Calculus of Variations and Integral Equations
The objectives of the discipline Calculus of Variations and Integral Equations is to develop knowledge of the basic tenets of the theory of integral equations and mastery of the respective solutions of problems and exercises, knowledge of the main provisions of the calculus of variations and the ability to use the concepts and methods of the theory in solving problems arising in theoretical and mathematical physics.
1. Integral equations 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.
2. Calculus of variations 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)