Asymptotic expansions of a boundary-value problem for a certain nonlinear partial differential equation
Consider the nonlinear, nonhomogeneous equation \[ \partial^ 2u/\partial t^ 2-\alpha(\partial u/\partial x)^ n\partial^ 2u/\partial x^ 2=\epsilon f(x,u,\partial u/\partial x,\partial u/\partial t,\partial^ 2u/\partial x^ 2,\nu t) \] where \(\alpha>0\), \(n+1=(2m+1)/(2r+1), m,r=0,1,...,t>0\), \(\epsilon\) a small parameter, \(f(x,u,\partial u/\partial x,\partial u/\partial t,\partial^ 2u/\partial x^ 2,\nu t)\) an analytic 2\(\pi\)-periodic (with respect to \(\nu\) t) function. Using special Ateb functions [see \textit{P. M. Senik}, Ukr. Mat. Zh. 21, 325-333 (1969; Zbl 0201.069)] the author constructs in a ''nonresonance'' case one- frequency asymptotic solutions of the above equation.
scientific article; zbMATH DE number 3859620