Integral transforms and Partial Differential Equations
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 principle.
- 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 dimensions. If time permits: Legendre polynomials and spherical functions.
- Pinchover Y.; Rubinstein J. Introduction to partial differential equations (in Hebrew), Department 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-Verlag, Berlin, 2001
- 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.