The course introduces students to mathematical methods widely used in modern engineering and applied sciences. It consists of three main parts: 1) Methods of Applied Linear Algebra (solving linear systems, LU, QR, SVD, and other factorizations, Principal Component Analysis, iterative methods, FFT, least squares, pseudo-inverse, etc.); 2) Statistical Methods and Data Analysis (mean, variance, probability; moments, covariance, Gaussian processes, Markov chains; regression, gradient descent; neural networks, machine learning); and 3) Applied Differential Equations (linear and nonlinear ordinary differential equations, stability and bifurcations of solutions, linear and nonlinear partial differential equations, hyperbolicity, characteristics, dispersion, reaction-diffusion phenomena, pattern formation).
The course is introductory by nature, covering a wide range of topics and methods of modern interest in applications. Its theoretical content is informal in style and most of the concepts will be illustrated with problems from engineering, physics, chemistry, and biology using numerical computations in Matlab and Python. The three parts of the course are aimed at: part 1 - using the right language that is crucial for understanding many computational techniques used in engineering; part 2 - learning important tools of analysis of results obtained either by computation or in experiments; and part 3 - learning the nature of key mathematical models that form the foundation of engineering and applied sciences.