Singular Integral Regularization and Neutrix Subtraction: Analytical and Numerical Approaches
Year: 2026
Type: article
Abstract: We have all often noticed that a common mathematical challenge recurs in certain areas, which is divergence, whether we are talking about fracture mechanics, quantum field theory, electrostatics, or singular integrals. A very common example is crack mechanics simulations, where the singular integrals of broken steel beams usually diverge, making numerical stability very difficult. In this paper, we analytically discuss the expansions of multi-dimensional singular integrals using Neutrix calculus, especially in the context of radial singularities. To do this, we isolate finite parts of a broad spectrum of regularization techniques such as Gaussian, rational, exponential, polynomial, Lorentzian, and hyperbolic methods and test their efficiency. The numerical results show that the closer one is to the singular region, the more important the magnitude of the integral can be: in one dimension, on the closed interval from 1 to 10, the value of the integral can vary between 1.845213 and 9.404645; in two dimensions between 9.454510 and 92.057897; in three dimensions between 62.435420 and 82.057897. When using algebraic-type regularizations with a value of \(n=1.2\) and a value of \(\epsilon=0.01\), the raw values of the integrals were similar, with Lorentzian 76.791266, Rational 78.411604, and Polynomial/Hyperbolic 80.304942 representing the values of the algebraic-type regularizations, respectively. However, using Gaussian and Exponential damping, the values of the algebraic-type regularizations were much smaller, with 3.403055 and 4.887252, respectively. The stable finite parts were given by the Lorentzian, Rational, Polynomial, and Hyperbolic methods, but it was found that the Gaussian damping was over-suppressed. Relative errors close to machine precision, in the Lorentzian case, which justify the errors of the numerical implementation, were also indicated by benchmark validation. This paper provides a comprehensive framework, an analytical derivation, and numerical simulations. This research could open new avenues for theoretical physics and computational engineering.
Source: Cureus Journal of Engineering.
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid:
OpenAlex ID: W7165032877