Announcements

Lecture No	Date	Content	Corresponding Reading material
1	Feb. 14	Review of ODEs: Order, linear ODEs, Linear operators, superposition of solutions, general solution of non-homogeneous linear ODEs; PDEs: Order, linear PDEs, some simple exactly solvable examples and their general solutions.	Pages 1-4 of Strauss
2	Feb. 16	Directional derivative and its meaning. First order linear PDEs and the method of characteristics.	Pages 6-9 of Strauss
3	Feb. 21	The solution of the most general first order linear PDE; Typical examples of second order linear PDEs.	10-Pages 10-20 of Strauss
4	Feb. 23	Classification of 2nd order linear PDEs; Well-posed problems	10-Pages 28-31 of Strauss
5	Feb. 28	Initial and boundary conditions; D'Alembert's solution of the wave equation in 1+1 dimensions,	10-Pages 20-25 & 33-37 of Strauss
6	Mar. 02	Causality and energy conservation for the 1+1 dimensional wave equation; Diffusion equation in 1+1 dimensions, Maximum principle with the proof of its weak version, the uniqueness theorem for solution of the Dirichlet problem for general non-homogeneous diffusion equation on a closed interval	10-Pages 20-25 & 39-44 of Strauss
7	Mar. 07	Another proof of the uniqueness theorem for solution of the Dirichlet problem for general non-homogeneous diffusion equation on a closed interval, Two different proofs of the stability of the solution of the Dirichlet problem for homogeneous diffusion equation with vanishing boundary conditions on a closed interval, Solution of the diffusion equation on R.	10-Pages 44-49 of Strauss
8	Mar. 09	Derivation of the solution of the diffusion equation on R using the method of Fourier transform, Solution for the initial condition being a box, an exponential function, and a Gaussian, General property of the solutions: causality and stability	Pages 49-52 of Strauss
9	Mar. 14	Solution of the diffusion equation on the half line with Dirichlet and Neumann boundary conditions at x=0; Solution of the diffusion equation with a source on the real line; Boundary-value/eigenvalue problem for -d^2/dx^2 on a closed interval with Dirichlet, Neumann, and Robin boundary conditions	Pages 57-61, 67-70 of Strauss
10	Mar. 16	Solution of the boundary-value/eigenvalue problem for -d^2/dx^2 on a closed interval with Dirichlet and Neumann boundary conditions. Solution of the wave equation for the vibrating string using the method of separation of variables; Fourier sine series.	Pages 84-89 and 104-105 of Strauss
11	Mar. 21	Solution of the diffusion equation with Neumann boundary conditions; Fourier sine and cosine series as odd and even extensions of the initial condition given in [0,L] to [-L,L] and periodic extensions of it to the whole real line.	Pages 105-115 of Strauss
Midterm Exam 1	Mar. 28
12	Mar. 30	General real and complex Fourier series; L^2-inner product and orthogonal functions; symmetric boundary conditions and symmetric Schroedinger operators acting on functions defined on a closed interval; Convergence of the Fourier series	Pages 115-131 of Strauss
Spring Break	Apr. 4-8
13	Apr. 11	Laplace's and Poisson's equations; Maximum Principle for Laplace's equation; Uniqueness theorem for the solution of the Dirichlet boundary-value problem for the Poisson's equations; Solution of the Laplace's equation in a rectangular region in the plane using the method of separation of variables.	Pages 152-162 of Strauss
14	Apr. 13	Solution of a Dirichlet problem for Laplace's equation in a rectangular parallelepiped using the method of separation of variables. Solution of the general Dirichlet problem for Laplace's equation on a disc in plane: Derivation of Poisson's formula.	Pages 163-168 of Strauss
15	Apr. 18	Mean-value property of maximum principle as corollaries of the Poisson's formula; Solution of the general Dirichlet problem for Laplace's equation on a circular wedge, annulus, and outside of a disc.	Pages 168-175 of Strauss
16	Apr. 20	Divergence theorem, Green's First Identity, a non-existence result for Neumann problem for the Poisson equation; Mean-value property and maximal principle for harmonic functions in R^3, A proof of the uniquenss theorem for Dirichlet problem for the Laplace equation using an energy argument; Dirichlet's principle	Pages 178-183 of Strauss
17	Apr. 25	Green's Second Identity, a representation formula for harmonic functions, Green's functions for the Laplace operator, general solution of the Dirichlet problem for Laplace and Poisson equations in terms of the Green's function, Green's function for half-space.	Pages 185-192 of Strauss
18	Apr. 27	Solution of the wave equation in three-dimensional Euclidean space: Derivation of Kirchhoff's formula	Pages 234-238 of Strauss
19	May 02	Calculus of Variations: Stationary, local minimum and maximum points of a differentiable function of several variables; Historical examples of variational problems: Catenary, Brachystochrone, Hamilton's principle and Kepler's problem, geodesics of a smooth surface	Pages 1-21 of van Brunt
20	May 04	Variational problems with a single variable and fixed boundary conditions: Review of Taylor series expansion for real-valued functions of several real variables, first variation and the Euler-Lagrange equation, Applications: geodesics in plane, a variational problem with no solution.	Pages 23-36 of van Brunt
21	May 09	Some special functionals: No explicit y or x-dependence of the integrand, examples: Solutions of the Catenary and Brachystochrone problems, Geodesics on a sphere	Pages 36-41 of van Brunt
Midterm Exam 2	May 11
22	May 16	A degenerate variational problem, invariance of Euler-Lagrange equation under coordinate transformations, generalization of the Euler-Lagrange equation to higher order derivatives	Pages 42-48 & 55-57 of van Brunt
23	May 18	Derivation of the invariant of x-independent 2nd order derivative variational problem; Variational problems with more than one dependent or independent variables: Derivation of the Euler-Lagrange equations, examples	Pages 57-69 of van Brunt
24	May 23	Isoperimetric variational problems: The finite-dimensional case (Method of Lagrange multipliers), examples	Pages 73-79 of van Brunt
25	May 25	Solution of the isoperimetric variational problems using the method of Lagrange multipliers, application to Dido's problem	Pages 83-93 of van Brunt

Note: The pages from the textbook and the reading material listed above may not include some of the subjects covered in the lectures.