Parametrically defined nonlinear differential equations and their solutions: Applications in fluid dynamics

The study deals with parametrically defined ordinary differential equations, practically unaddressed in the literature. It finds the general solutions for three classes of first- and second-order nonlinear ODEs of this kind. The solutions are further used to construct new exact solutions to the equations of an unsteady axisymmetric boundary layer with pressure gradient on a body of revolution of arbitrary shape. Also the paper suggests a short list of essential problems for nonlinear ODEs and PDEs defined parametrically that need to be addressed in the future.

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

[2]  Taha Aziz,et al.  Remark on classical Crane's solution of viscous flow past a stretching plate , 2016, Appl. Math. Lett..

[3]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[4]  P. Clarkson,et al.  New similarity solutions of the unsteady incompressible boundary-layer equations , 2000 .

[5]  Ahmed Alsaedi,et al.  Flow of a viscous fluid over an impermeable shrinking sheet , 2013, Appl. Math. Lett..

[6]  Wai How Hui,et al.  Similarity solutions of the two-dimensional unsteady boundary-layer equations , 1990, Journal of Fluid Mechanics.

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

[8]  E. Kamke.  Differentialgleichungen : Lösungsmethoden und Lösungen , 1977 .

[9]  Von mises- and crocco-type transformations: Order reduction of nonlinear equations, RF-pairs, and Bäcklund transformations , 2010 .

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

[11]  G. Burde.  New similarity reductions of the steady-state boundary layer equations , 1996 .

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

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

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

[15]  A. Polyanin,et al.  Unsteady axisymmetric boundary-layer equations: Transformations, properties, exact solutions, order reduction and solution method , 2015 .

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

[17]  A. Cañada,et al.  Handbook of differential equations , 2004 .

[18]  Xiaoping Xu,et al.  Algebraic Approaches to Partial Differential Equations , 2012, 1205.6535.

[19]  A. Aksenov,et al.  Reductions of the stationary boundary layer equation with a pressure gradient , 2013 .