Lopatkin S.A., Reizlin V.I. Additional Chapters on Mathematics

Study aid. — Tomsk: Tomsk Polytechnic University, 2008. — 126 p. — ISBN 5-98298-315-2.The study aid contains computational methods, basics of edge problems, solution for ordinary differential and partial derivative equations as well as some issues on functional analysis.
The study aid is developed in the framework of Innovative Educational Programme of TPU on the direction «Power-saving, basic, special, and industrial discharge, radiation, and plasma-beam technologies». The manual is intended for training students majoring in the course 140200 «Power Electrical Engineering» and studying the master programme «Technology and Physics of High Voltage».Introduction.
Solving the Edge Problems for Ordinary Differential Equations and Systems.
Shooting method.
Method of finite differences or the mesh method.
Semi-analytical methods of edge problem solving.
Collocation method.
Galerkin’s method.
Numerical Solution of Partial Derivative Equations.
Difference schemes. Fundamental issues.
Convergence, approximation and stability of difference schemes.
Difference schemes for parabolic equations.
The solution of Cauchy problem.
Stability of two-layer difference schemes.
Difference schemes for the equations of an elliptic type.
Construction of difference approximation for the Poisson’s equation.
Different edge problems and approximation of edge conditions.
The construction of difference scheme in case of Dirichlet’s problem for Poisson’s equation.
Matrix sweep method.
Iteration method of difference solution method for Dirichlet’s problem.
Difference schemes for simple equations of hyperbolic type.
Solving Cauchy problem.
Solving mixed problem.
Method of finite elements (MFE).
General remarks.
Discretization of area and numbering of nodes.
Linear interpolar polynomials.
One-dimensional simplex element.
Two-dimensional simplex element.
Local system of coordinates.
Two-dimensional L-coordinates.
Aggregation of elements into ensemble.
Finding the equations for element with the help of Galerkin’s method.
Example. Calculation of one-dimensional temperature field in a homogeneous rod.
Two-dimensional equations of the field theory.
Appendix. Elements of Functional Analysis.
Transformations.
Vector space.
Basis of vector space.
Coordinate transformation.
Metrics and norm.
Banach space.
Hilbert space.
Orthogonality and the theories of Fourier.
Basis of Hilbert space.
Linear operations.
Linear operator matrix.
Сonvergence method.
Spectral radius of operator.
References.