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{\large \bf Partial Differential Equations, Math 436, Fall 1996}
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{\bf Instructor:} Mark Alber
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{\bf Text:}
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Mark A. Pinsky, Partial Differential Equations and Boundary-Value Problems
with Applications, Second Addition, McGraw-Hill.
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{\bf Syllabus:}
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1. Short review of ODE's and dynamical systems
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2. Examples of PDE's with applications. Mathematical modeling.
   Linear second-order PDE's of hyperbolic, elliptic and parabolic type.
   Laplace's equation, Wave equation, Heat equation,
   Telegraph equation, Poisson equation.
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3. Initial and boundary conditions. Method of separation of variables.
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5. Functional series: power series, Taylor series and Furier series.
   Uniform confergense.
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6. Sturm-Liouville eigenvalue and eigenfunction problem. Orthogonal complete
   sets of functions. Bessel's inequality.
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7. The Fourier theorem. Generalized Fourier Series. Gibb's phenomenon.
   Double Fourier Series.  Fourier series in cylindrical and spherical
   coordinates. Poisson integral formula.
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8. Vibrating string. Fundamental frequencies and harmonics. Resonance.
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9. The d'Alembert solution.
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10. Fourier series method for the Heat, Wave and Laplace equations.
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11. Fourier transform.  Application to the the Heat and Wave equations.
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12. Laplace transform. The complex inversion integral.
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13. Generalized functions. Delta function.
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14.  Green's function for ODE's. Green's function for PDE's.
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15. Nonlinear integrable PDE's and Inverse Scattering Transform (IST) method.
    Periodic and soliton solutions of the KdV equation.
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16. Shock solutions of the nonlinear PDE's.
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17. Numerical methods for PDE's.

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