Abstract:

The first part of the course is dedicated to study of techniques for

solving systems of linear equations. Several basic methods are presented:

Gauss elimination, matrix solution, Cramer rule.

The last two methods involve matrix arithmetic,

which is also presented in this part. The second part discusses vector spaces and linear transformations. Most attention is given

to the matrix approach. Matrix diagonalization issues summarize this part.

Abstract:

Sorting differential equations.

First-order differential equations.

Linear differential equations of order n:

homogeneous and non-homogeneous equation, Wronskian homogeneous equations with constant coefficients. Separation into homogeneous and non-homogeneous problem, method of unknown coefficients and method of variation of parameters.

Language Problems - Sturm Liouville Theory: Defining a close-knit operator, finding the operator's self-values and self-func tions and proving their properties.

System of Linear Differential Equations Order 1: Solving the homogeneous system using eigenvalues and eigenvectors of the matrix.

Abstract:

Derivation of the wave equation.

D’Alembert solution for an infinite string, wave bouncing from a clamped and a free end of a string.

Well-posedness. Classification of second order linear problems.

Canonical forms. Laplace equation.

Solution of the wave equation on a bounded interval by separations of variables.

Uniqueness of the solution using the energy method. The maximum principle.

Separation of variables to Laplace equation in a rectangular and in a circle.

The heat equation. The maximum principle for the heat equation.

Solution of the inhomogeneous problem.

Solution of partial differential equations using Integral transforms.

Abstract:

Interpolation and approximations methods, error analysis.

Numerical integration, differentiation.

Ordinary differential equations solution, Solutions to non-linear equations.