Introduction to Partial Differetial Equations

1. The Fourier transform: convolutions, the inversion formula, Plancherel’s theorem, Hermite functions, tempered distributions. The Poisson summation formula. The Fourier transform in R^n.
2. The Laplace transform. Connections with convolutions and the Fourier transform. Laguerre polynomials. Applications to ODE’s. Uniqueness, Lerch’s theorem.
3. Classification of the second order PDE: elliptic, hyperbolic and parabolic equations, examples of Laplace, Wave and Heat equations.
4. Elliptic equations: Laplace and Poisson equations, Dirichlet and Neumann boundary value problems, Poisson kernel, Green’s functions, properties of harmonic functions, Maximum principle
5. Analytical methods for resolving partial differential equations: Sturm-Liouville problem and the method of separation of variables for bounded domains, applications for Laplace, Wave and Heat equations including non-homogenous problems. Applications of Fourier and Laplace transforms for resolving problems in unbounded domains.

Bibliography

1. Stein E. and Shakarchi R., Fourier analysis, Princeton University Press, 2003.
2. Korner T.W., Fourier analysis, Cambridge University Press, 1988.
3. Katznelson Y., An Introduction to Harmonic Analysis, Dover publications. 4. John, Partial differential equations, Reprint of the fourth edition. Applied Mathematical Sciences, 1. Springer-Verlag, New York, 1991.
4. Evans Lawrence C. Partial Differential Equations, Second Edition.
5. Gilbarg D.; Trudinger N. S. Elliptic partial differential equations of second order, Reprint of the 1998 edition. Classics in Mathematics. Springer-Ver lag, Berlin, 2001.
6. Zauderer E. Partial differential equations of applied mathematics, Second edition. Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1989. xvi+891 pp. ISBN: 0-471-61298-7.

Introduction to partial differential equations ? The first order equations: a linear equation, a quasilinear equation, resolving the initial value problem by the method of characteristic curves. ? Classification of the second order equations: elliptic, hyperbolic and parabolic equations, exam- ples of Laplace, Wave and Heat equations. ? Elliptic equations: Laplace and Poisson?s equations, Dirichlet and Neumann boundary value problems, Poisson?s kernel, Green?s functions, properties of harmonic functions, Maximum prin- ciple. ? Analytical methods for resolving partial differential equations: Sturm-Liouville problem and the method of separation of variables for bounded domains, applications for Laplace, Wave and Heat equations including non-homogenous problems. Applications of Fourier and Laplace transforms for resolving problems in unbounded domains. ? Heat equation: initial value problem in unbounded domain, basic formula for the solution, initial-boundary value problems in bounded domains, Maximum principle. ? Wave equation: Dalamber formula, non-homogenous equation, Wave equation in higher dimen- sions. ? If time permits: Legendre polynomials and spherical functions. Literature: ? Pinchover Y.; Rubinstein J. Introduction to partial differential equations (in Hebrew), Depart- ment of mathematics, Technion, 2011, ? John F. Partial differential equations, Reprin t of the fourth edition. Applied Mathematical Scien ces, 1. Springer-Verlag, New York, 1991, ? Evans Lawrence C. Partial Differential Equations, Second Edition, ? Gilbarg D.; Trudinger N. S. Elliptic partial differential equations of second order, Reprint of the 1998 edition. Classics in Mathematics. Springer-Ver lag, Berlin, 2001, ? Zauderer E. Partial differential equations of applied mathematics, Second edition. Pure and Appli ed Mathematics (New York). A Wiley-Interscience Public ation. John Wiley & Sons, Inc., New York, 1989. xvi+891 pp. ISBN: 0-471-61298-7. 1