Introduction to ordinary differential equations and partial differential equations,

probability and statistics and geometry.

Degree competences (all transversal):

G001. Learn to pose and solve problems correctly.

G005. Be able to organize, plan and learn autonomously.

G006. Be able to analyze, synthesize and reason critically.

G008. Be able to present ideas, problems and scientific results orally and in writing.



All Mathematics module competences (all generic):

CM01. Appreciate mathematical abstraction and redirect it for the concrete calculation.

CM03. Be able to organize a logical discourse with mathematical support.

CM02. Approach correctly and solve problems involving the main concepts of Classical Physics, Chemistry and Electronics and their applications.

Programme



1. Introduction to differential equations

Definition, classification. Concepts of existence, uniqueness and methods for obtaining solutions.



2. First order ordinary differential equations

Definition. Geometric meaning. Exact equations, separate variables. Integrating factors; separable and linear equations. Transformation methods: homogeneous and Bernoulli equations.



3. Higher order ordinary differential equations

Reduction of order. Linear equations. Dependence and linear independence of functions. Linear homogeneous equations: fundamental solution system and Liouville formula. Complete linear equations: variation of constants and Cauchy method. Dirac Delta as a generalized function and elementary solution. Concept of distribution.



4. Systems of ordinary differential equations

Reduction to an equation. First integral. Linear homogeneous and complete systems. Exponential of matrices.



5. Laplace transformation

Definition and basic properties. Convolution Application to initial value problems for linear equations and systems of linear equations.



6. Power Series solutions

Regular and singular regular points. Frobenius method. Special functions: Hermite, Bessel, Legendre.



7. Nonlinear equations and stability theory

Stability concept. Balance points. Stability of linear systems. Linear stability Conservative systems.



8. Sturm-Liouville and Green's function

Spaces of functions and developments in sets of orthogonal functions. Problems with values ​​at the border. Theory of Sturm-Liouville. Fourier series.



9. Partial differential equations

Introduction to partial differential equations. Boundary problems and separation of variables. Use of integral transformations in the resolution of boundary problems. Characteristics in second order equations: classification.



10. Probability

Introduction to probability. Basic discrete distributions. Probability distributions. Moments. Random variable functions. Characteristic function. Central limit theorem.



11. Statistics

Statistics Estimators Estimation by confidence intervals.



12. Introduction to geometry

Geometry of curves. Geometry of surfaces.

Lectures on theoretical aspects, and practical problem-solving sessions.

• Final Assessment System
• Tools and qualification percentages:
  • Written test to be taken (%): 100

- Written exam including problem-solving exercises.



- There will be a first term exam in January . Those students with at least a pass (5 out of 10) may choose to only sit the part corresponding to the second term in the ordinary call (final) exam. The marks from the partial exam will not be carried over to the resit (extraordinary call) exam.



- The exams may contain an eliminatory part.



- Not taking the ordinary call (convocatoria ordinaria) exam equals giving up the call (renuncia a la convocatoria).

- Written exam including problem-solving exercises.

A level of B2 or higher is recommended to attend courses taught in English.

Basic bibliography

* K. F. Riley, M. P. Hobson, and S.J. Bence Mathematical Methods for Physics and Engineering Cambridge University Press (3d rev. ed. 2006))



* M. D. Greenberg Foundations of applied mathematics Prentice-Hall (1978)



* J. Mathews and R.L. Walker Mathematical methods of physics Benjamin (1970)



* H.F. Weinberger Ecuaciones diferenciales en derivadas parciales Reverté (1986)



* W. E. Boyce y R. C. DiPrima Ecuaciones diferenciales y problemas con valores en la frontera 4[tm] Ed., Limusa (1998)



* L. Elsgoltz Ecuaciones diferenciales y calculo variacional URSS (1994)



* P. Z. Peebles Probability, random variables, and random signal principles McGraw-Hill (1987)



* A. V. Pogoriélov, "Geometría diferencial", URSS