Differential Equations

1. To know and manage the concepts and techniques described in the contents of the subject.

2. To understand the relation between real problems and their mathematical modelling in terms of differential equations.

3. To classify and solve the most common ordinary differential equations, especially in the linear case, and their application to the mathematical modelling of chemical engineering process.

4. To study the main analytical methods to solve differential equations.

5. Understanding the need to use numerical methods to obtain a numerical approximation of the solution to an initial value problem when it cannot be solved by analytical techniques.

6. To use MATLAB to solve problems of ODEs and to check the results obtained.

1. Introduction to Ordinary Differential Equations (ODEs)
Motivation. Basical concepts: type, order and linearity. General and particular solution. Singular solutions. Existence and uniqueness of solution for initial value problem of first order. Some engineering problems leading to ODEs.

2. First order differential equations
Separable differential equation. Exact equations. Integrating factor. Linear equations. Homogeneous equations. Applications of first order ordinary differential equations in chemical engineering.

3. Introducition to the numerical solution of ODEs
Motivation. Numerical solution of initial value problem of first order.Euler's method. Second order Runge-Kutta methods.
Applications.

4. Second and higher order differential equations
Second order linear equations. Homogeneous linear equations with constant coefficients. General solution. Nonhomogeneous linear equations with constant coefficients. Method of undetermined coefficients and method of variation of parameters. Higher order linear differential equations. Applications. Numerical solution of differential equiations of higher order.

5. Resolution of linear systems of ODEs. Laplace Transform.
Definition of the Laplace transform. Calculation and properties of the Laplace transform. Inverse Laplace transform. Application to solving linear systemsof differential equations. Applications in chemical engineering.

6. Introduction to Partial Differential Equations (PDEs)
Definition of PDE. Order and solution. Second order linear PDEs. Examples. Method of separation of variables. Introduction to the finite difference method.

BASIC BIBLIOGRAPHY:

• NAGLE, R. Kent, SAFF, Edward B., 2005. Ecuaciones diferenciales y problemas con valores en la frontera. 8ª ed. México: Pearson Education. ISBN 978-968-444-483-6. Bibliotecas USC. Sinaturas: 1202 360 1, 1202 360 2, A ES 155 A 1

• NAGLE, R. Kent, SAFF, Edward B., SNIDER A., 2019. Fundamentals of Differential Equations. 9ª ed. Harlow: Pearson Education. ISBN 9781292240992. Biblioteca ETSE: Sinaturas: A012 13 C, A012 13 D, A012 13 E

Available Electronically (PreLo):

• NAGLE, R. Kent, SAFF, Edward B., SNIDER A. David., 2013. Fundamentals of Differential Equations. Harlow: Pearson. [Recurso electrónico]

• NAGLE, R. Kent, SAFF, Edward B., SNIDER A. David, 2005. Ecuaciones diferenciales y problemas con valores en la frontera. 4ª ed. México: Pearson. [Recurso electrónico]


ADDITIONAL BIBLIOGRAPHY:

• BOYCE, William E., DIPRIMA, Richard C., 2010. Elementary Differential Equations and Boundary Value Problems. 9th ed. New York: Wiley. ISBN 978-0-470-39873-9

• CUTLIP, Michael B., SHACHAM, Mordechai, 2000. Problem solving in chemical engineering with numerical methods. New Jersey: Prentice Hall International Series in the Physical and Chemical Engineering Sciences. ISBN 0-13-862566-2

• SIMMONS, George F., 2002. Ecuaciones diferenciales con aplicaciones y notas históricas. 2ª ed. Madrid: McGraw-Hill. ISBN 84-481-0045-X

• ZILL, Dennis G., CULLEN, Michael R., 2008. Matemáticas avanzadas para ingeniería I: ecuaciones diferenciales. 3ª ed. México: McGraw-Hill. ISBN 9789701065143

To contribute to achieve the generic skills and competences listed in the Report of bachelor’s degree in Chemical Engineering of the USC. Specifically:

General and basic skills
CB.1. Knowledge and understanding in a field of study starting from the basis of general secondary education, and it is typically at a level which, although it is supported by advanced textbooks, includes some aspects which require knowledge from the forefront of the field of study.
CG.3. Knowledge in basic and technological topics enabling to learn new methods and theories. Ability to adapt to new situations.
CG.4. Ability to solve problems with initiative, decision making, creativity, critical thinking and to communicate and transmit knowledge, skills and abilities in the field of industrial engineering.

