On Runge–Kutta Methods Based on the Operators J±

Using an integral method based on indefinite integrals J±, we will give new numerical methods for tackling a variety of initial value problems. Within the scope of this study, a number of example cases of stiff and non-stiff initial value problems are examined in order to validate the newly developed computational approaches. The approximation sequence and the number of discretization steps may be more easily controlled with the help of this new technique. The methods provide an a priori evaluation of the approximation error, facilitating access to an optimal solution. The approach entails modeling indefinite integrals by discretization using conformal maps or the roots of orthogonal polynomials. We will analyze the techniques employed for several basis functions, including Sinc, Sinc-Gaussian, Lagrange, and generalized Lagrange polynomials. Utilizing these collocation approaches, matrices A+ for implicit Runge–Kutta procedures are automatically generated. In addition to being novel in the realm of Sinc techniques, this approach is also novel in the methodology of polynomial approximation. It eliminates the combinatorial problems that are linked with Radau techniques while also generalizing them.