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Contribute to the student's knowledge of some mathematical models and techniques used in other areas of knowledge.

Improve the student's knowledge in Mathematical Analysis and develop their skills in dealing with models and in problem solving.

Basic knowledge of Linear Algebra and Real Analysis in several variables. Knowledge of Analytical Geometry and of Complex Analysis is also desirable, however, lack of the latter may be compensated with further work by the student.

Each one of the topics below will be illustrated with examples of application to different areas of knowledge to be worked out by the students.

1 - Linear ordinary differential equations in R^n: flow, exponential of a linear operator, phase portraits on the plane, equilibria, stability.

2 - Nonlinear ordinary differential equations in R^n: flow, phase portraits, equilibria, stability. Linearisation of a vector field, Lie derivative and Liapunov functions. Stable manifold and Grobman-Hartman theorems (without proof), bifurcations.

3 - Classical linear partial differential equations: treatment as an eigenvalue problem, Dirichilet and Neumann problems on a rectangle.

4 - Complex analysis and partial differential equations: elementary functions of a complex variable, image of subsets of C by the elementary functions, holomorphic functions, Cauchy-Riemann conditions, conformal maps and harmonic functions. Use of conformal maps to solve Laplace's equation, homographic maps. 5- Pattern formation in partial differential equations: reaction-diffusion equations, Turing instability.

Presentation of techniques either in classroom or in distance learning using platforms like moodle and zoom.
Spin learning with student's active participation for problem solving and discussion of examples.
The latter will take place either in personally attended sessions, or in distance sessions with active student participation, in case, for instance, of epidemiological restrictions.

The final mark is the sum of the following two components:


C1 - presentation of examples in writing and in spin-learning classes - 10 points. Consists in solving previously prescribed exercises.


C2 - final written exam - 10 points.

In the resit period the final mark will be obtained in an exam worth 20 points.

For final marks (in both exam periods) over 16 points an oral exam may be required.

May only be done in the resit period.