Integrable dissipative nonlinear second order differential equations via factorizations and Abel equations

Abstract

We emphasize two connections, one well known and another less known, between the dissipative nonlinear second order differential equations and the Abel equations which in its first kind form have only cubic and quadratic terms. Then, employing an old integrability criterion due to Chiellini, we introduce the corresponding integrable dissipative equations. For illustration, we present the cases of some integrable dissipative Fisher, nonlinear pendulum, and Burgers-Huxley type equations which are obtained in this way and can be of interest in applications. We also show how to obtain Abel solutions directly from the factorization of second-order nonlinear equations.

A class of exact solutions of the Liénard-type ordinary nonlinear differential equation

“…Similar results were previously obtained in [42], where the integral form of the Chiellini integrability condition were explicitly formulated. A particular integrable case of the Liénard equation can be obtained for the case k = 0.…”

“…In fact, the Chiellini condition has been recently used in [38][39][40][41][42][43][44] for the study of Abel differential equations and second-order differential equations reducible to an Abel-type equation.…”

“…In [42] the differential Chiellini integrability condition was reformulated in an integral form, and the general form of the solution of the Abel equation was obtained in a simpler form. In [44] the Chiellini integrability condition of the first-order Abel equation of the first kind dy/dx = f (x)y 2 + g(x)y 3 was extended to the case of the general Abel equation of the form…”

Exact analytical solutions of the Susceptible-Infected-Recovered (SIR) epidemic model and of the SIR model with equal death and birth rates

“…The mathematical properties of the Abel type equation, and its applications, have been intensively investigated in a series of papers [27][28][29][30][31]. Note that when the average death rate µ is zero, in view of Eq.…”

Hypercomplex analysis and integration of systems of ordinary differential equations

“…From an historical standpoint, we believe that Abel's equation of the first kind should be called Riccati–Liouville equation, and Abel's equation of the second kind must be renamed Abel–Appell equation. Abel's equations arise in the integration of several models of dissipative systems . Our main task here is to lift Liouville–Appell theory of integration of Abel's equation of the first kind where the overdot is differentiation with respect to t an the functions z and c i 's are valued in a commutative hypercomplex, to systems that we shall call Abel‐type systems.…”