In this paper, we are concerned with the solution of the third-order nonlinear differential equation f″′+ff″+βf′(f′-1)=0, satisfying the boundary conditions f(0)=a∈R, f′(0)=b<0, and f′(t)→λ, as t→+∞, where λ∈{0,1} and 0<β<1. The problem arises in the study of the opposing mixed convection approximation in a porous medium. We prove the existence, nonexistence, and the sign of convex and convex-concave solutions of the problem above according to the mixed convection parameter b<0 and the temperature parameter 0<β<1.

Abstract. The similarity differential equation f + f f + β f ( f − 1) = 0 with β > 0 is considered. This differential equation appears in the study of mixed convection boundary-layer flows over a vertical surface embedded in a porous medium. In order to prove the existence of solutions satisfying the boundary conditions f (0) = a 0 , f (0) = b 0 and f (+∞) = 0 or 1, we use shooting and consider the initial value problem consisting of the differential equation and the initial conditions f (0) = a , f (0) = b and f (0) = c . For 0 < β 1 , we prove that there exists a unique solution such that f (+∞) = 0 , and infinitely many solutions such that f (+∞) = 1 . For β > 1 , we give only partial results and show some differences with the previous case.Mathematics subject classification (2010): 34B15, 34C11, 76D10.

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In this paper, we will study the existence of strong solutions for a nonlinear system of partial differential equations arising in convective flow, modeling a phenomenon of mixed convection created by a heated and diving plate in a porous medium saturated with a fluid. The main tools are Schäfer’s fixed-point theorem, the Fredholm alternative, and some theorems on second-order elliptic operators.