Guías Docentes Electrónicas
1. General information
ECTS credits:
Academic year:
First quarter
Main language:
Second language:
Use of additional languages:
English Friendly:
Web site:
2. Pre-Requisites
To achieve the learning objectives of the course, previous knowledge and skills are assumed to have been acquired during undergraduate studies. In particular, it is necessary to have a solid knowledge of ordinary differential equations and familiarity with partial differential equations, as well as basic knowledge of mathematical analysis and topology.

Also, it is very convenient to be acquainted with scientific text editors (e.g., LaTeX, Microsoft Word, Scrivener, LyX, MathCast, Notepad, Writebox, Writer, Google Docs, etc) and to be familiar with the use of numerical and symbolic calculation software (Matlab, Mathematica, Python, Octave, Maple, etc).
3. Justification in the curriculum, relation to other subjects and to the profession

In all countries around us there is a clear tendency towards the creation of interdisciplinary studies. Given the interdisciplinarity of modern science, versatile graduated students come out, who also adapt better to changing technologies and markets, so that technological transfer processes are improved. In many fields of Physics or Mathematics a series of mathematical concepts (fractals, chaos, bifurcations, attractors, solitons, complex systems, interfaces, cellular automata, pattern formation, catastrophes, critical phenomena, self-similarity, self-criticality, scale invariance, renormalization group, ...) have recently been extended today to some of the most promising scientific research lines. At present the relationship between Physics and Mathematics and other sciences is providing important perspectives and new exploration pathways for the future. The comprehension of reality through its modeling is a fascinating and motivating challenge in nearby fields such as Engineering, Biology, Medicine, Economics, Ecology or Telecommunications. One of the purposes of this course is to strengthen and provide the necessary tools that allow connecting with these lines of work, creating the teaching structures that facilitate learning by means of a problem-solving approach in these areas.

At present, it seems commonly accepted that the great challenge of physics and mathematics in the 21st century, as well as the international repertoires and calls, is the interaction with biology and medicine, which FisyMat intends to promote with a specialty or module. In some countries a term that includes part of the previous ideas begins to be generic: mathematical or physical engineering (also bioengineering). Our point of view, regardless of the denomination, is that this program from physics and mathematics is a commitment to a return to the essence of the origins of science: the knowledge of reality and the resolution of problems that is the basic idea of an integral science, without borders.

The program of this course aims to familiarize students with the modeling of complex physical systems in which a large number of particles or dynamic agents can interact through partial differential equations. Special attention will be given to problems originated in Kinetic Theory and Fluid Mechanics. Likewise, a theoretical basis of the variational formulation and analytical resolution techniques will be provided. Since PDEs appear in almost any field of Mathematics and Physics and lately it is gaining importance in other fields such as Biology, Medicine or Economics, it is a fundamental subject.

This course will also present a great interrelation with other subjects such as Dynamic Systems, Numerical Methods, Optimization and Biomathematics, to name just a few examples.

4. Degree competences achieved in this course
Course competences
Code Description
CB07 Apply the achieved knowledge and ability to solve problems in new or unfamiliar environments within broader (or multidisciplinary) contexts related to the area of study
CB10 Have the learning skills which allow to continue studying in a self-directed or autonomous way
CE01 Solve physical and mathematical problems, planning their solutions based on the available tools and time and resource constraints
CE05 Know how to obtain and interpret physical and/or mathematical data that can be applied in other branches of knowledge
CE07 Ability to understand and apply advanced knowledge of mathematics and numerical or computational methods to problems of biology, physics and astrophysics, as well as to build and develop mathematical models in science, biology and engineering
CG03 Present publicly the research results or technical reports, to communicate the conclusions to a specialized court, interested persons or organizations, and discuss with their members any aspect related to them
CG05 Gain the ability to develop a scientific research work independently and in its entirety. Be able to search and assimilate scientific literature, formulate hypotheses, raise and develop problems and draw conclusions from the obtained results
CT03 Develop critical reasoning and the ability to criticize and self-criticize
CT05 Autonomous learning and responsibility (analysis, synthesis, initiative and teamwork)
5. Objectives or Learning Outcomes
Course learning outcomes
Carry out a critical analysis of a scientific article that addresses topics related to the course
Deepen aspects of modeling through the study of different interaction nuclei that represent shock, coagulation, fragmentation or dispersion phenomena
Learn non-linear analysis techniques to study the qualitative behavior of problem solutions from Kinetic Theory. This will allow to identify the qualitative and analytical differences between dispersion and diffusion models.
The previous point implies that the student will be able to handle easily specialized literature in EDPs
Additional outcomes
Not established.
6. Units / Contents
  • Unit 1: Introduction to PDEs in Science and Engineering
  • Unit 2: Models of Transport Equations in Kinetic Theory and Fluid Mechanics
  • Unit 3: Variational Formulation
  • Unit 4: Analytic Methods for Solving PDEs in Transport Models
  • Unit 5: Nonlinear Transport PDEs
7. Activities, Units/Modules and Methodology
Training Activity Methodology Related Competences ECTS Hours As Com R Description *
Class Attendance (theory) [ON-SITE] Lectures 1.2 30 N N N
Class Attendance (practical) [ON-SITE] Problem solving and exercises 0.6 15 N N N
Final test [ON-SITE] Assessment tests 0.2 5 Y Y Y
Writing of reports or projects [OFF-SITE] Project/Problem Based Learning (PBL) 2 50 Y Y N
Study and Exam Preparation [OFF-SITE] Self-study 2 50 N N N
Total: 6 150
Total credits of in-class work: 2 Total class time hours: 50
Total credits of out of class work: 4 Total hours of out of class work: 100

