THE EXISTENCE AND NON-EXISTENCE OF CONVEX OR CONVEX-CONCAVE SOLUTIONS OF A DIFFERENTIAL EQUATION RESULTING FROM FLUID MECHANICS

Abstract

A similar problem was recently considered with f^' (t)→λ as t→ T_c, (T_c is the maximum value for the existence or no existence of f_c), λ∈ {0,1,±∞} and 0<β<1. In this paper, we study same the equation (with β<0) and we establish the existence of solutions satisfying the boundary conditions f (0) =a∈R,f'(0) =b<0 and f'(+∞) ∈ {0,1}. In order to do that, we utilize the shooting technique, then we consider the initial value problem consisting of the differential equation such that f (0)=a∈R,f'(0) =b<0 and f"(0)=c≥0. We accompany our results by some numerical illustrations.