Public University of Navarre

Academic year: 2024/2025 | Previous academic years:  2023/2024  |  2022/2023  |  2021/2022  |  2020/2021 
Bachelor's degree in Industrial Engineering at the Universidad Pública de Navarra
Course code: 252301 Subject title: MATHEMATICS III
Credits: 6 Type of subject: Mandatory Year: 2 Period: 1º S
Department: Estadística, Informática y Matemáticas
PALACIAN SUBIELA, JESUS FCO. (Resp)   [Mentoring ]

Partes de este texto:


Module/Subject matter

Industrial common training module / Applied mathematics to engineering



Laplace transforms. Properties. Transforms calculation. Inverse Laplace transform. Properties of Laplace transforms. Application to the solution of ordinary differential equations and integro-differential equations. Applications in engineering.
Fourier series. Related integrals. Real and complex forms.
Fourier transforms. Properties. Transforms calculation. Inverse Fourier transform. Properties of Fourier transforms. Application to the solution of boundary differential equations. Applications in engineering.


General proficiencies

  • CG3: Background in basic and technological subjects, enabling the student the learning of new theories and methods to, providing him/her enough versatility to adapt to new situations


Specific proficiencies

  • CFB1: Ability to solve mathematical problems arising in Engineering. Aptitude to apply knowledge about linear algebra, geometry, differential geometry, differential and integral calculus, ordinary and partial differential equations, numerical methods, numerical algorithms, statistics and optimisation.

  • CFB3: Basic knowledge on use and computer programming, operating systems, data bases and software application in engineering.


Learning outcomes

At the end of the training period the student is able to:

  • R1- Get some knowledge on basic topics on numerical and power series, improperios integration and parametric integration.
  • R2 - Handle basic concepts in complex analysis such as representation of complex numbers, complex functions of a real variable and elementary complex functions.
  • R3 - Get some knowledge of Laplace and Fourier transforms as well as acquire basic foundations of Fourier series analysis.
  • R4 - Get basic concepts and terminology off partial differential equations. Classify second order linear partial differential equations.
  • R5 - Solve engineering problems modeled through ordinary differential equations applying Laplace transforms.
  • R6 - Decompose and process signals applying methods based on Fourier series.
  • R7 - Solve some second order linear partial differential equations applying the method of separation of variables and integral transforms.



Methodology - Activity Attendance Self-study
A-1 Exposition/Participative Classes  43  
A-2 Practical classes  13  6
A-3 Individual practice and study time    75
A-4 Exams and evaluation activities  4  
A-5 Tutorials    9
Total  60  90






Continuous assessment

Weight (%) It allows
test resit
required grade
R1, R2, R3, R5 Mid-term exam (lessons 1, 2, 3): solving problems with a similar level of difficulty of those given in the class 50% yes 4
R3, R4, R6, R7 Exam of lessons 4, 5, 6: solving problems with a similar level of difficulty of those given in the class 50% yes 4


Resit exam

Weight (%) It allows
test resit
required grade
R1, R2, R3, R4, R5, R6, R7 Final exam for the students that did not pass the subject through continuous assessment 100% no 5





1. Supplements of Calculus     

1.1 Numerical sequences and series. Series convergence tests

1.2 Power series. Convergence tests. Taylor series

1.3 Improper integrals. Convergence tests. Parametric integrals: derivation

1.4 Eulerian functions 

1.5 Lab/class practice


2.  Introduction to Complex Numbers 

2.1 The set C of complex numbers

2.2 Binomial and polar forms. Modulus and argument. Euler formula

2.3 Polynomial functions. Fundamental Theorem of Algebra

2.4 Rational functions. Zeroes and poles

2.5 Complex-valued functions of a real variable: derivation and integration. Complex functions of a complex

variable: elementary examples

2.6  Lab/class practice


3.  Laplace transform     

3.1 Definition and existence conditions

3.2 Inverse Laplace transforms

3.3 Fundamental properties

3.4 Convolution and impulse. Transfer functions

3.5 Application to ordinary differential equations, initial value problems and integro-differential equations

3.6 Lab/class practice


4.  Fourier series

4.1 Hilbert spaces, orthonormal sequences and general Fourier series

4.2 Trigonometric Fourier series

4.3 Convergence theorems. Parseval´s identity

4.4 Periodic extensions, odd and even periodic extensions

4.5 Lab/class practice


5.  Fourier transform

5.1 Definition and examples. Fourier Integral Theorem. Riemann-Lebesgue Theorem

5.2 Basic properties

5.3 Convolution

5.4 Applications: solution of ordinary differential equations

5.5 Lab/class practice


6.  Partial differential equations: separation of variables and transform methods

6.1 Introduction to partial differential equations

6.2 Second order linear partial equations: classification and examples

6.3 Heat equation, wave equation and Laplace equation

6.4 Boundary problems. Method of separation of variables

6.5 Sturm-Liouville Theory: application to the solution of linear partial differential equations of second order

6.6 Transform methods for solving partial differential equations in unbounded domains:

potential equation, heat transmission and vibrations

6.7 Lab/class practice





Access the bibliography that your professor has requested from the Library.


  1. López García, J.L. y Yanguas Sayas, P.: Mathematics III: Integral Transforms, Fourier Series & PDEs, 2016. 
  2. Andrews, L.C. y Shivamoggi, B. K.: Integral transforms for engineers and applied mathematicians, MacMillan, 1999.
  3. Kreyszig, E.: Matemáticas avanzadas para Ingeniería, Limusa, 2012.
  4. Nagle, R. K., Saff, E. B. y Snider, D. A.: Ecuaciones diferenciales y problemas con valores en la frontera, Pearson Educación, 2008.



  1. Bracewell, R. N.: The Fourier transform and its applications, McGraw-Hill, 2000.
  2. O'Neil, P. V.: Matemáticas avanzadas para Ingeniería, Thomson, 2014.







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