Numerical Methods for Singular Differential Equations

Abstract

Singular differential equations (SDEs) arise in various fields of science and engineering, presenting unique challenges for numerical solution techniques due to the presence of singularities or discontinuities in the differential equation or its solution. This paper provides a comprehensive review of numerical methods developed for solving singular differential equations, encompassing both ordinary and partial differential equations. The review covers a wide range of numerical techniques, including finite difference methods, finite element methods, spectral methods, collocation methods, and boundary element methods, among others. Each method is discussed in detail, highlighting its strengths, weaknesses, and applications in solving different types of singular differential equations. Additionally, recent advances in adaptive and meshless numerical methods tailored for handling singularities are examined.