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# BRINGING PDEs to life

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## María José Cano

on 9 April 2015

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#### Transcript of BRINGING PDEs to life

BRINGING
partial differential equations
TO LIFE
María José Cano
Francisco Esquembre
Eliseo Chacón
A classroom experience with a set of interactive PDE simulations.
María José Cano
Francisco Esquembre
Eliseo Chacón
Finite elements methods
Ordinary Differential Equations
Partial Differential Equations
To create Interactive Simulations on a easy way
Teaching
PDEs
Programming effort
Out-of-curriculum topics
Pedagogical challenges
Taking advantages from computers
Cons
Solutions are functions, hard to mentally visualize.
They model a physical phenomena
Sophisticated resolution

Visualization and Interactivity
How to build
this simulation?
Easy Java Simulations (EJS)
FreeFem++
Variational formulation
1. Description
2. Model
3. View
Laplace operator: elastic membrane
Diffusion operator: heat equation
Results
Allaire, G. (2007) Numerical Analysis and Optimization: An introduction to Mathematical Modelling and Numerical Simulation. OUP Oxford
Thank you for your attention
D'Alambert operator: vibrant membrane
Activities
Number of mesh points -- precision
Values of f -- Concavity/convexity
Values in Dirichlet boundary conditions: constants and analytical functions
Neumann boundary conditions
Diffusion: k resistance of the environment against the diffusion.
Transport: transport velocity and Reynold's number
Reaction: a > 0 destruction, a<0 creation of material
What values of a, d1 and k are necessary to have
a) an stable system in which the temperature of the 90% of the block is 0ºC?
b) an stable system in which the temperature of the 70% of the block is 0ºC?

Partial Differential Equations
How teachers see them
How my mother see them
How students see them
How we want to show them
Classroom experience
Detailed documentation
2h of not passive students classes
Simulations available on the web
Three models: elastic membrane,
heat equation and
vibrant membrane.
Conclusions
To analyze the solution of different cases
From a solution to guess the equations
Conclusions
Thank you for
to be continued...
EJS
Visualization
Interactivity