Course guide of Mathematical Methods 2 (2671125)

It is recommended that the student has taken the following subjects: Linear Algebra and Geometry, Mathematical Analysis and Mathematical Methods for Physics I.

• Methods of solving ordinary differential equations and systems.
• Partial differential equations. The method of separation of variables.
• Special functions.

• To know the fundamental results of the theory of Differential Equations.
• To know some of the applications of the ordinary differential equations in different fields in Physics, especially in Classical Mechanics, Electromagnetism and Quantum Physics.
• To understand how special functions arise in the study of ordinary differential equations and understand how to apply them.
• To know the fundamental results of the theory of Partial Differential Equations.
• To know some applications of the theory of Partial Differential Equations the in different fields in Physics, especially in Classical Mechanics, Electromagnetism and Quantum Physics.

Differential equations

• Lesson 1. Ordinary differential equations of first order. Methods of integration.
• Lesson 2. Ordinary linear differential equations of higher order. Systems of linear equations.
• Lesson 3. Solving differential equations by power series.

Special functions

• Lesson 4. Basic special functions.
• Lesson 5. Hypergeometric and Bessel functions.

Partial differential equations

• Lesson 6. Partial differential equations of interest in physics: The method of separation of variables.
• Lesson 7: The wave equation, the heat equation and the Laplace equation.
• Lesson 8. Introduction to the Sturm-Liouville problem.

Seminars:

1. Kepler's laws.
2. The Laplace transform.
3. Sturm's theory of separation of zeros.
4. The wave equation in two and three dimensions. Huygens principle.
5. Green's functions.
6. Euler's equations of fluids.
7. The multidimensional Schrödinger equation. Application to the infinite square well.
8. The multidimensional Schrödinger equation. Application to the three-dimensional harmonic oscillator.
9. The vibrating equation in two dimensions.

• D.G. Zill, M.R. Cullen, Differential Equations with Boundary-Value Problems, Cengage Learning, 2009.
• M. Abramowitz, I. A. Stegun, Handbook of mathematical functions, Dover, 1975.
• L. C. Andrews, Special functions of mathematics for engineers, Oxford Science Publications, 1998.
• W.E. Boyce, R.C. DiPrima, Elementary differential equations and boundary value problems, Wiley 2012.
• L. C. Evans, Partial Differential Equations, AMS, 2002.
• V. Nikiforov, V. Uvarov, Special functions of mathematical physics (Birkhäuser Verlag, 1988).
• I. Peral, Primer curso de Ecuaciones en derivadas parciales. Addison-Wesley, Wilmington, 1995.
• C. Henry Edwards, David E. Penney, David T. Calvis, Differential Equations and Boundary Value Problems: Computing and Modeling, Pearson Education 2015.
• C. Henry Edwards, David E. Penney, David Calvis, Differential Equations and Linear Algebra, Pearson 2017.
• E. Rainville, Intermediate Differential Equations, MacMillan, 1964.
• G.F. Simmons, Ecuaciones diferenciales con aplicaciones y notas históricas. McGraw Hill, 1993.
• W. A. Strauss, Partial differential equations, an introduction, New York, John Wiley and Sons, 2008.

• F. Brauer y Nohel, Ordinary Differential Equations with Applications, Harper & Row, 1989.
• C. Carlson, Special Functions of Applied Mathematics, Academic Press.
• R. K. Nagle, E. B. Saff y A.D. Snider, Ecuaciones diferenciales y problemas con valores en la frontera, Pearson Educación, 2005.
• F.W. Olver, Asymptotic and Special functions, Academic Press, 1974.
• R.D. Richtmyer, Principles of Advanced Mathematical Physics, vol. 1, Springer-Verlag, 1978.

• Notes by Prof. R. Ortega “Métodos Matemáticos de la Física IV”: http://www.ugr.es/~rortega/M4.htm
• Notes by Prof. M. Calixto “Métodos Matemáticos de la Física II”: https://www.ugr.es/~calixto/MMII.pdf

• MD01. Theoretical classes

In general, the attendance to lectures is not compulsory without being an impediment to apply the evaluation criteria described below.

In order to evaluate the knowledge and competences acquired by the students, the following criteria will be used with the indicated percentages:

• Written examination including basic questions and problems/exercises. This will count 70% of the total score. It will be required to obtain a mark of at least 4 over 10 in this item.
• Homework and seminars done individually or in groups. This covers all work and seminars made by the students during the course (exercises and solving proposed problems). Importance will be given to the work itself, the slides presentation and the defense. Participation, attitude and personal work in all programmed activities will be considered. The final score for this part will count 30% of the final score.

The final score will be the sum of the weighted scores obtained in the different aspects of the evaluation system.

• It will be in written form and will consist of questions and problems/exercises to guarantee that the student can get the total score from it (100%).

Besides the above-mentioned evaluation procedure, the students will be allowed to apply for a unique evaluation in the terms established in the regulation of evaluation of students at the University of Granada, approved on May the 20th of 2013.

The test consists of a written examination that includes theory and problems on the list of topics of the curse, similar to the extraordinary assessment sesion, where the student can get the total score from it (100%).

All regarding evaluation will be applied according to the “Normativa de evaluación y calificación de los estudiantes” existing at the University of Granada, which can be found at:

http://www.ugr.es/~minpet/pages/enpdf/normativaevaluacionycalificacion.pdf