Difference between revisions of "MAT 5673"
m (→Topics List) |
|||
Line 88: | Line 88: | ||
Introduction to Fourier series | Introduction to Fourier series | ||
|| | || | ||
− | + | Infinite series | |
|| | || | ||
* Orthonormal systems of functions, spectral method for the wave and heat equation | * Orthonormal systems of functions, spectral method for the wave and heat equation | ||
Line 98: | Line 98: | ||
Schroedinger equation | Schroedinger equation | ||
|| | || | ||
− | + | Complex numbers | |
|| | || | ||
* Basic properties of Schroedinger equation, particle in a potential well | * Basic properties of Schroedinger equation, particle in a potential well | ||
Line 104: | Line 104: | ||
|Week 10 | |Week 10 | ||
|| | || | ||
− | + | Complex numbers | |
|| | || | ||
Qualitative properties of PDE's | Qualitative properties of PDE's | ||
Line 118: | Line 118: | ||
Introduction to numerical methods for PDE (optional) | Introduction to numerical methods for PDE (optional) | ||
|| | || | ||
− | + | Derivatives, Calculus | |
|| | || | ||
* Basic finite difference schemes for first-order quasilinear equations, CFL condition | * Basic finite difference schemes for first-order quasilinear equations, CFL condition | ||
Line 124: | Line 124: | ||
|Week 12 | |Week 12 | ||
|| | || | ||
− | + | Matrices, Linear Algebra | |
|| | || | ||
Introduction to the Laplace and Poisson equation | Introduction to the Laplace and Poisson equation | ||
Line 138: | Line 138: | ||
Introduction to the Calculus of Variations | Introduction to the Calculus of Variations | ||
|| | || | ||
− | Differentiation of an integral with respect to a parameter | + | Differentiation of an integral with respect to a parameter, parametric surfaces |
|| | || | ||
* Compute the variational derivative of a functional | * Compute the variational derivative of a functional |
Revision as of 07:21, 23 March 2023
Course description
Partial differential equations arise in many different areas as one tries to describe the behavior of a system ruled by some law. Typically, this has to do with some physical process such as heat diffusion in a material, vibrations of a bridge, circulation of fluids, the behavior of microscopic particles or the evolution of the universe as a whole. Modeling by means of partial differential equations has been successful in other disciplines as well, like in the case of the Black-Scholes equation for stock options pricing and the Hodgkin–Huxley equations for firing patterns of a neuron. Partial differential equations are an important tool in Applied Math and Pure Math. This course gives an introduction to PDE's in the setting of two independent variables.
Topics List
Date | Sections | Topics | Prerequisite Skills | Student Learning Outcomes |
---|---|---|---|---|
Week 1 |
|
Introduction and classification of PDE, Calculus review |
Multivariable Calculus, Chain Rule |
|
Week 2 |
|
Applied examples of PDE |
Multivariable Calculus, Chain Rule |
|
Week 3 |
|
The method of characteristics for first-order quasilinear equations |
Multivariable Calculus, Chain Rule |
|
Week 4 |
|
The method of characteristics for first-order fully nonlinear equations |
Multivariable Calculus, Chain Rule |
|
Week 5 |
|
Heat and wave equation on the whole real line |
Differentiation of integrals with respect to a parameter, integration by parts |
|
Week 6 |
|
Initial-boundary value problem for heat and wave equation I |
Partial derivatives, chain rule |
|
Week 7 |
|
Initial-boundary value problem for heat and wave equation II, introduction to Fourier series |
Partial derivatives, chain rule |
|
Week 8 |
|
Introduction to Fourier series |
Infinite series |
|
Week 9 |
|
Schroedinger equation |
Complex numbers |
|
Week 10 |
Complex numbers |
Qualitative properties of PDE's |
Differentiation of integrals with respect to parameter |
|
Week 11 |
|
Introduction to numerical methods for PDE (optional) |
Derivatives, Calculus |
|
Week 12 |
Matrices, Linear Algebra |
Introduction to the Laplace and Poisson equation |
|
|
Week 13 |
|
Introduction to the Calculus of Variations |
Differentiation of an integral with respect to a parameter, parametric surfaces |
|
Week 14 |
|
Review, advanced topics |
|
|