Unit 13 of 13
Existence Constructions and Advanced Extensions
- 4 lessons
- Videos 75 to 78
The course closes with a constructive existence proof and three introductions to further study, on perturbation methods, delay differential equations and differential-algebraic equations.
Picard iteration writes an initial-value problem as an integral equation and builds the solution by successive approximation, and a contraction estimate proves convergence on a short interval. Perturbation methods expand in a small parameter, multiple scales avoid secular terms, and a boundary layer needs its own inner approximation. A delay equation needs a whole history function as initial data and is solved by the method of steps. A differential-algebraic equation joins rates with constraints that the initial data must satisfy, and differentiating a constraint can reveal a hidden one.
Every lesson will be one video. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.
Existence by Successive Approximation
Integral Equations and Picard Iteration
How to write an initial-value problem as an integral equation, build its solution by Picard iteration, prove convergence with the sup norm and a contraction estimate, and tell local from global existence.
Video 75
Perturbation, Delay and Differential-Algebraic Equations
Perturbation Methods and Multiple Scales
How a regular expansion in a small parameter is built, why secular terms and the boundary layers of singular perturbations break it, and how multiple scales and composite approximations repair it on an interval of validity.
Video 76
Delay Differential Equations
Why a delay differential equation needs a whole history function as initial data, how the method of steps builds its solution, why its state is infinite-dimensional, and how characteristic roots reveal delay-induced instability.
Video 77
Differential-Algebraic Equations
How a differential-algebraic equation joins rates with algebraic constraints, how to tell differential from algebraic variables, and how to check consistent initial values, hidden constraints and the differentiation index.
Video 78