Approximate solution of some perturbed boundary value problems
An analysis of elastic oscillatory systems with distributed and slightly variable parameters is often reduced to the solution of the linear boundary value problem of Sturm-Liouville type \(((1+\epsilon \sigma (x))X')'+\lambda\) \(2(1+\epsilon \delta (x))X=0\), \(X(0)=X(1)=0\), \(x\in [0,1]\), \(\epsilon\in [0,\epsilon]\), \(\lambda\) \(2\geq 0\), where \(\epsilon\) is a small parameter characterizing the nonhomogeneity. The authors analyze some reasons for the appearance of secular terms in asymptotic expansions for eigenvalues and eigenfunctions in (A) and construct uniform (with respect to the index number n) expansions enabling to avoid them. They reduce (A) to a system in ``amplitude-phase'' variables and develop an asymptotic procedure based on a correction of the phase \(\psi_ n\) which is in general similar to conventional methods in the theory of nonlinear oscillations. An approximation of functions \(f(y)\in C_ 0^{(2)}\) using the approximate basis of the problem (A) is also studied. Afterwards this technique is applied to Dirichlet's problem describing the oscillations of nonhomogeneous elastic systems with clamped ends.
scientific article; zbMATH DE number 4033207