Applications of Fourier and Laplace transforms

Introduction to partial differential equations. Differential equations which are important in applications are classified and solved. Elliptic, parabolic and hyperbolic differential equations are treated. The following mathematical tools are introduced: Laplace transforms, Fourier series, separation of variables, calculus of variation, methods of characteristics.

Mathematical tools for the engineer

Mathematical tools of an engineer

Examples of linear diffusion problems, diffusion-reaction systems: Turing instability and its application to model animal coat markings. Population models with or without diffusion.

Examples of partial differential equations. Linear partial differential equations. Separation of variables. Fourier series, Fourier transform, Laplace transform. Applications to solving commonly encountered linear partial differential equations (Laplace's Equation, Heat Equation, Wave Equation).

Semilinear parabolic and elliptic problems: existence by the method of sub- and supersolutions, finite blow up, finite vanishing time, traveling waves. Optimal bounds for solutions, critical values. Dead cores in elliptic problems.