Partial Differential Equations

Code:

M4038

Acronym:

M4038

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2018/2019 - 1S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Master in Mathematical Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
M:A_ASTR	5	Plano de Estudos oficial desde_2013/14	1	-	6	56	162
			2
M:ENM	4	Official Study Plan since 2013-2014	1	-	6	56	162
			2

Teaching language

Portuguese

Objectives

This course is intended for students to acquire basic knowledge of the theory and numerical treatment of partial differential equations.

Learning outcomes and competences

It is expected that on completing this curricular unit the student will know:

a. to solve first order partial differential equations by the method of characteristics;

b. to solve second semilinear and quasilinear partial differential equations;

c. to apply Cauchy-Kovalevskaya’s theorem to a Cauchy problem;

d. to use numerical methods for solving partial differential equations.

Working method

Presencial

Program

1. First order equations: method of characteristics, existence and uniqueness of solution of the Cauchy problem (quasilinear and general cases), geometrical solutions and complete integral.

2. Second order equations: semilinear equations (parabolic, hyperbolic, elliptic) and reduction to canonical form, quasilinear equations, method of characteristics for the Cauchy problem, propagation of singularities, mixed Cauchy problem for the wave equation.

3. Systems of first order edp's: characteristics, canonical form, existence and uniqueness of solution for hiperbolic semi-linear systems. Hyperbolic quasi-linear systems.

4. Cauchy-Kovalevskaya theorem.

5. Numerical treatment (finite differences, finite elements) of partial differential equations.

Mandatory literature

LeVeque Randall J.; Finite difference methods for ordinary and partial differential equations. ISBN: 978-0-898716-29-0

Complementary Bibliography

Quarteroni Alfio; Numerical approximation of partial differential equations. ISBN: 978-3-540-85267-4
Johnson Claes; Numerical solution of partial differential equations by the finite element method. ISBN: 0-521-347-580
John F.; Partial differential equations. ISBN: 0-387-90021-7
Garabedian P. R.; Partial differential equations. ISBN: 0-8284-00325-2
Evans Lawrence C.; Partial differential equations. ISBN: 0-8218-0772-2

Teaching methods and learning activities

Exposition of the syllabus contents and resolution and discussion of exercises. Group work focusing on the part of the numerical treatment of partial differential equations.

All resources are available for students at the unit’s web page.

 

Evaluation Type

Distributed evaluation without final exam

Assessment Components

designation	Weight (%)
Teste	50,00
Trabalho escrito	50,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	86,00
Frequência das aulas	56,00
Trabalho escrito	20,00
Total:	162,00

Eligibility for exams

The assessment consists of two components: test (10 points) + project (10 points). The project focuses on the part of the numerical treatment of partial differential equations. A minimum of 4 points is required in each of the assessment's components.

Calculation formula of final grade

test's mark+project's mark