An Orr-Sommerfeld equation with a first-order differential turning point (Q1768959)

This paper deals with the construction of asymptotic expansions of solutions of ordinary differential equations with turning points on the closed interval \([0,l]\). In particular, the author is interested in the classical Orr-Sommerfeld equation with a first-order differential turning point. The construction of such asymptotic solutions is of interest in singular perturbation theory. The author makes a first important step towards the construction of such solutions by describing the construction of a uniform asymptotic expansion of a solution of the equation \[ \varepsilon^5y^{(4)}(x,\varepsilon)+ \varepsilon^3a_3(x)y'''(x,\varepsilon)+ \varepsilon^2a_2(x)y''(x,\varepsilon)+ a_1(x)y'(x,\varepsilon)+ a_0(x) y(x,\varepsilon)=h(x), \] where \(\varepsilon\) tends to \(0\) for \(x\) in \([0,1]\) and \(a_i(x)\) and \(h(x)\) are \(C^\infty[I]\) for \(i=0,1,2,3\) and \(a_1(x)=x\widetilde{a}(x)\) with \(\widetilde{a}(x)>0\) for \(x\) in \([0,1]\).