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:

Fourier series: expansion to Fourier series on a finite interval,

Fourier coefficients. Complex representation of Fourier series,

the convergence of the series, Dirichlet func tion,

convergence in a jump discontinuity. Gibbs phenomena.

Parseval’s identity. Differentiation and integration of Fourier series.

Fourier transform, definition, properties and the transform table.

Applications of Fourier transform in signal processing and

in solutions of differential equations. Laplace transform and its applications

in solving ordinary differential equations.

Abstract:

Complex integration : Contour Integrals. Residue theory.