Mathematics 4

Course objectives

The primary goal is to get acquainted with mathematical modeling of phenomena in mechanics that are subject to other courses and provide basic math apparatus used in these courses. The topics include:

Matrix theory aimed towards methods of solutions of large systems of linear equations that occur in the course of numerical solutions of mechanics problems.
Boundary value problems for ordinary differential equations, namely those that model beam behavior.
Selected partial differential equations: Laplace, heat and wave equations.

Outline

Matrices, eigenvalues, eigenvectors. Norms of matrices and vectors. Large systems of linear equations.
Symmetric and positive-definite matrices. Condition number. Direct methods of linear equations systems solution.
Variational principle for linear equations systems. Iterative methods of linear equations systems solution. The conjugate gradient method as a direct and iterative method.
Initial value problem for ordinary differential equation. Fundamental system and general solution.
Dot product and orthogonality of continuous functions. Boundary value problem. Eigenvalues and eigenfunctions of a boundary value problem. Existence and uniqueness of a solution.
The energy of an equation. Variational principle.
Application: The deflection of simply supported and clamped beams. Energy and stability.

Requirements for Exam

Eigenvalues, spectrum of a matrix, spectral radius, Gershgorin Theorem; norms; symmetric and positive definite matrices; Cholesky factorization; condition number; energy and variational principle; convergence of Steepest Descent and Conjugate Gradients methods.
Linear differential equations; null space, fundamental system, Wronski determinant, reduction of order; Cauchy problems, variation of constants; equations with constant coefficients (also higher order)
Boundary value problems; eigenvalues, eigenfunctions, eigenspaces; solvability. Linear differential operators; bounded, positive and weakly positive operators; eigenvalues and eigenfunctions; Fredholm Theorem, solvability.
Energy of Au=f. Variational principle. Stability.

Literature and resources

Collection of Problems
Notes on Eigenvalues
Notes on Norms
Notes on Iterative Methods
Czech-English Math Dictionary
K. Rektorys: Solving ordinary and partial boundary value problems in science and engineering, CRC Press 1999, ISBN 0-8493-2552-8.
As relevant literature in English is not easy to get for a reasonable cost, the main resource for your study will be lecture notes.
A great source of information is Wikipedia.
There will be handouts and sample problems available here.
Sample problems