Approximate Solutions and Estimate of Galerkin Method for Variable Third-Order Operator-Differential Equation

Abstract

The paper considers a variable third-order operator-differential equation in a separable Hilbert space. Under certain assumptions, it is proved that this ODE has a unique solution. The proof is based on a classical Galerkin discretization of the separable Hilbert space in term of certain eigenfunctions. The approximation quality of the Galerkin approximations can be controlled in terms of the eigenvalues.
We deduce estimates for the convergence rate of the approximate solutions to the exact one. An example provided as application to the investigated method.