A regularization technique to overcome the Ill-posedness arising in specific engineering models: formulation, implementation, error analysis, and some engineering applications

This article presents a regularization technique for the stable numerical solution of first-kind Fredholm integral equations, which frequently arise in the mathematical modeling of engineering and physical science problems. The proposed technique combines an approximation framework based on a special representation of the triangular function vector forms and their properties with a stabilization strategy to convert the original ill-posed problem into a well-posed algebraic system. Another notable advantage is the reduced computational cost of the proposed technique, as it eliminates the need for performing any integrations during the setup of the algebraic system. Detailed error analysis and convergence proofs are provided, offering rigorous theoretical guarantees for the method’s performance. Numerical experiments on test problems demonstrate the efficiency, stability, and high accuracy of the proposed technique, especially when compared with other regularization methods. Furthermore, the proposed technique is applied to analyze some engineering models, including electromagnetic scatterers and thin-wire antennas. In all cases, the results show excellent agreement with full-wave simulations performed using Altair FEKO software. These findings confirm the robustness, versatility, and computational effectiveness of the proposed regularization strategy for practical ill-posed problems.