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+ | ====== Mathematical Toys ====== | ||

+ | * Develop toy mathematical models of physical phenomena | ||

+ | * Focus on differential and difference equations | ||

+ | * Each toy model accompanied by numerical and analytical insight | ||

+ | |||

+ | ===== Fluid Mechanics ===== | ||

+ | |||

+ | * [[mathtoys:brownian|From Brownian motion to diffusion]] | ||

+ | * Spreading droplets | ||

+ | * Dripping paint | ||

+ | * A traffic jam (telegraph and Burger's) | ||

+ | * Transition to turbulence (Orr-Sommerfeld) | ||

+ | * Ship waves | ||

+ | * Porous flow, oil extraction, and viscous fingering | ||

+ | * Capillary and gravity waves on a stream | ||

+ | * Shock waves | ||

+ | * Flight, Kutta-Joukowski, conformal mapping | ||

+ | * Coriolis effect | ||

+ | * Non-dispersive water waves (Boussinesq) | ||

+ | |||

+ | |||

+ | ===== Solid Mechanics ===== | ||

+ | |||

+ | * Waves on a string | ||

+ | * Wrinkling and buckling | ||

+ | * Crack propogation | ||

+ | * Earthquakes: P-waves and S-waves | ||

+ | * Vibrational modes of a plate (biharmonic equation) | ||

+ | |||

+ | ===== Dynamics ===== | ||

+ | * [[mathtoys:rotatinghoop|Rotating hoops and sliding beads]] | ||

+ | * [[mathtoys:eulerwobble|Euler wobble and free rotations]] | ||

+ | * Tops | ||

+ | |||

+ | ===== Biology, chemistry, and ecology ===== | ||

+ | * The leopard and its spots (Gray-Scott) | ||

+ | * Metastability (Allen-Cahn) | ||

+ | * Chemical systems | ||

+ | * Modeling the brain | ||

+ | * Predator prey systems | ||

+ | * Spinodal decomposition (Cahn-Hilliard) | ||

+ | ===== Light and electromagnetism ===== | ||

+ | * Shortest distances and geodesics | ||

+ | * Airy equation, rays, caustics | ||

+ | * Scattering light and Helmholtz | ||

+ | * Green's functions and method of images | ||

+ | * Superconductivity (Ginzburg-Landau) | ||

+ | |||

+ | ===== Economics and finance ===== | ||

+ | |||

+ | * [[:mathtoys:blackscholes|Black-Scholes model]] | ||

+ | |||

+ | ===== Sociology and psychology ===== | ||

+ | |||

+ | ===== Computers and technology ===== | ||

+ | * Image sharpening (Backwards heat equation, Perona-Malik) |