Cross-disciplinary skills
CT.1. Capacity for analysis and synthesis.
CT.6. Troubleshooting.
CT.7. Decision making
CT.13 Ability to apply knowledge in practice.
CT.19. Autonomous learning.

Achieve specific competences described in the basic module grade memory. More precisely:
Specific skills
CF.1. Ability to solve mathematical problems that may arise in engineering. Ability to apply the
knowledge on:
CF.1.2. Differential equations and partial differential equations.
CF.1.3. Numerical methods, numerical algorithms.

The students enrolled in the course will have content related material on the website of the subject hosted in the ​Learning Management System for teaching.

It will follow the general methodological instructions listed below:

- Expository teaching: classes in which the teacher shows, on the blackboard or with the help of multimedia, the content specified in program area. Skills CB1 and CG3.
- Seminars: interactive classes in which problems will be solved. Skills CG.4, CT.6, CT.7, CT.13, FB.1.2.
- Computer practices: interactive classes in which applied problems related to the contents of the course will be solved with the help of Matlab package. These classes will be held in the computer room. Skills CT.6, CT.7, CT.13, FB.1.3.
- Tutorials: interactive classes in which students may discuss, present or solve with the teacher any question related to the development of the subject. Skills: CG.4, CT.6, CT.7, CT.13. Special emphasis in skills CT.1 and CT.19.

he evaluation system follows the general criteria indicated in the chemical engineering degree program. Each student's score will be based on continuous assessment and final examination.

In each evaluation opportunity, students will take a final written exam on the official dates set by the center. To pass the course, it will be necessary to achieve at least a 3 out of 7 on this test.

For continuous assessment, two individual tests will be taken into account: a written test halfway through the semester related to the theoretical and practical contents of the subject, and a test in the computer lab using Matlab, related to the contents of the completed practical sessions.

The grades for the continuous assessment will be communicated to the student before the final written exam and will be saved for the second assessment opportunity if necessary. In the case of the Matlab test, the grade will only be saved for the second opportunity if it is equal to or higher than 0.75 points (50% of the total test grade). Otherwise, the test must be retaken on the same day as the officially scheduled date for the second assessment opportunity, following the written test.

These tests together with the continuous assesment will evaluate all the basic and general competencies (CB.1, CG.3, CG.4), transverse (CT.1, CT.6, CT.7, CT.13 , CT.19) and specific one (FB.1).

The weight of each of the activities in the student's final grade and the evaluation of each skill in the different activities is specified below:

- Written exam cualification (ET): 70% (7 points over 10) To pass the course the student should obtain at least 3/7 on this exam. Assessment of skills CB1, CG3, CG4, CT.1, CT.6, CT.7, CT.13, F.B.1.2
- Work in the computer room (EP): 15% (1.5 points over 10). Assessment of skills CT.6, CT.7, CT.13, FB.1.3
- Works, seminars and tutorials (EC): 15% (1.5 points over 10). Assessment of skills CT.1, CT.13, CT.19, FB.1.2

Those students not attending any of the exams will be qualified as "absent" (no presentado).

In cases of fraudulent performance of exams or tests, the provisions of the Regulations on the assessment of students' academic performance and review of grades shall be applied.

1.Class time: 51h
- theory and problems: 39h
- MATLAB laboratory: 10h
- Tutorials in small groups: 2h

2.Self study. 66h
- theory and problems: 50h
- tutorials in small groups: 10h
- Matlab laboratory: 4h
- Individual tutorials: 2 h

3. Exam preparation, written and laboratory examination: 33 h
Total: 150h

1. Taken and passed the subject of mathematics taugth in the first year.
2. Homogeneous distribution of the study time along the course.
3. Class attendance and active participation.
4. Take advantage of the tutorial hours.