In this course, we study elliptic Partial Differential Equations (PDEs) with variable …
In this course, we study elliptic Partial Differential Equations (PDEs) with variable coefficients building up to the minimal surface equation. Then we study Fourier and harmonic analysis, emphasizing applications of Fourier analysis. We will see some applications in combinatorics / number theory, like the Gauss circle problem, but mostly focus on applications in PDE, like the Calderon-Zygmund inequality for the Laplacian, and the Strichartz inequality for the Schrodinger equation. In the last part of the course, we study solutions to the linear and the non-linear Schrodinger equation. All through the course, we work on the craft of proving estimates.
Fall: Fundamental solutions for elliptic, hyperbolic and parabolic differential operators. Method of …
Fall: Fundamental solutions for elliptic, hyperbolic and parabolic differential operators. Method of characteristics. Review of Lebesgue integration. Distributions. Fourier transform. Homogeneous distributions. Asymptotic methods. Spring: Sobolev spaces. Fredholm alternative. Variable coefficient elliptic, parabolic and hyperbolic linear partial differential equations. Variational methods. Viscosity solutions of fully nonlinear partial differential equations. The main goal of this course is to give the students a solid foundation in the theory of elliptic and parabolic linear partial differential equations. It is the second semester of a two-semester, graduate-level sequence on Differential Analysis.
This 10-minute video lesson provides an introduction to 2nd order, linear, homogeneous …
This 10-minute video lesson provides an introduction to 2nd order, linear, homogeneous differential equations with constant coefficients. [Differential Equations playlist: Lesson 13 of 45]
This 8-minute video lesson continues the discussion from the previous video and …
This 8-minute video lesson continues the discussion from the previous video and shows how to find the general solution. [Differential Equations playlist: Lesson 14 of 45]
This 6-minute video lesson continues the discussion from the previous videos and …
This 6-minute video lesson continues the discussion from the previous videos and introduces some initial conditions to solve for the particular solution. [Differential Equations playlist: Lesson 15 of 45]
This 9-minute video lesson continues the discussion from the previous videos and …
This 9-minute video lesson continues the discussion from the previous videos and provides another example using initial conditions. [Differential Equations playlist: Lesson 16 of 45]
This 10-minute video lesson looks at what happens when the characteristic equations …
This 10-minute video lesson looks at what happens when the characteristic equations has complex roots. [Differential Equations playlist: Lesson 17 of 45]
This 10-minute video lesson continues the discussion of what happens when the …
This 10-minute video lesson continues the discussion of what happens when the characteristic equation has complex roots. [Differential Equations playlist: Lesson 18 of 45]
This 10-minute video lesson concludes the series on complex roots of characteristic …
This 10-minute video lesson concludes the series on complex roots of characteristic equations and demonstrates an example with initial conditions. [Differential Equations playlist: Lesson 19 of 45]
This 10-minute video lesson provides intuiton on exact equations including the chain …
This 10-minute video lesson provides intuiton on exact equations including the chain rule using partial derivatives (not a proof; more intuition). [Differential Equations playlist: Lesson 4 of 45]
The laws of nature are expressed as differential equations. Scientists and engineers …
The laws of nature are expressed as differential equations. Scientists and engineers must know how to model the world in terms of differential equations, and how to solve those equations and interpret the solutions. This course focuses on the equations and techniques most useful in science and engineering.
This 8-minute video lesson provides another example of using substitution to solve …
This 8-minute video lesson provides another example of using substitution to solve a first order homogeneous differential equations. [Differential Equations playlist: Lesson 12 of 45]
This 8-minute video lesson concludes the previous video on integrating factors by …
This 8-minute video lesson concludes the previous video on integrating factors by solving the equation. [Differential Equations playlist: Lesson 10 of 45]
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