On the choice of denominator functions and convergence of NSFD schemes for a class of nonlinear SBVPs

Abstract

We propose two novel non-standard finite difference (NSFD) schemes for a class of non-linear singular boundary value problems (SBVPs). One of the basic idea of NSFD schemes is that the step size Δt is replaced by a non-linear function, e.g., sinΔt, tanΔt etc. In the present work, we propose to include one more parameter in the non-linear denominator functions (DFs), e.g., in place of tanΔt, we propose to use ktan(Δt/k), where k∈[c,∞) with c∈R+ which further accelerates the accuracy and reduces the CPU time. We have applied this idea to solve generalized class of non-linear doubly singular BVPs. We observe that there exist some optimal values of k that give much better results. We determine the value of k by experimenting with various DFs on the proposed NSFD schemes. Further it will be interesting to see if we can find out optimal value of k analytically and whether this optimal value is unique. Convergence of the proposed NSFD schemes has also been proved. The results have been verified by several test examples. The results have been compared with other existing methods and matlab bvp4c solver, which shows the applicability and effectiveness of the proposed NSFD schemes.