Advanced Mathematics

Course: Advanced Mathematics
Course Number: 251 - 15331
ECTS credits: 6

PREREQUISITES/STUDENTS ARE EXPECTED TO HAVE COMPLETED:
Calculus I, Calculus II, Linear Algebra

COMPETENCES AND SKILLS THAT WILL BE ACQUIRED AND LEARNING RESULTS:

The aim of this course is to provide students the basic tools for understanding and solving initial and boundary value problems for ordinary and partial differential equations using analytical and numerical techniques. To achieve this goal, the students must acquire a range of expertise and capabilities.

SPECIFIC LEARNING OBJECTIVES (PO a):
- To understand the principle of superposition for solving linear differential equations.
- To solve elementary ordinary differential equations by separation of variables and other methods.  
- To know and use numerical methods to solve initial and boundary value problems for ordinary differential equations.
- To distinguish between elliptic, hyperbolic and parabolic partial differential equations and which initial or boundary conditions are appropriate for them.
- To understand how to apply separation of variables and the Fourier method to solve initial-boundary value problems for the equations of Mathematical Physics.
- To understand the finite difference method and how to use it to solve initial-boundary value problems for the equations of Mathematical Physics.
SPECIFIC ABILITIES (PO a, k):
- To understand what is an ordinary differential equation and know how to apply techniques of solving differential equations in different contexts.
- To understand boundary value problems associated to linear ordinary differential equations and the main analytical and numerical techniques to solve them.
- To understand what is a partial differential equation, the classification of second order linear partial differential equations and the associated initial and boundary value problems.
- To understand the separation of variables technique, the role of the resulting eigenvalue problems and the principle of superposition to solve initial-boundary value problems for the equations of Mathematical Physics.
- To understand how to use finite difference methods to solve numerically initial-boundary value problems for the equations of Mathematical Physics and ability to use software packages.

 GENERAL ABILITIES (PO a, g, k):
- To understand the necessity of abstract thinking and formal mathematical proofs.
- To acquire communicative skills in mathematics.
- To acquire the ability to model real-world situations mathematically,  with the aim of solving practical problems.
- To improve problem-solving skills.  
(PO: a)

DESCRIPTION OF CONTENTS:

1 .- First-order ordinary differential equations. Existence and uniqueness. Numerical methods: Euler, Runge-Kutta.
2 .- Second order linear ordinary differential equations. Inhomogeneous equations: variation of parameters, undetermined coefficients, reduction of order. Linear oscillator and resonance.
3.- Systems of linear ordinary differential equations. Two dimensional homogeneous system, classification of fixed points and phase portrait. Homogeneous systems and superposition. Inhomogeneous systems and variation of parameters.
4.- Nonlinear systems. Examples. Autonomous systems and direction fields. Stability and linearization.
5.- Boundary value problems for second order linear ordinary differential equations. Shooting method and finite differences.
6.- Eigenvalue problems. Solutions obtained by power series. Fourier series and properties. Sturm-Liouville eigenvalue problems.
7.- Introduction to partial differential equations. Classification of second-order linear PDEs. Initial and boundary value problems from physical models. Separation of variables and Fourier method.  
8 .- Finite difference methods for the heat and the wave equation. Stability analysis. Finite difference methods for the Poisson equation.

LEARNING ACTIVITES AND METHODOLOGY:

Theory (3.0 credits. PO a, g).
Problem sessions working individually and in groups (3.0 credits. PO a, g).

Collective office hours might be offered if the professor deems them to be appropriate.

ASSESSMENT SYSTEM:

We follow a continuous-assessment system plus a final exam:

 - The continuous-assessment part consists in a written examination contributing with weight 40% to the final mark. The mid-term examination will take place, approximately, at two thirds of the semester and it will be held in regular class hours, according to the current regulations.

 - The final exam (contributing with weight 60% to the final mark) will be held at the end of the semester.

BIBLIOGRAPHY:

    G. F. Simmons, . Differential Equations with Applications and Historical Notes, 2nd ed.. McGraw Hill. 1991
    G.F. Carrier, C.E. Pearson. Ordinary Differential Equations. SIAM (SIAM Classics in Applied Mathematics vol. 6). 1991
    R. Haberman. Elementary applied partial differential equations. 3rd ed.. Prentice Hall. 1998
    R.L. Borrelli, C.S. Coleman. Differential equations: A modeling perspective. 2nd ed. . Wiley. 2004
    W.E. Boyce, R.C. Di Prima. Elementary differential equations and boundary value problems. 8th ed. . John Wiley . 2009