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.

In the case of using AI tools for the development of the course, the student must adopt an ethical and responsible use of such tools. The recommendations contained in the document "Recommendations for the use of artificial intelligence at the UGR", published at this location, must be followed:
https://ceprud.ugr.es/formacion-tic/inteligencia-artificial/recomendaciones-ia#contenido0

• 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.
• Lesson 4. Basic special functions. Hypergeometric and Bessel functions.
• Lesson 5. Partial differential equations of interest in physics: The method of separation of variables.

The practical component of the course will be devoted to the resolution of problem sheets related to the contents of the subject, as well as to the preparation and presentation of group seminars aimed at broadening the scope of the course. These seminars are intended to explore specific aspects of the syllabus in greater depth, present relevant applications, and complement the students’ practical training. The seminars/workshops may address, among others, the following topics:

1. Kepler's Laws.
2. The Laplace Transform.
3. Sturm's Theory on the Separation of Zeros.
4. The Wave Equation in Two and Three Dimensions. Huygens' Principle.
5. Green's Functions.
6. Euler's Equations for 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 Membrane 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.
• 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.
• 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.
• L. C. Evans, Partial Differential Equations, AMS, 2002.
• 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.
• 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.
• E. Rainville, Intermediate Differential Equations, MacMillan, 1964.
• 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 5 over 10 in this item.
• Coursework and seminars. This component will account for 30% of the final grade. Of this percentage, 20% will correspond to the presentation and defense of seminars at the end of the semester, while the remaining 10% will assess continuous work, participation, attitude, and the student’s personal effort in the scheduled training activities, such as exercises, problem solving, oral or written questions, classroom participation, and other individual or group assignments or tests.

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

Assessment Due to Incidents:

Students who are unable to attend final assessment exams (ordinary, extraordinary and single final) or officially scheduled evaluations outlined in the Course Guide may request special assessment due to extenuating circumstances. This is permissible under the conditions specified in Article 9 of the University of Granada's Regulations on Student Assessment and Grading, and must follow the procedure detailed therein.

• 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%) and cover all learning outcomes..

In accordance with the UGR's Regulations on Student Assessment and Grading, a single final assessment is available for students who are unable to participate in the continuous assessment method due to any of the reasons stipulated in Article 8. To opt for this single final assessment, students must submit a request via the electronic portal within the first two weeks of the course's instruction, or within two weeks following their enrollment if it occurs later. Exceptions may be made for overriding unforeseen circumstances that arise after these initial periods. The request must clearly state and provide evidence for the reasons preventing their participation in the continuous assessment system.

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%).

Students with Specific Educational Support Needs (SESN):

Following the recommendations from the CRUE and the UGR's Secretariat for Inclusion and Diversity, the systems for acquiring and assessing competencies outlined in this teaching guide will be applied in accordance with the principle of universal design. This approach aims to facilitate learning and the demonstration of knowledge, aligning with the needs and functional diversity of the student body. Teaching methodology and assessment will be adapted for students with SESN, in line with Article 11 of the UGR's Regulations on Student Assessment and Grading, published in the Official Bulletin of the UGR No. 112, dated November 9, 2016. UGR Inclusion and Diversity For students with disabilities or other SESN, the tutoring system must be adapted to their needs, in accordance with the UGR's Inclusion Unit recommendations. Departments and Centers must establish appropriate measures to ensure that tutoring sessions are held in accessible locations. Furthermore, faculty may request support from the University's competent unit when special methodological adaptations are required.

Información de interés para estudiantado con discapacidad y/o Necesidades Específicas de Apoyo Educativo (NEAE): Gestión de servicios y apoyos (https://ve.ugr.es/servicios/atencion-social/estudiantes-con-discapacidad).