PDE Models of Reality

How to build mathematical models using calculus.
18 Nov 2014 to 27 Feb 2015

A partial differential equation (PDE) is a rather advanced mathematical concept, founded on concepts in real analysis, functional analysis and the calculus. It has esoteric connections to rather advanced mathematics such as distribution theory, differential geometry and even algebraic topology. Surprisingly, this seemingly esoteric mathematical concept is used to describe almost all real phenomena. Everything from how sugar dissolves in a cup of tea to the complex majesty of a lightning strike can be modeled as a PDE.

In this course, we endeavor to teach students how to build models of reality. The basic concept in getting from an intuitive understanding of reality to the PDE formulation is what is known as the “continuum approximation” — a procedure which evades a precise exposition in words, but which the student will develop a feel for by the end of this course.

Lecturer: Udayan Kanade.

Venue: Noumenon Multiphysics.

List of Lectures:

18 Nov 2014Local linearization; gradient
19 Nov 2014Tank model; rate of change, density, flux; zero inflow
20 Nov 2014Tank model; change of variables; const. inflow
21 Nov 2014Diffusion
24 Nov 2014Diffusion equation
25 Nov 2014Diffusion equation in 3D
26 Nov 2014A simple solution
27 Nov 2014Rotational invariance of gradient
28 Nov 2014Rotational invariance of the Laplacian
02 Dec 2014Differential geometry, tangent spaces and tangent bundle
04 Dec 2014Axis-independent representation of the Laplacian
05 Dec 2014Weak form formulation of ODEs
08 Dec 2014Derivation of boundary conditions using the weak form
09 Dec 2014Weak form formulation of PDEs
11 Dec 2014Vibration model; wave equation
12 Dec 2014Wave equation for heterogenous media
16 Dec 2014Replacing a finite domain with an infinite domain; phasors
17 Dec 2014Solution of wave equation using phasors
18 Dec 2014Method of separation of variables; plane waves
23 Dec 2014Solutions of initial value problem using separation of variables; Fourier transform
24 Dec 2014Solution of non-homogenous differential equations using solutions to homogenous differential equations
25 Dec 2014Diffusion with forcing function
26 Dec 2014Heat conduction with continuous heating
30 Dec 2014Non-homogeneous to homogeneous differential equation
31 Dec 2014Steady state diffusion equation with discontinous forcing function; boundary conditions obtained from weak form
01 Jan 2015Wave equation with one initial condition; non-uniqueness of solution
05 Jan 2015Wave equation with displacement and velocity initial conditions; Fourier transform method
08 Jan 2015Flux of momentum
13 Jan 2015Definitions of stress
14 Jan 2015Linearity of stress
27 Jan 2015The stress tensor
28 Jan 2015Stress tensor for uni-directional forces
29 Jan 2015Symmetry, groups, morphisms, applications to PDEs
30 Jan 2015Properties of the uni-directional stress matrix - symmetry, singularity, rank, nullity
02 Feb 2015General stress-like situations, description through intensity
04 Feb 2015Symmetry of the general stress tensor
05 Feb 2015Mechanical statics PDE
09 Feb 2015Integral form of the statics PDE
10 Feb 2015Taylor Theorem in one dimension
11 Feb 2015Taylor Theorem in multiple dimension
16 Feb 2015Conversion of Stress PDE from integral form to differential form
18 Feb 2015Torque balance of body
19 Feb 2015Torque balance gives symmetry of stress matrix
20 Feb 2015Static Fluids, stress model, pressure
23 Feb 2015PDE for static fluids, purely gravitic solutions, proofs of: liquids seek their own level, Archimedes’ principle, hydrostatic head
25 Feb 2015Galilean relativity, inertial frame of reference
27 Feb 2015Non-inertial frame of reference, pseudo-force, fluid in a rotated container - parabolic surface

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MAY 2024