A new multi-stage method for the numerical solutions of fully implicit nonlinear differential systems

This paper presents a novel technique for solving fully implicit nonlinear systems of ordinary differ-ential equations, which are typically challenging to address using existing numerical methods and symbolic tools like Mathematica, Matlab or Maple. Our approach applies the Differential Transform Method (DTM) directly to the im-plicit differential system and utilizes an important property of Adomian polynomials, leading to a simple and efficient algorithm. The main advantage of our method is that it require the solution of only one algebraic system compared to implicit numerical method which require the solution of nonlinear algebraic system at every step. Furthermore, our technique does not require the differential system to be in an explicit form. The DTM generates the exact solution as a convergent power series. To enlarge the interval of convergence, we have developed a multistage DTM algorithm that enables accurate numerical solutions over larger intervals. The effectiveness of our method is demonstrated through several numerical examples that conventional tools cannot solve, showcasing its ability to compute both exact and numerical solutions for implicit nonlinear differential systems efficiently.