Mathematics / advanced
Equations for distributed change
Classify and solve model PDEs, then examine weak formulations, energy estimates, characteristics, and numerical discretizations.
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Module 01 advanced
Equations for distributed change
Classify and solve model PDEs, then examine weak formulations, energy estimates, characteristics, and numerical discretizations.
- 01
PDE classification and well-posedness
Read the highest derivatives, specify the data, and decide what a meaningful solution claim must include.
30 min - 02
Characteristics and transport
Follow information along curves and locate the point where a smooth transport description can fail.
30 min - 03
Heat equation and smoothing
Derive diffusion from flux, evolve Fourier modes, and separate smoothing from loss of total heat through a boundary.
32 min - 04
Wave equation and energy
Follow finite-speed propagation and use energy flux to understand reflection, uniqueness, and stable initial data.
32 min - 05
Laplace equation and boundaries
Use mean values, flux compatibility, and boundary data to understand harmonic equilibria and their uniqueness.
31 min - 06
Weak solutions and Sobolev spaces
Move derivatives onto test functions and distinguish a distributional derivative from an integrable weak derivative.
35 min - 07
Variational methods and finite elements
Turn a coercive weak problem into a finite-dimensional approximation and inspect exactly what the discrete system enforces.
36 min - 08
Conservation laws and shocks
Continue a nonlinear balance law beyond characteristic crossing and use entropy to choose the physically admissible weak solution.
36 min - 09
Finite differences: consistency and stability
Separate local truncation error from error amplification and apply the Lax equivalence principle within its actual hypotheses.
36 min - 10
Nonlinear PDEs and existence questions
Separate local existence, global continuation, weak limits, and regularity, and test broad claims against explicit nonlinear examples.
38 min