# 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 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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