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

Abstract

In this study, an effective numerical method based on Taylor expansions is presented for boundary valueproblems. 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 degreesof freedom. It is shown that the order refinement procedure of the IELDTM does not affect the degrees of freedom. Apriori error analysis of the current method is constructed and order conditions are presented in a detailed analysis. Thetheoretical order expectations are verified for nonlinear BVPs. Stability of the IELDTM is investigated by followingthe analysis of approximation matrices. To illustrate efficiency of the method, qualitative and quantitative results arepresented for various challenging BVPs. It is tested that the current method is reliable and accurate for a broad rangeof problems even for strongly nonlinear and singularly perturbed BVPs. The produced results have revealed that theIELDTM is more accurate than the existing ones in literature.