Rational Approximation Method for Stiff Initial Value Problems

While purely numerical methods for solving ordinary differential equations (ODE), e.g., Runge–Kutta methods, are easy to implement, solvers that utilize analytical derivations of the right-hand side ODE, such as Taylor series method, outperform them in many cases. Nevertheless, method is not well-suited stiff problems since it explicit and A-stable. In our paper, we present a numerical-analytical based on rational approximation ODE solution, which naturally A- A(α)-stable. We describe consider issues order, stability, adaptive step control. Finally, through examples, prove superior performance when highly problems, comparing with same accuracy order.