Singular perturbation of a boundary value problem for quasilinear third-order ordinary differential equations involving two small parameters (Q5937079)

The authors study the existence of solutions to the singularly perturbed quasi-linear differential equation \(\varepsilon y''' + \mu f(x,y) y'' = g(x,y,y',\varepsilon,\mu)\), with the boundary conditions \(y(0) = a(\varepsilon,\mu)\), \(y'(0) = b(\varepsilon,\mu)\) and \(y'(1) = c(\varepsilon,\mu)\). Here, \(\varepsilon > 0\) and \(\mu > 0\) are small parameters, and some regularity assumptions on \(f\) and \(g\) are made. In particular, the problem is assumed to have a unique solution \({\overline y}(x)\) in \([0,1]\) for \(\varepsilon=\mu=0\). The authors prove the existence of unique solutions in \([0,1]\) in the perturbed case. Moreover, they construct formal asymptotic expansions of the solutions and give an estimate on the remainder.