Ordinary Differential Equations
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The usual caveat applies: at the time of recording, inevitably, minor typographical errors were also incorporated. These have not been corrected within the video files. However, the PDF slides corresponding to each lecture can be downloaded further below. There, known errors have been corrected. Further corrections may be sent to horst@sjtu.edu.cn.
7. Differential Operators and Types of Solutions
The general form of a linear differential operator. The conjunct and the formal adjoint. Green's formula and Lagrange's identity. Classical and weak solutions to differential equations.
8. Causal Fundamental Solutions and Initial Value Problems
Fundamental solution to ODEs. Causal fundamental solutions and how to find them. A review of initial value problems, including existence and uniqueness of solutions and independence of solutions via the Wronskian. A general solution formula.
9. Second-Order Boundary Value Problems for ODEs
Some basic methods for solving boundary value problems for second-order equations with different types of boundary conditions.
10. Adjoint Boundary Value Problems
Adjoint boundary value problems and Green functions. A new perspective on the solution formula. Higher-order equations.
11. Solvability Conditions and Modified Green's Functions
Not every boundary value problem is solvable. A condition for the existence and uniqueness of solutions. Problems that do not always have a solution may not have a Green function. So-called modified Green functions can then be constructed and used in solution formulas instead.