Solution of the Cauchy problem for ordinary differential equations using the collocation and least squares method with the Pade approximation

A new method for solving the Cauchy problem for an ordinary differential equation is proposed and implemented using the collocation and least squares method of increased accuracy. It is based on the derivation of an approximate nonlinear equation by a multipoint approximation of the problem under consideration. An approximate solution of the problem in the form of the Pade approximation is reduced to an iterative solution of the linear least squares problem with respect to the coefficients of the desired rational function. In the case of nonlinear differential equations, their preliminary linearization is applied. A significant superiority in accuracy of the method proposed in the paper for solving the problem over the accuracy of the NDSolve procedure in the Mathematica system is shown. The solution of a specific example shows the superiority in accuracy of the proposed method over the fourth-order Runge–Kutta method. Examples of solving the Cauchy problem for linear and non-linear equations with an accuracy close to the value of rounding errors during operations on a computer with numbers in the double format are given. It is shown that the accuracy of solving the problem essentially depends on the complexity of the behavior of the values of the right-hand side of the equation on a given interval. An example of constructing a spline from pieces of Pade approximants on partial segments into which a given segment is divided is given in the case when it is necessary to improve the accuracy of the solution.