A new implicit-explicit local differential method for boundary value problems

Boundary value problems (BVPs) of differential equations arise in many disciplines such as physics, chemistry, engineering, finance, mathematical biology and so on. Analytical solutions are often not available for most of those problems. While some series-based techniques are capable of producing semianalytical solutions for BVPs, convergence of those methods is largely dependent on the global smoothness of exact solutions, as observed from examples discussed in the literature [38, 42]. In addition, some BVPs involving significant local behaviors such as sharp discontinuities or boundary layers are areas where various notable difficulties are encountered [36, 37]. In such cases, any analytical or numerical approach to such problems should be well defined. Numerical methods for boundary value problems are mainly divided into two categories: direct methods