Both Emden-Fowler and generalized Emden-Fowler nonlinear ordinary differential equations (ODEs) are reduced to Abel's equation of the second kind by means of admissible functional transformations. Since in Part I a mathematical technique is developed leading to the construction of exact analytic solutions of the above Abel equation, it follows that the Emden-Fowler equations admit exact analytic solutions too. In this sense several basic particular nonlinear ODEs in mathematical physics are examined.
In this work it is shown that, by a series of admissible functional transformations, the generalized Blasius equation in fluids can be exactly reduced to a three-term generalized Emden–Fowler equation. Furthermore, the restricted in axisymmetric flows and simplified forms of this equation can be reduced to (i) two-term generalized Emden–Fowler equations; (ii) generalized Emden–Fowler equations; (iii) Emden–Fowler equations of the normal form; and (iv) Abel equations of the second kind. By means of a recently developed mathematical solution methodology (Panayotounakos, Fifth Greek Congress on Mechanics, Xania, Crete, 22–25 June 2004, Hellas, Greece), we provide exact analytic solutions for the simplified as well as for the restricted forms of the above-mentioned Blasius equations. Thus, it is proved that important, unsolvable in exact form problems in nonlinear fluid dynamics now can be analytically solved.
Two kinds of second-order nonlinear ordinary differential equations (ODEs) appearing in mathematical physics and nonlinear mechanics are analyzed in this paper. The one concerns the Kidder equation in porous media and the second the gas pressure diffusion equation. Both these equations are strongly nonlinear including quadratic first order derivatives (damping terms). By a series of admissible functional transformations we reduce the prescribed equations to Abel's equations of the second kind of the normal form that they do not admit exact analytic solutions in terms of known (tabulated) functions. According to a mathematical methodology recently developed concerning the construction of exact analytic solutions of the above class of Abel's equations, we succeed in performing the exact analytic solutions of both Kidder's and gas pressure diffusion equations. The boundary and initial data being used in the above constructions are in according with each specific problem under considerations.
We prove that the first-order nonlinear differential system governing the plastic spin response in simple shear (Dafalias’s equations) is reduced to an equivalent equation of the Abel normal form by means of admissible functional transformations. In similar Abel equations result also the original and the generalized Volterra differential systems, describing the problem of two populations conflicting with one another. The above reduced Abel equations do not admit exact analytic solutions in terms of known (tabulated) functions, since only very special cases of these types of equations can be analytically solved in parametric form. We provide a mathematical solution methodology leading to the construction of exact implicit analytic solutions of the above-mentioned type of equations. Since the plastic spin nonlinear differential system results in a special unsolvable form of Abel’s equation of the normal form, we perform the exact implicit analytic solution of this system too.
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