Course: 2021/2022

Applied differential calculus

(15975)

Requirements (Subjects that are assumed to be known)

Calculus (Course 1 - Semester 1)
Linear Algebra (Course 1 - Semester 1)

The specific competences of the subject have been divided into three sections:
KNOWLEDGE:
- Know how to solve linear and nonlinear ordinary differential equations of first order and interpret results.
- Know how to solve linear ordinary differential equations of second order.
- Know how to calculate the Laplace transform and how to use it to solve differential equations.
- Know how to solve systems of linear differential equations of first order.
- Understand the concept of Fourier series and using it to solve differential equations.
- Know how to use numerical methods to compute approximate solutions of non-linear differential equations.
SPECIFIC ABILITIES:
- Increase the level of abstraction.
- To be able to solve practical problems using differential equations.
GENERAL ABILITIES:
- Ability to communicate orally and in writing correctly using signs and the language of Mathematics.
- Ability to model a real situation described in words by differential equations.
- Ability to interpret the mathematical solution of a problem, their reliability and limitations.

Skills and learning outcomes

Description of contents: programme

1.- First order differential equations:
a. Introduction.
b. Linear equations.
c. Separable equations.
d. Exact equations.
e. Homogeneous equations.
2.- Second order differential equations.
a. Linear and nonlinear equations.
b. Homogeneous and non-homogeneous linear Equations.
c. Reduction of order.
d. Euler-Cauchy equations.
3.- The Laplace Transform:
a. Definition. Properties.
b. Application to differential equations.
4.- Systems of differential equations:
a. Linear and nonlinear systems.
b. Vectorial representation.
c. Eigenvalues and linearization.
5. Fourier series and separation of variables:
a. Basic results.
b. Fourier Sine and Cosine Series.
c. Applications of Fourier series to partial differential equations.
6.- Numerical methods:
a. Euler method.
b. Runge-Kutta method.
c. Boundary value problems.

Learning activities and methodology

1.- Teaching in big or aggregate groups. Lectures sessions (3 ECTS).
2.- Face-to-face teaching in small groups. Problem sessions with individual and group work (3 ECTS).
Office hours: Each teacher offers a number of office hours according to the regulations of the Carlos III University. In particular, a minimum of one hour per group with the time schedule compatible with the students.

Assessment System

- % end-of-term-examination 60
- % of continuous assessment (assigments, laboratory, practicals...) 40

Basic Bibliography

- Boyce, William E.. Elementary differential equations and boundary value problems . John Wiley & Sons,.
- Simmons, George Finlay. Differential equations with applications and historical notes.. McGraw-Hill.
- Zill, Dennis G.. Ecuaciones diferenciales con aplicaciones de modelado . International Thomson.

- Manuel Carretero, Luis L. Bonilla, Filippo Terragni, Segei Iakunin, Rocío Vega · Curse OCW-UC3M Applied Differential Calculus : http://ocw.uc3m.es/matematicas/applied-differential-calculus

Additional Bibliography

- Haberman, Richard . Elementary applied partial differential equations with Fourier series and boundary value problems 3rd ed. Prentice Hall.
- Gockenbach, Mark S.. Partial differential equations : analytical and numerical methods. SIAM.
- Kiseliov, Aleksandr I.. Problemas de ecuaciones diferenciales ordinarias . Mir.
- Weinberger, Hans F. . A first course in partial differential equations with complex variables and transform methods. Dover.

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The course syllabus may change due academic events or other reasons.