Learning Outcomes
Upon completion of the course the students should be able to:
1. Solve linear differential equations of second or higher order and systems of 1st order linear differential equations.
2. Use Laplace transform (its properties and properties of inverse Laplace transform) for solving physical problems, e.g. circuits.
3. Perform calculations with complex numbers and use elementary complex functions as a tool of geometric transformations. Also, use complex integration as a tool for qualitative study of two dimensional vector fields (circulation-flux, complex potential). Finally, uo use complex analysis techniques as a tool in signal processing and PDE’s.
4. Use methods of Fourier Analysis (Fourier series and Fourier integrals) for solving two dimensional Dirichlet's prolem inside a rectangular, or a disk or in the upper half plane, or one dimensional heat and wave equations by applying the method of separation of variables.
Course Content (Syllabus)
Linear differential equations of second and higher order: definitions and solutions. A short introduction to non linear differential equations. Systems of differential equations. Applications.
Laplace transform and inverse Laplace transforms. Properties and applications.
Elements of complex analysis: complex derivative and integration, Cauchy’s theorem and Cauchy’σ integral formula, Laurent series, residuals, harmonic functions.
A short introduction to Fourier Analysis: Fourier series and Fourier integral.
A short introduction to partial differential equations. The method of separation of variables. Two dimensional Dirichlet problem iside a rectangular, disk, or upper half plane. One dimensional heat and wave equations.
Keywords
Differential equations, Laplace transform, complex numbers, Fourier Analysis, method of separation of variables.