Exact boundaries for the analytical approximate solution of a class of first-order nonlinear differential equations in the real domain

Abstract

The paper gives a solution to one of the problems of the analytical approximate method for one class first order nonlinear differential equations with moving singular points in the real domain. The considered equation in the general case is not solvable in quadratures and has movable singular points of the algebraic type. This circumstance requires the solution of a number of mathematical problems. Previously, the authors have solved the problem of the influence of a moving point perturbation on the analytical approximate solution. This solution was based on the classical approach and, at the same time, the area of application of the analytic approximate solution shrank in comparison with the area obtained in the proved theorem of existence and uniqueness of the solution. Therefore, the paper proposes a new research technology based on the elements of differential calculus. This approach allows to obtain exact boundaries for an approximate analytical solution in the vicinity of a moving singular point. New a priori estimates are obtained for the analytical approximate solution of the considered class of equations well in accordance with the known ones for the common area of action. These results complement the previously obtained ones, with the scope of the analytical approximate solution in the vicinity of the movable singular point being significantly expanded. These estimates are consistent with the theoretical positions, as evidenced by the experiments carried out with a non-linear differential equation having the exact solution. A technology for optimizing a priori error estimates using a posteriori estimates is provided. The series with negative fractional powers are used.

Keywords

• moving singular points
• nonlinear differential equation
• cauchy problem
• exact boundaries of a domain
• a priori and a posteriori errors
• analytical approximate solution