Galoitica: Journal of Mathematical Structures and Applications GJMSA 2834-5568 10.54216/GJSMA https://www.americaspg.com/journals/show/3087 2022 2022 Department of Mathematics, University of Rijeka, City of Rijeka, Croatia Stipan Stipan Jadavpur University, Department Of Mathematics, Kolkata, India Barbara Charchekhandra This paper is devoted to find the solution of the Vanishing Neutral Differential Equations (VNDEs), where we review the ARCHI code for solving neutral differential equations, and then an improvement of this code will be presented in this paper to solve problems of VNDEs. This improvement is done by suggesting a new hybrid method of special interpolants with iteration procedure of RK method. We will analyze the convergence of the suggested new method by proving the following criteriaBn(t)≤TH(c1+(c2L3)(1-L3 ))exp(LT(1-L3 ))Where the solution is convergent , we have  and when and the derivative of the solution is also convergent for VNDEs. 2024 2024 01 11 10.54216/GJMSA.0110101 https://www.americaspg.com/articleinfo/33/show/3087