Mathematical Methods in Mechanics
The areas of study are integral equations and calculus of variations.
Integral equations area involves: classification of integral equations, methods of solving Volterra integral equations of both the first and the second kind, some approaches to solutions of Fredholm integral equations (degenerative kernels, Fredholm resolvent, continuous kernels), theory of Fredholm integral equations with symmetric kernels, Hilbert-Schmidt theorem, application to Sturm-Liouville problem, green function
Calculus of variations area involves: functionals and proximity of functions, Lagrange’s lemma. Necessary condition for an extremum. Increment of a functional and its variation. Transversality conditions. Functionals of several dependent variables. Functionals dependent on higher derivatives. Functionals dependent on functions of several independent variables. Conditional extremum. Isoperimetric problem. Sturm-Liouville systems as variational problems. Lagrange problem. Direct methods for solving variational problems (Ritz’s, Galerkin’s and Kantorovich’s methods)