Horner’s Scheme for investigation of solutions of differential equations with polynomial right-hand side
We present a method for investigating solutions of systems of ordinary differential equations with polynomial right-hand side. Similar systems are of long-term interest for applications, because many process models have different physical, biological and economical natures described by these systems. The standard methods of numerical analysis are usually applied obtaining system solutions with the polynomial right-hand side, disregarding the specific form of the right-hand side. We suggest a different method starting from the fact that the right side of the equation appears to be a multidimensional polynomial. The relative simplicity of the right-hand side of the system under consideration made it possible to construct by this method approximate analytic solutions in the form of functions not only of time but of the initial conditions as well. In contrast to the majority of known methods, the latter made it possible in many cases to directly trace the systematic computational error. The implementation of the method is based on the construction of a discrete dynamical system along the solutions of the original system with subsequent use of the generalized Horner’s Scheme. The computation peculiarity of Horner’s Scheme lies in the fact that in many cases the scheme allows us to reduce the number of machine operations required for computation of the polynomial in comparison with the ordinary computation process. The second peculiarity of the generalized Horner’s Scheme lies in the fact that there is a good decomposition of computation process that allows us to make calculations in parallel on independent nodes. According to computational experiments, this enables us to reduce computation time hugely even in the simplest cases while retaining required accuracy.