Course Details:
Introduction, PDEs as mathematical models, first order PDEs: method of characteristics, quasilinear first order PDEs.(8 lectures)
Classification of second order PDEs, Laplace equation: fundamental solution, mean-value formulas, The maximum principle, Poisson equation, properties of harmonic functions, Green’s functions, energy methods.(12 lectures)
Heat equation: Diffusion, Fourier transforms, energy methods. (6 lectures)
Wave equation: one-dimensional wave equation, d'Alembert’s formula, higher-dimensional wave equation, energy estimates. (8 lectures)
Fourier series, boundary value problems, separation of variables. (8 lectures)
Learning Outcomes:
1. Explain concepts and theory of basic methods for solving various types of partial differential
equations.
2. Obtain energy estimates for solutions of several PDEs.
3. Use Green functions to construct solutions of several nonhomogeneous PDEs.
4. Prove properties of harmonic functions using mean-value property.
5. Use the maximum principle to prove a priori estimates and to derive gradient estimates.
6. Apply the Fourier method to solve nonhomogeneous versions of heat and wave equations.
Text/Reference Books:
Text Books:
1. Walter A. Strauss, Partial Differential Equations: An Introduction, John Wiley & Sons; 2nd
edition, 2008, ISBN-10: 0470054565, ISBN-13: 978-0470054567
2. Lawrence C. Evans, Partial Differential Equations, American Mathematical Society; 2nd Revised
edition, 2010, ISBN-10: 0821849743, ISBN-13: 978-0821849743
3. Qing Han, A Basic Course in Partial Differential Equations, American Mathematical Society;
2011, ISBN-10: 0821852558, ISBN-13: 978-0821852552
References:
1. András, Vasy, Partial Differential Equations: An Accessible Route through Theory and
Applications: American Mathematical Society, 2015, ISBN-10: 1470418819, ISBN-13: 978-
1470418816
2. Gerald B. Folland, Introduction to partial differential equations, Princeton University
Press, Princeton, NJ, second edition, 1995.
3. Michael E. Taylor. Partial differential equations: Basic theory, Texts in Applied Mathematics, Vol.
23, Springer-Verlag; 1996, ISBN-10: 0387946543, ISBN-13: 978-0387946542
4. Yehuda Pinchover and Jacob Rubinstein, An Introduction to Partial Differential Equations,
Cambridge University Press; 2005, ISBN-10: 052161323X, ISBN-13: 978-0521613231
5. Fritz John, Partial Differential Equations (Applied Mathematical Sciences), Springer; 4th ed.,
1991, ISBN-10: 0387906096, ISBN-13: 978-0387906096
6. Jurgen Jost, Partial Differential Equations, Graduate Texts in Mathematics 214, Springer; 2007,
ISBN-10: 1441923802, ISBN-13: 978-1441923806
7. Hörmander, L. Lectures on Nonlinear Hyperbolic Differential Equations, Springer; 1997, ISBN-
10: 3540629211, ISBN-13: 978-3540629214
8. Courant, R. and Hilbert, D. Methods of mathematical physics. Vol. II: Partial differential
equations, Wiley VCH; 1989, ISBN-10: 9780471504399, ISBN-13: 978-0471504399
9. Michael Taylor, Partial Differential Equations I: Basic Theory; Springer, 2011, ISBN-10:
1461427266, ISBN-13: 978-1461427261