Novel Stochastic Method for Multidimensional Fredholm Integral Equations

Integral equations have broad applications in various fields, such as applied mathematics, physics, engineering, geophysics, electricity and magnetism, kinetic theory of gases, quantum mechanics, mathematical economics, and queuing theory. Therefore, it is essential to develop and explore efficient and reliable approaches for solving integral equations. For multidimensional problems, existing biased stochastic algorithms based on a finite number of integrals suffer from the high dimensionality effect since they are based on quadrature points. Thus, advanced unbiased algorithms are required to solve multidimensional problems, which we propose in this paper. We introduce a new unbiased stochastic method for solving multidimensional Fredholm integral equations of the second kind, which we analyze and compare to the old unbiased stochastic algorithm for both one-dimensional and multidimensional problems. This research aims to enhance the understanding of unbiased stochastic algorithms and improve their effectiveness and reliability for solving multidimensional integral equations.