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Integral Transforms and Differential Equations in Partial Derivatives

Code: 107622
Credits: 6
2026/2027
Degree programme Type Course
Physics OB 2

Contact lecturer

Name :
Carles Sanchez Alonso
Email :
carles.sanchez@uab.cat

Teaching staff

Emili Bagan Capella

Group languages

You can consult this information at the end of the document.

Prerequisites

A good knowledge of single-variable and multivariable calculus, basic linear algebra, and the resolution of first-order ordinary differential equations and higher-order ordinary differential equations with constant coefficients is recommended.


Objectives

To provide the tools needed to solve ordinary and partial differential equations that arise in Physics using Fourier series, integral transforms, power series, special functions and Sturm–Liouville theory. To teach students how to apply these methods to problems with initial and boundary conditions and to the modelling of different physical phenomena. To highlight that integral transforms are also a fundamental tool for the analysis of functions, signals and physical systems, beyond the resolution of differential equations.


Learning outcomes

  • CM23 (Adapt the advanced mathematical strategy when addressing a complex problem determined from an analytical point of view.) Adapt the advanced mathematical strategy when addressing a complex problem determined from an analytical point of view.
  • KM22 (Identify the advanced concepts of vector and subspace space, bilinear and scalar and tensor product, and the methodology of endomorphism diagonalisation.) Identify the advanced concepts of vector and subspace space, bilinear and scalar and tensor product, and the methodology of endomorphism diagonalisation.
  • KM23 (Describe the basic concepts of calculus and analysis and the different methods of solving differential equations in their different typologies.) Describe the basic concepts of calculus and analysis and the different methods of solving differential equations in their different typologies.
  • SM19 (Apply the knowledge acquired in advanced mathematics to the resolution of mathematical problems, as well as to physical problems with mathematical representation.) Apply the knowledge acquired in advanced mathematics to the resolution of mathematical problems, as well as to physical problems with mathematical representation.

Contents

1. Fourier Series

  • Inner product, norm, and orthogonality.
  • Orthogonal systems of functions.
  • Fourier series expansions.
  • Convergence, Parseval's identity, and the Gibbs phenomenon.
  • Applications to physics.

2. Fourier Transform

  • The Fourier transform as the infinite-interval limit of Fourier series.
  • Definition of the Fourier transform and its inverse.
  • Basic properties.
  • Convolution.
  • Differentiation and translation.
  • Dirac delta.
  • Applications to the solution of differential equations and physical problems.

3. Laplace Transform

  • Definition and basic properties.
  • Inverse Laplace transform.
  • Applications to the solution of ordinary differential equations and systems of ordinary differential equations.
  • Applications to physics.

4. Solving Differential Equations by Power Series

  • Ordinary points and regular singular points.
  • Power series solutions.
  • Frobenius method for second-order differential equations.

5. Special Functions and Orthogonal Polynomials

  • Legendre and Bessel equations.
  • Legendre polynomials and Bessel functions.
  • Rodrigues' formulas.
  • Basic recurrence relations.
  • Orthogonality and normalization.
  • Applications to physics.

6. Sturm–Liouville Theory

  • Self-adjoint operators.
  • Regular and singular eigenvalue problems.
  • Eigenvalues and eigenfunctions.
  • Orthogonality.
  • Expansions in terms of eigenfunctions.
  • Connection with Fourier series.
  • Applications to physics.

7. Partial Differential Equations

  • Elementary classification of PDEs.
  • Wave, Laplace, and heat equations.
  • Separation of variables.
  • Solution of problems in Cartesian, cylindrical, and spherical geometries.


Learning activities and methodology

Title Hours ECTS Learning outcomes
Seminars 8 0.32 CM23, KM22, KM23, SM19
Study of theoretical concepts and methods 54 2.16 CM23, KM22, KM23, SM19
Theoretical lectures 28 1.12 CM23, KM22, KM23, SM19
Problem solving 27 1.08 CM23, KM22, KM23, SM19
Problem-solving sessions 14 0.56 CM23, KM22, KM23, SM19

This course consists of lectures, problem-solving sessions, and seminars:

Lectures: The definitions, theoretical results, and solution methods covered in the course are presented and illustrated with examples of their application to mathematical and physical problems.

Problem-solving sessions: Problems related to the course contents are solved and discussed, with the aim of applying the techniques introduced in the lectures.

