## Applied partial differential equations

• ECTS credits: 6
• Semester: 1
• University: University of L'Aquila
• Objectives:

Students will know basic of properties (existence, uniqueness, etc.) and techniques (characteristics, separation of variables, Fourier methods, Green's functions, similarity solutions, etc.) to solve basic PDEs (conservation laws, heat, Laplace, wave equations).

• Topics:

Integral curves and surfaces of vector fields. First order partial differential equations. Linear and quasi linear partial differential equations (PDEs) of first order. Method of characteristics. The initial value problem: existence and uniqueness. Development of shocks.

The Cauchy-Kovalevsky theorem. Linear partial differential operators and their characteristic curves and surfaces. Methods for finding characteristic curves and surfaces. The initial value problem for linear first order equations in two independent variables. Holmgren's uniqueness theorem. Canonical form of first order equations. Classification and canonical forms of second order equations in two independent variables. Second order equations in two or more independent variables. The principle of superposition.
The divergence theorem and the Green's identities. Equations of Mathematical Physics.

LAPLACE'S EQUATION AND HARMONIC FUNCTIONS Elementary harmonic functions. Separation of variables. Inversion with respect to circles and spheres. Boundary value problems associated with Laplace's equation. Representation theorem. Mean value property. Maximum principle. Harnack’s inequality and Liouville’s theorem. Well-posedness of the Dirichlet problem. Solution of the Dirichlet problem for the unit disc. Fourier series and Poisson's integral. Analytic functions of a complex variable and Laplace's equation in two dimensions. The Neumann problem.

GREEN'S FUNCTIONS. Solution to the Dirichlet problem for a ball in three dimensions. Further properties of harmonic functions. The Dirichlet problem in unbounded domains. Method of electrostatic images.

THE WAVE EQUATION. Cauchy problem. Energy method and uniqueness. Domain of dependence and range of influence. Conservation of energy. One-dimensional wave equation. D’Alembert formula. Characteristic parallelogram. Non homogeneous equation and Duhamel’s method. Multi-dimensional wave equation. Well posed problems. Fundamental solution (n=3) and strong Huygens’ principle. Kirchhoff formula. Method of descent. Poisson?s formula (n=2). Wave propagation in regions with boundaries. Uniqueness of solution of the initial-boundary value problem. Separation of variables. Reflection of waves.

THE HEAT EQUATION. Heat conduction in a finite rod. Maximum principle and applications. Solution of the initial-boundary value problem for the one dimensional heat equation. Method of separation of variables. The initial value problem for the one dimensional heat equation. Fundamental solution. Non homogeneous case and Duhamel’s method. Heat conduction in more than one space dimension.

• Books:

E. C. Zachmanoglou and Dale W. Thoe, lntroduction to Partial Differential Equations with Applications. Dover Publications, Inc.. 1986. ISBN 0-486-65251-3

L.C. Evans, Partial Differential Equations. American Mathematical Society. 2010. Second edition, ISBN-13: 978-0821849743

S. Salsa, Partial Differential Equations in Actions: from Modelling to Theory. Springer-Verlag Italia. 2008. ISBN 978-88-470-0751-2

W. A. Strauss, Partial Differential Equations, Student Solutions Manual: An Introduction. John Wiley & Sons, LTD. 2008. Second edition, ISBN-13: 978-0470260715

W. A. Strauss, Partial Differential Equations: an introduction. John Wiley & Sons, LTD. 2007. Second edition, ISBN-13 978-0470-05456-7