A rigorous convergence analysis in the [Formula: see text]-space establishes the method’s theoretical foundation. Numerical experiments demonstrate the superior accuracy, stability, and computational efficiency of the proposed method across problems with diverse solution behaviors, confirming its effectiveness and practical value.