As: Assessable training activity
Com: Training activity of compulsory overcoming
R: Rescheduling training activity

8. Evaluation criteria and Grading System
  Grading System  
Evaluation System Face-to-Face Self-Study Student Description
Theoretical papers assessment 100.00% 0.00% Problem handouts
Total: 100.00% 0.00%  

Evaluation criteria for the final exam:
The final grade of the ordinary call will correspond to the weighted average of all the problem handouts proposed throughout the semester. The course will be passed when the obtained grade is greater than or equal to 5.0. In case of not reaching that grade, the student will have to do a global exam of the Course.
Specifications for the resit/retake exam:
In case of not having obtained a grade equal to or higher than 5.0 in the ordinary call, the student will have to do a global exam of the Course in the extraordinary call.
Specifications for the second resit / retake exam:
Evaluation criteria not defined
9. Assignments, course calendar and important dates
Not related to the syllabus/contents
Hours hours

Unit 1 (de 5): Introduction to PDEs in Science and Engineering
Activities Hours
Class Attendance (theory) [PRESENCIAL][Lectures] 2
Class Attendance (practical) [PRESENCIAL][Problem solving and exercises] 1
Writing of reports or projects [AUTÓNOMA][Project/Problem Based Learning (PBL)] 5
Study and Exam Preparation [AUTÓNOMA][Self-study] 5

Unit 2 (de 5): Models of Transport Equations in Kinetic Theory and Fluid Mechanics
Activities Hours
Class Attendance (theory) [PRESENCIAL][Lectures] 4
Class Attendance (practical) [PRESENCIAL][Problem solving and exercises] 2
Writing of reports or projects [AUTÓNOMA][Project/Problem Based Learning (PBL)] 10
Study and Exam Preparation [AUTÓNOMA][Self-study] 10

Unit 3 (de 5): Variational Formulation
Activities Hours
Class Attendance (theory) [PRESENCIAL][Lectures] 10
Class Attendance (practical) [PRESENCIAL][Problem solving and exercises] 5
Writing of reports or projects [AUTÓNOMA][Project/Problem Based Learning (PBL)] 15
Study and Exam Preparation [AUTÓNOMA][Self-study] 15

Unit 4 (de 5): Analytic Methods for Solving PDEs in Transport Models
Activities Hours
Class Attendance (theory) [PRESENCIAL][Lectures] 8
Class Attendance (practical) [PRESENCIAL][Problem solving and exercises] 4
Writing of reports or projects [AUTÓNOMA][Project/Problem Based Learning (PBL)] 12
Study and Exam Preparation [AUTÓNOMA][Self-study] 12

Unit 5 (de 5): Nonlinear Transport PDEs
Activities Hours
Class Attendance (theory) [PRESENCIAL][Lectures] 6
Class Attendance (practical) [PRESENCIAL][Problem solving and exercises] 3
Final test [PRESENCIAL][Assessment tests] 5
Writing of reports or projects [AUTÓNOMA][Project/Problem Based Learning (PBL)] 8
Study and Exam Preparation [AUTÓNOMA][Self-study] 8

Global activity
Activities hours
10. Bibliography and Sources
Author(s) Title Book/Journal Citv Publishing house ISBN Year Description Link Catálogo biblioteca
Bird R.B., Stewart W.E., Lightfoot, E.N., Klingenberg, D.J. Introductory Transport Phenomena John Wiley & Sons 978-1-118-77552-3 2015  
Brenn G. Analytical Solutions for Transport Processes: Fluid Mechanics, Heat and Mass Transfer Springer 978-3-662-51421-4 2017  
Brezis H. Functional Analysis, Sobolev Spaces and Partial Differential Equations Springer 978-0-387-70913-0 2011 Ficha de la biblioteca
Debnath L. Nonlinear Partial Differential Equations for Scientists and Engineers Birkhäuser 978-0-8176-8264-4 2012 Ficha de la biblioteca
Evans L.C. Partial Differential Equations American Mathematical Society 978-0-8218-9474-3 2010 Ficha de la biblioteca
Haberman R. Applied Partial Differential Equations with Fourier Series and Boundary Value Problems Pearson Education 978-0-321-79705-6 2013 Ficha de la biblioteca
Hauke G. An Introduction to Fluid Mechanics and Transport Phenomena Springer 978-1-4020-8536-9 2008  
Kasman A. Glimpses of Soliton Theory: The Algebra and Geometry of Nonlinear PDEs American Mathematical Society 978-0-8218-5245-3 2010  
Logan J.D. An Introduction to Nonlinear Partial Differential Equations John Wiley & Sons 978-0-470-22595-0 2008  
Logan J.D. Applied Partial Differential Equations Springer 978-3-319-12492-6 2015  
Myint-U T., Debnath L. Linear Partial Differential Equations for Scientists and Engineers Birkhäuser 987-0-8176-4393-5 2007 Ficha de la biblioteca
Perthame B. Transport Equations in Biology Birkhäuser Verlag 978-3-7643-7841-7 2007  
Salsa S. Partial Differential Equations in Action: From Modelling to Theory Springer-Verlag 978-3-319-15092-5 2015 Ficha de la biblioteca
Salsa S., Vegni F.M.G., Zaretti A., Zunino P. A Primer on PDEs: Models, Methods, Simulations Springer-Verlag 978-88-470-2861-6 2013  
Soto R. Kinetic Theory and Transport Phenomena Oxford University Press 978-0-19-871606-8 2016  

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