Parametrically defined nonlinear differential equations, differential-algebraic equations, and implicit ODEs: Transformations, general solutions, and integration methods

The study deals with nonlinear ordinary differential equations defined parametrically by two relations; these arise in fluid dynamics and are a special class of coupled differential–algebraic equations. We propose a few techniques for reducing such equations, first or second order, to systems of standard ordinary differential equations as well as techniques for the exact integration of these systems. Several examples show how to construct general solutions to some classes of nonlinear equations involving arbitrary functions. We specify a procedure for the numerical solution of the Cauchy problem for parametrically defined differential equations and related differential–algebraic equations. The proposed techniques are also effective for the numerical integration of problems for implicitly defined equations.

[1]  G. Latta,et al.  Ordinary differential equations and their solutions , 1960 .

[2]  Cleve Moler,et al.  Mathematical Handbook for Scientists and Engineers , 1961 .

[3]  A. Polyanin,et al.  Handbook of Exact Solutions for Ordinary Differential Equations , 1995 .

[4]  Andrei D. Polyanin,et al.  Parametrically defined nonlinear differential equations and their solutions: Applications in fluid dynamics , 2016, Appl. Math. Lett..

[5]  H. Fernholz.  Boundary Layer Theory , 2001 .

[6]  Ernst Hairer,et al.  The numerical solution of differential-algebraic systems by Runge-Kutta methods , 1989 .

[7]  Linda R. Petzold,et al.  Numerical solution of initial-value problems in differential-algebraic equations , 1996, Classics in applied mathematics.

[8]  J. Butcher.  The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods , 1987 .

[9]  Uri M. Ascher,et al.  Computer methods for ordinary differential equations and differential-algebraic equations , 1998 .

[10]  W. Rheinboldt,et al.  Theoretical and numerical analysis of differential-algebraic equations , 2002 .

[11]  D. Zwillinger.  Handbook of differential equations , 1990 .

[12]  A. Polyanin,et al.  On order reduction of non-linear equations of mechanics and mathematical physics, new integrable equations and exact solutions , 2012 .

[13]  Granino A. Korn,et al.  Mathematical handbook for scientists and engineers , 1961 .

[14]  A. Polyanin,et al.  Handbook of Nonlinear Partial Differential Equations , 2003 .

[15]  G. Saccomandi.  A remarkable class of non-classical symmetries of the steady two-dimensional boundary-layer equations , 2004 .

[16]  W. Schiesser.  Computational Mathematics in Engineering and Applied Science: ODEs, DAEs, and PDEs , 1993 .

[17]  A. Polyanin,et al.  On RF-pairs, Bäcklund transformations and linearization of nonlinear equations , 2012 .

[18]  L. G. Loĭt︠s︡i︠a︡nskiĭ.  Mechanics of liquids and gases , 1966 .