Seminars: Students solve problems in the classroom, individually or in small groups, under the supervision of an instructor.


Annotation: within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.

Assessment

Continuous assessment activities

Title Weight Hours ECTS Learning outcomes
Recovery exam 90% 5 0.2 CM23, KM22, KM23, SM19
Seminars / problem assignments 10% 8 0.32 CM23, KM22, KM23, SM19
Second partial exam 45% 3 0.12 CM23, KM22, KM23, SM19
First partial exam 45% 3 0.12 CM23, KM22, KM23, SM19

Continuous assessment

In each of the two course blocks, j = 1, 2, the activities carried out during the seminars will receive a grade S_j, marked out of 10. The corresponding midterm examination will receive a grade E_j, also marked out of 10. The grade for each block, N_j, will be calculated according to the formula:

N_j = S_j/10 + (9/10) E_j

That is, the grade for each block will be the weighted average of the corresponding midterm examination (90%) and seminar activities (10%).

The final continuous assessment grade will be calculated as the geometric mean of the grades for the two blocks:

N = sqrt(N_1 N_2)

If this grade is equal to or greater than 5.0, the student will have passed the course through continuous assessment. If it is below 5.0, or if the student wishes to improve the grade obtained, they may take the resit examination.

Resit examination

The resit examination will consist of two parts, j = 1, 2, one for each of the two course blocks, each marked out of 10.

For each block, if the student submits the corresponding part of the resit examination, the grade obtained, R_j, will replace the corresponding block grade N_j. If the student chooses not to submit it, they will retain the grade N_j obtained through continuous assessment.

The final grade for each block, M_j, will therefore be:

M_j = N_j, if part j of the resit examination is not submitted.

M_j = R_j, if part j of the resit examination is submitted.

The final course grade will again be calculated as the geometric mean of the final grades for the two blocks:

N_final = sqrt(M_1 M_2)

Single assessment

Students who have opted for the single assessment system must take a final examination covering the entire course syllabus. This examination will account for 100% of the final grade and will not include seminar activities. It will be held on the same date, at the same time, and in the same location as the second midterm examination of the continuous assessment system.

Students who have opted for single assessment will also be entitled to take the same resit examination as students following the continuous assessment system. To be eligible, they must have taken the final examination. Under this assessment system, the resit examination grade will account for 100% of the final grade.

Legend

S_j: Seminar activity grade.

E_j: Midterm examination grade.

N_j: Block grade under continuous assessment.

R_j: Resit examination grade for the block.

M_j: Final block grade after the resit examination.

Note

Any irregularity committed during an assessment activity (academic fraud, plagiarism, or unauthorized use of AI, unless such use is expressly authorized in the course guide) that may lead to a significant alteration of the student's grade will result in that assessment activity being graded 0. If the course guide establishes that passing the course requires obtaining a minimum grade in that assessment activity, or if multiple irregularities are detected in assessment activities for the same course, the final course grade will be 0. Without prejudice to these academic consequences, the corresponding disciplinary proceedings may also be initiated against the student.


Bibliography

  • Course notes prepared by Emili Bagan and Carles Sánchez and made available to students through the Virtual Campus.
  • Elementary Differential Equations and Boundary Value Problems, W. E. Boyce & R. C. DiPrima, John Wiley & Sons (2012).
  • Mathematical Methods for Physicists, G. B. Arfken, H. J. Weber & F. E. Harris, Academic Press.
  • Partial Differential Equations: An Introduction, W. A. Strauss, John Wiley & Sons.


Software

Basic knowledge of Python.


Course groups and languages

The information provided is provisional until November 30. After this date, you will be able to consult the language of each group through this link. To access the information, you will need to enter the course CODE

Type of teaching Group Language Semester Shift
(TE) Theory 1 Catalan first semester morning-mixed
(PAUL) Classroom practices 1 Catalan/Spanish first semester morning-mixed
(TE) Theory 2 Catalan first semester morning-mixed
(PAUL) Classroom practices 2 Catalan/Spanish first semester morning-mixed
(SEM) Seminars 11 Catalan/Spanish first semester morning-mixed
(SEM) Seminars 12 Catalan/Spanish first semester morning-mixed
(SEM) Seminars 21 Catalan/Spanish first semester morning-mixed
(SEM) Seminars 22 Catalan/Spanish first semester morning-mixed