The Integrating Factors for Riccati and Abel Differential Equations

Abstract

We can recast the Riccati and Abel differential equations into new forms in terms of introduced integrating factors. Therefore, the Lie-type systems endowing with transformation Lie-groups S L(2, R) can be obtained. The solution of second-order linear homogeneous differential equation is an integrating factor of the corresponding Riccati differential equation. The numerical schemes which are developed to fulfil the Lie-group property have better accuracy and stability than other schemes. We demonstrate that upon applying the group-preserving scheme (GPS) to the logistic differential equation, it is not only qualitatively correct for all values of time stepsize h, and is also the most accurate one among all numerical schemes compared in this paper. The group-preserving schemes derived for the Riccati differential equation, second-order linear homogeneous and non-homogeneous differential equations, the Abel differential equation and higher-order nonlinear differential equations all have accuracy better than O(h 2 ).

Comparison criteria for the Abel equation of 1st kind

“…This equation has important applications in questions of natural sciences (e.g., in the Friedman model, in the study of conditions for existence of limit cycles for dynamic systems [Hilbert's 16th problem]) and many works are devoted to it (see [1][2][3][4][5][8][9][10][11] and cited works therein).…”