syllabus | timetable | problems | notes

## ASTR 5540 Mathematical Methods Spring 2012: Syllabus

### Content

The APS department has agreed on a model syllabus (pdf) for Mathematical Methods. The department recommends that the course include an emphasis on numerics and “practical” techniques that you should find useful in your research. We will use Mathematica both in class and in problem sets.
1. Root-finding. Polynomial interpolation, Simpson's rule. Spline. Rational approximation.
2. Numerical integration of ODEs. Runge-Kutta. Stability, explicit/implicit methods, stepsize control.
3. Ist order ODEs. Analytic methods. Linear ODEs, Green function.
4. 2nd and higher order ODEs. Linear ODEs, homogeneous and particular solutions, Green function, Wronskian. Series solution.
5. Eigenfunction methods for linear differential equations. Fourier transform, spherical transform, Bessel transform. Separation of variables. Hilbert space, orthogonality, Hermitian operators.
6. Recurrence relations, stability.
7. Fourier series, continuous Fourier transform, discrete Fourier transform, FFT, convolution. Harmonic oscillator, WKB. Correlation function, power spectrum, shot noise.
8. Particle-mesh algorithm.
9. Fluid equations, perturbations, sound waves. Shocks, self-similarity, Riemann problem. ID Lagrangian hydro code.
10. 1st order PDE characteristics. Characteristics of fluid equations.
11. 2nd order PDE characteristics. Boundary conditions, hyperbolic, parabolic, elliptic. Wave equation, diffusion equation, Schrödinger equation, Poisson equation.
12. Numerical solution of PDEs. Discretization. Accuracy, stability. Leap frog, Lax-Wendroff, Crank-Nicolson.

### Texts

There is no required text, since no single text encompasses the mix of analytics and numerics covered in the course. Books I have found most useful myself are:

### Problem Sets

Problem sets will be assigned (almost) every week, and will usually be due the following Tuesday. They are intended to be frequent, and not too long. There will be no problem sets in the last two weeks of the semester.

### Projects

Each of you will carry out a numerical project, and make a short presentation of it during the last two weeks of classes.