Universidad Pública de Navarra - Nafarroako Univertsitate Publikoa

Module/Subject matter

Industrial common training module / Applied mathematics to engineering

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Contents

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.

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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

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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.

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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.

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Methodology

Methodology - Activity	Attendance	Self-study
A-1 Exposition/Participative Classes	41
A-2 Practical classes	13	6
A-3 Individual practice and study time		75
A-4 Exams and evaluation activities	6
A-5 Tutorials		9
Total	60	90

 

 

 

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Languages

English

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Evaluation

The system designed for evaluation has three different options to pass the course, namely, option 1: pass the two partial exams that will be held in the regular session of classes; option 2: pass the ordinary exam that will be held in January; option 3: pass the extraordinary exam that will be held in January. In the tables given below more details are given about percentages, the lessons and related questions. 

 

Option 1

 

Learning outcome	Evaluation system	Weight (%)	Possibility of resit
R1, R2, R3, R5	Partial exam (lessons 1,2, 3): problems solving with the same level of difficulty of those given in the class	50%	yes, passing the ordinary or extraordinary final exams
R3, R4, R6, R7	Partial exam (lessons 4, 5, 6 and 7): problems solving with the same level of difficulty of those given in the class	50%	yes, passing the ordinary or extraordinary final exams

 

 

Option 2

 

Learning outcome	Evaluation system	Weight (%)	Possibility of resit
R1, R2, R3, R4, R5, R6, R7	Final exam for the students that did not pass the subject through option 1 problems solving with the same level of difficulty of those given in class.	100	Yes

 

 

Option 3

 

Learning outcome	Evaluation system	Weight (%)	Possibility of resit
R1,R2,R3,R4,R5,R6,R7	Final exam for those who has not passed the course using options 1 and 2: problems solving with the same level of difficulty of those given in the class	100%	no

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Agenda

1.  Calculus Supplements     

1.1 Numerical sequences and series. Series convergence tests

1.2 Power series. Convergence tests

1.3 Taylor series

1.4 Improper integrals. Convergence tests. Parametric integrals: derivation

1.5 Eulerian functions. Special functions

1.6 Lab 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 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 practice

 

4.  Fourier series

4.1 Trigonometric Fourier series

4.2 Convergence theorems. Parseval´s identity

4.3 Periodic extensions, odd and even periodic extensions

4.4 Hilbert spaces, orthonormal sequences and generalised Fourier series

4.5 Lab practice

 

5.  Problems of Sturm-Liouville

5.1 Eigenvalues and eigenfunctions of linear differential operators

5.2 Regular and periodic problems. Properties of the solutions

5.3 Introduction to singular problems: practical examples, related special functions

5.4 Lab practice

 

6.  Partial differential equations: separation of variables

6.1 Basics on 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 Lab practice

 

7.  Fourier transform

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

7.2 Basic properties

7.3 Convolution

7.4 Applications: solution of differential equations in unbounded domains, potential equation, heat transmission

and vibrations

7.5 Lab practice

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Bibliography

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

BASIC BIBLIOGRAPHY:

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, 1988.
3. Kreyszig, E.: Matemáticas avanzadas para Ingeniería, Limusa, 2000.
4. Nagle, R. K., Saff, E. B. y Snider, D. A.: Ecuaciones diferenciales y problemas con valores en la frontera, Pearson Educación, 2005.

 

SUPPLEMENTARY BIBLIOGRAPHY:

1. Bracewell, R. N.: The Fourier transform and its applications, McGraw-Hill, 1986.
2. O'Neil, P. V.: Matemáticas avanzadas para Ingeniería, Thomson, 2004.

 

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Location

Aulario building

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