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

Abstract

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 In this study, an effective numerical method based on Taylor expansions is presented for boundary value problems. This method is arbitrary directional and called as implicit-explicit local differential transform method (IELDTM). With the completion of this study, a reliable numerical method is derived by optimizing the required degrees of freedom. It is shown that the order refinement procedure of the IELDTM does not affect the degrees of freedom. A priori error analysis of the current method is constructed and order conditions are presented in a detailed analysis. The theoretical order expectations are verified for nonlinear BVPs. Stability of the IELDTM is investigated by following the analysis of approximation matrices. To illustrate efficiency of the method, qualitative and quantitative results are presented for various challenging BVPs. It is tested that the current method is reliable and accurate for a broad range of problems even for strongly nonlinear and singularly perturbed BVPs. The produced results have revealed that the IELDTM is more accurate than the existing ones in literature.