Course: 2019/2020

Applied differential calculus

(15975)

Students are expected to have completed

Calculus.
Linear Algebra.

Competences and skills that will be acquired and learning results. Further information on this link

KNOWLEDGE (PO a - RA1.1):
- Solving linear differential equations and interpret results.
- Understand the concept of stability.
- Know how to plot slope fields.
- Know how to calculate Laplace transforms and how to use them to solve differential equations.
- Know how to solve systems of linear differential equations of first order.
- Understand the concept of Fourier series and using them to solve differential equations.
- Know how to use numerical methods to compute approximate solutions of first order non-linear systems of differential equations.
SPECIFIC ABILITIES (PO a - RA1.1):
- Increase the level of abstraction.
- To be able to solve practical problems using differential equations.
GENERAL ABILITIES (PO a - RA1.1):
- 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.

Description of contents: programme

1.- First order differential equations:
a. Linear Equations.
b. Separable Equations.
c. Qualitative Technique: Slope Fields. Equilibrium and Phase line. Bifurcations.
2.- Second Order Differential Equations.
a. Nonlinear and linear Equations.
b. Homogeneous Linear Equations.
c. Reduction of Order.
d. Euler-Cauchy Equations.
3.- Laplace transformations:
a. Definition.
b. Application to differential equations.
c. Convolution.
4.- Systems of differential equations:
a. Linear and Nonlinear Systems.
b. Vector representation.
c. Eigenvalues and linearization.
5. Fourier series and separation of variables:
a. Basic results.
b. Fourier Sine and Cosine Series.
c. Convergence of Fourier series.
d. Applications of Fourier series to Differential Equations.
6.- Numerical methods:
a. Euler method.
b. Runge-Kutta method.
c. Solution of boundary value problems.

Learning activities and methodology

Lectures sessions: 3.0 ECTS credits (PO:a - CGB1 - RA1).
Problem sessions: 3.0 ECTS credits (PO:a - CGB1 - RA1).)
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 2nd ed.. McGraw-Hill.
- Zill, Dennis G.. Ecuaciones diferenciales con aplicaciones de modelado . International Thomson.

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.