Review Sheet for Final Exam
Math 312
May 7, 1996

1. You should be able to define, know, etc., the following:
   1. Piecewise continuous, piecewise smooth.
   2. Fourier coefficients.
   3. Inner product, norm.
   4. Heat equation, wave equation, Laplace's equation
   5. Pointwise, uniform, and convergence.
   6. Know the integral formula for the partial sum of the Fourier series of a function.
   7. Explain why the eigenfunctions of certain second order differential operators (with boundary conditions) are orthogonal.
   8. Understand how the eigenfuctions of certain second order differential operators (with boundary conditions) are used to solve some partial differential equations. You should also understand when the separation of variables technique will not work.
   9. Fourier transform, Sine and Cosine transforms: properties of these transforms and their inversion formulas.
   10. What properties do some of the kernel functions have that enable us to infer that they behave like functions.
2. Know how to do the following:
   1. Mathematically model a heat conduction or vibrating string problem.
   2. Use the separation of variables technique to solve heat conduction or vibrating string types of problems.
   3. Compute the Fourier series of a function on the interval [-l,l].
   4. Compute the Fourier sine or cosine series of a function on the interval [0,l].
   5. Explain why the Fourier series of a ``nice'' function converges.
   6. Determine to what values, if any, the Fourier series converges.
   7. Solve various boundary value problems in more than one space variable.
   8. Determine the eigenvalues and eigenfunctions of second order linear differential operators with boundary conditions.
   9. Be able to solve nonhomogeneous heat and wave equations, with nonhomogeneous boundary conditions.
   10. Be able to solve Laplace's equation on various regions in , e.g., disks, rectangles, and annuli.
   11. Be able to solve the heat and wave equations on various regions in , e.g., disks, rectangles, and annuli, and .
   12. Use the Fourier transform to solve various partial differential equations in unbounded regions.
3. Some sample problems.
   1. Let . Graph the periodic extension of f. On what intervals is this extended function smooth? On what intervals is this extended function piecewise smooth. Find , where is the third partial sum of the Fourier series of f. Does the Fourier series converge to f uniformly? pointwise? To what values does the Fourier series converge?
   2. Find the Fourier series expansion for the function on the interval .
   3. State Bessel's inequality for Fourier series, and explain in a general setting why it is true.
   4. What is Bessel's inequality for the function on the interval ?
   5. Using separation of variables solve the problem , where k is a constant, and , for 0 < t, and u(x,0)=f(x), for 0 < x < l. Explain what you do any why the ``linearity'' of the problem is essential.
   6. Solve the problem:

      

   7. Solve the problem:

      

   8. Solve the problem:

      

   9. Solve the problem:

      

   10. Be able to verify some of the properties of the Fourier transform.

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