Unbounded Solutions of Third-order Infinite Interval Problems Relevant to the Laminar Mixing Layer

Abstract

In this paper, we consider the existence, uniqueness, and qualitative properties of solutions of nonlinear third-order two-point boundary value problems on the half-line by using the shooting method together with the Bolzano’s theorem and the maximum principle. Meanwhile, the nonlinear third-order three-point boundary value problems on the whole real line are studied by employing the matching method. As applications, we present the existence, uniqueness, and qualitative properties of solutions of the Blasius equations u‴+uu″ = 0 with one of the following boundary conditions

$$\begin{aligned}&u^{\prime}(-\infty)=C,\ u(0)=0,\ u^{\prime}(+\infty)=1, \qquad\text{with}\; 0 \leq C< 1,\\&u^{\prime}(-\infty)=C,\ u^{\prime\prime\prime}(0)=0,\ u^{\prime}(+\infty)=1, \quad\;\,\,\text{with}\; 0 \leq C < 1,\end{aligned}$$

which arise from the laminar mixing layer between two parallel flows with different velocities.