FROM MATHEMATICS
TO REALITY

# 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.

Venue: Noumenon Multiphysics.

List of Lectures:

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

This lecture series occurred in the past. If you wish to hold a similar lecture series at your venue, please send an email to training@noumenonmp.com.

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