Course information

General on models: Types, reliability, construction. Mechanical models. Population dynamics (single-species, multi-species, Lotka-Volterra competitive models). Lanchester combat models. Ecological – Biological models. Dimensional Analysis: Buckingham s pi theorem. Normalization. Perturbation Methods: Model construction, regular and singular perturbation, boundary layer analysis. Calculus of Variations: Model construction, variational problems (brachistochrone), Euler – Lagrange equation, Hamilton s principle, isoperimetric problems, geodesics. Traffic models. Elliptic problems: Gravitational field. Electromagnetism. Acoustics. Electrochemical plating. Hyperbolic problems: Traveling waves. Telegraph equation, pantograph. Scattering. Parabolic problems: Electromagnetism. Heat and mass transfer. Probabilistic heat model. Economic model. Wave phenomena in continuous media: Linear and nonlinear waves. Burger s, KdV equations, mathematical models of continuous media. Stochastic models. Prerequisite knowledge: Mathematical Analysis, Differential Equations, Mathematica, Matlab. Convex sets and convex functions. Fréchet derivatives and directional derivatives. Extrema. Existence and uniqueness theorems. Basic necessary and sufficient conditions for optimality. Lagrange and Kuhn-Tucker-Lagrange multiplier theorems. Quadratic functions. Least Squares Methods and applications. Golden Section, Gradient, Conjugate Gradient, Newton, Frank-Wolfe, Projected Gradient, Penalty, Gradient-Penalty Methods. Applications in Optimal Control.