**MTH ****405: Partial Differential Equations**** ****(4)**

**Pre-requisites**: MTH 306 Ordinary Differential Equations

*Learning Objectives*:

This is an introductory course in partial differential equations for students majoring in mathematics. After discussing the solutions of first-order linear and quasi-linear equations in considerable detail we introduce the Cauchy problem for first and higher order equations and then briefly discuss the Cauchy-Kovalevski existence theorem and Holmgren's uniqueness theorem. We follow this by a study of second-order linear equations; here the goal is to understand the solutions of the three prototypical equations, Laplace, Wave and the Heat equation, in the classical set-up.

*Course Contents*:

First-order equations: linear and quasi-linear equations, general first-order equation for a function of two variables, Cauchy problem, envelopes

Higher-order equations: Cauchy problem, characteristic manifolds, real analytic functions, Cauchy-Kovalevski theorem, Holmgren’s uniqueness theorem

Laplace equation: Green’s identity, Fundamental solutions, Poisson’s equation, Maximum principle, Dirichlet problem, Green’s function, Poisson’s formula

Wave equation: spherical means, Hadamard’s method, Duhamel’s principle, the general Cauchy problem

Heat equation: initial-value problem, maximum principle, uniqueness, regularity

*Suggested Books*:

- F. John,
*Partial differential equations*, 4th edition, Springer, 1982 - G. B. Folland,
*Introduction to Partial differential equations*, 2nd edition, Princeton University Press, 1995 - J. Rauch,
*Partial differential equations*, Springer, GTM 128, 1991 - L. Evans,
*Partial differential equations,*American Mathematical Society GSM series, 1998

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