Future Students > PhD Program

PhD Program

Program

of entrance exam in graduate school of Lviv National Ivan Franko University in the specialties 01.01.07 – Computational Mathematics and 01.05.02 – Mathematical Modeling

1. Function spaces and linear continuous operators

Linear normalized, Banach and Hilbert spaces, examples. Continuous linear operators and inverse operators. A criterion of the existence of bounded continuous operator.

Orthogonal projections, their properties. The best approximation problem.

Adjoint and self-adjoint operators in Hilbert space. Continuous linear functionals, dual space, Hahn – Banach theorem. Riesz Hilbert space representation theorem.

2. Numerical methods of linear algebra

Vectors, matrices and operations on them. Determinants and their properties. The system of linear algebraic equations and their analysis. Gauss method. Iterative methods for solving systems of linear algebraic equations. Terms of convergence of iterative methods.

Linear operators in a finite-dimensional space and their matrix representation. Characteristic polynomial, eigenvalues ​​and eigenvectors of a linear operator. Adjoint and self-adjoint operators. Quadratic forms, reducing them to the canonical form.

Methods of maximum modulo eigenvalue finding. Method of powers, scalar product method. Methods of solving the complete eigenvalue problem​​. The method of Jacobi. Bisection method. Householder transformation.

3. Methods for solving nonlinear equations

Iteration and its convergence. Secant line method. Bisection method. Iteration method for systems of equations. Newton’s method and its modifications. Descent method.

4. Interpolation

Interpolation problem substance and convergence of interpolation process. Finite difference and difference ratios. Presentation of the interpolation polynomial in Lagrange, Newton and Gauss form. Approximation using spline functions. Spline interpolation.

5. Numerical integration

General interpolation quadrature. Quadrature formula with equidistant nodes. The simplest quadrature Newton–Cotes formulae: trapezoid rule, Simpson’s rule. Quadrature formulae of the highest algebraic degree of precision.

6. Boundary value problems for elliptic equations

Elliptic type equations and boundary value problems for them (electrostatics, heat conduction, rod torsion). Dirichlet, Neumann and mixed boundary conditions. Well-posed boundary problem. The maximum principl.

Difference methods for solving boundary value problems. Approximation, stability and convergence.

Variational formulation of problems: minimization problem, problem for variational equation. The existence and uniqueness of solution, its boundedness.

Ritz method and its convergence. Properties of Ritz approximations. Galerkin method. Its convergence.

Finite element approximations, convergence speed estimate.

Iterative methods for solving grid equations.

7. Initial-boundary value problems for parabolic equations

Parabolic type equations and applications that imply them. (conductivity, impurities diffusion). Formulation of boundary and initial conditions. Well-posed problems. The maximum principle.

Difference methods for solving parabolic problems. Approximation, stability and convergence. Explicit and implicit schemes. Conditionally and unconditionally stable scheme.

Variational formulation of initial-boundary value problem. Energy equation. Solution uniqueness. Galerkin half-discretization. One step recursive schemes for solving half-discrete problems, their stability and convergence.

8. Initial-boundary value problem for hyperbolic equations

Hyperbolic equation and its applications that implies it (acoustics, string vibration). Difference methods for hyperbolic problems. The variation problem formulation. Energy equation. Kinetic, potential energy and their dissipation. Solution uniqueness and boundedness.

Galerkin half-discretization. Methods of solving half-discrete problems, their stability and convergence.

9. Numerical methods of solving integral equations

Method of replacing the integral with quadrature sum. Solving integral equations by replacing the kernel with degenerate kernel.

10. Methods of optimization and operations research

Necessary and sufficient conditions for function extremum. Conditional extremes. Methods for finding the absolute extremum: gradient method, Newton method, conjugate gradient method. Linear programming problem and simplex algorithm for solving it. Numerical methods for nonlinear programming: the penalty functions method. Optimization methods based on sequential options analysis.

11. Applied programming. Software for scientific research

Computers generations. Main groups and structural features of modern computers. The algorithm concept. Algorithmic languages​​. The principles of programming concept. Structured and modular programming.

Description of the main parts of mathematical computer software. The operating system functions and the operating modes.

Stages of solving the problem and its control. Programming system and its main functions: broadcast, diagnostics, debugging and editing.

Application software. Databases and their classification. Application packages. Functional and system package content. Methods of algorithmic languages description.

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