The subject "Differential Equations and Numerical Methods" is a second-year subject of the Bachelor's Degree of Biomedical Engineering. It is one of the five subjects of that degree assigned to the Applied Mathematics Department, and it belongs to the so-called Advanced Sciences Module. It contains the basic aspects of the Differential Equations and Numerical Analysis theory. Its main goal is to give the basic formation necessary to deal with other subjects.



Knowledge about differential equations is, for example, the starting point to solve Maxwell's Equations, which govern the generation, transmission, and reception of electromagnetic waves, which students will develop in applications based in communications, radars and microwaves.



Knowledge about numerical method will able students to solve equations that are not solvable analytically. These equations could be nonlinear equations, linear systems, differential equations or integrals.



A big part of the subject is based on concepts learnt in the first-year subject Algebra. Concepts and methods developed in Calculus I and Calculus II are also fundamental. Then, although no compulsory prerequisites apply to this subject, having been successful in the previously mentioned subjects is recommendable.

RAG05-The graduate will be able to identify the concepts and methods related to mathematics that are applicable in the field of engineering.



RAT1-The graduate will be able to solve problems with initiative, decision-making, creativity and critical reasoning.



G003 Knowledge of basic and technological subjects, enabling students to learn new methods and theories, and giving them the versatility to adapt to new situations.



T001 Ability to solve problems with initiative, decision-making, creativity, critical reasoning, respecting the principles of universal accessibility and design for all people.



M01FB01 Ability to solve mathematical problems that may arise in engineering. Ability to apply knowledge of: linear algebra; geometry; differential geometry; differential and integral calculus; differential and partial differential equations; numerical methods; numerical algorithms; statistics and optimisation.

CHAPTER 0: Introduction to numerical methods



CHAPTER 1: Numerical method for nonlinear equations and linear sistems



CHAPTER 2: First-order ordinary differential equations: analytical and numerical calculations.



CHAPTER 3: Systems of linear first-order ordinary differential equations



CHAPTER 4: Higher order linear ordinary differential equations



CHAPTER 5: The Laplace transform



CHAPTER 6: The Fourier transform



CHAPTER 7: Interpolation, integration and approximation of functions

The competences that the student must acquire in this module will be got through different activities carried out in lectures and computer laboratory sessions.



In the lectures the theoretical concepts will be dealt with and some exercises will be solved and others will be proposed to be solved by the students.



In the computer laboratory sessions, students will use a mathematical software (Matlab) to solve problems developed in the lectures.



Students will have access to several course materials through the virtual course (eGela).



A more personal tracking of students' learning could be made in the office hours.

• Continuous Assessment System
• Final Assessment System
• Tools and qualification percentages:
  • Written test to be taken (%): 80
  • Computer lab tests (Matlab) (%): 20

By default, at the beginning of the course all the students are enrolled in the continuous assessment system.



That evaluation will consist of:



- first written exam (Chapters 0, 1, 2 and 3), which will take place after finishing these chapters:40%.



- second written exam (Chapters 4, 5, 6 and 7), which will take place coinciding with the official

date established for this subject in the calendar of exams in the ordinary call: 40%.



- computer laboratory tests: 20%. They will take place during the computer laboratory sessions.



Thus, the final grade will be calculated as follows:



0.4*grade of the first written exam + 0.4*grade of the second written exam + 0.2*grade of the computer laboratories.



Not taking an exam, test, etc. will result in a grade of zero in that test





REQUIREMENTS TO PASS THE SUBJECT:



To pass the subject the final grade should be greater than or equal to 5 over 10 and, simultaneously, the following requirements must be fulfilled:



- a minimum weighted average grade of 4 over 10 in the written parts.



- a minimum average grade of 3.5 over 10 in the computer laboratories.



Those students, who having an average grade greater than or equal to 5 over 10, do not satisfy some of the previous conditions will fail the call with grade of 4.9.





NOTES:



1: RENUNCIATION TO THE CONTINUOUS EVALUATION.



Students have the right to be evaluated with a final evaluation, no matter if they have participated in the continuous evaluation or not. For that, the students must present in writing the renunciation to the continuous evaluation to their lecturer, not later than week 9 from the beginning of the term, according to the academic calendar of the School.



In this case the exam will consists of two parts: a written part and a computer part. Grades will be distributed as follows:



- Written part: 80%. This exam will contain all the knowledge seen in the theoretical lectures.



- Computer laboratories exam (about all the practicals made during the course): 20%



To pass the subject the final grade should be greater than or equal to 5 over 10 and, simultaneously, the following requirements must be fulfilled:



- a minimum grade of 4 over 10 in the written exam.



- a minimum average grade of 3.5 over 10 in the computer laboratories exam.



Those students, who having an average grade greater than or equal to 5 over 10, do not satisfy some of the previous conditions will fail the call with grade of 4.9.



2: RENUNCIATION TO THE CALL.



- Not attending the exam in the official date of exams will imply the automatic renunciation to

the corresponding call.

The exam will have two parts: a written part and one computer part. Grades will be distributed as follows:



- Written part about theoretical knowledge: 80%



- Computer laboratories exam (about the entire computer practical made during the course): 20%



To pass the subject the final grade should be greater than or equal to 5 over 10 and, simultaneously, the following requirements must be fulfilled:



- a minimum grade of 4 over 10 in the written exam



- a minimum average grade of 3.5 over 10 in the computer laboratories exam.



Those students who, having an average grade greater than or equal to 5 over 10, do not satisfy some of the previously mentioned requirements will be graded as "SUSPENSO, 4.9" ("FAILED, 4.9").





RENUNCIATION TO THE CALL:



Not taking the exam will be enough to renounce to this second call.



OTHER CLARIFICATIONS:



- Those students, who attended the ordinary call and failed it, but got an average weighted grade greater than or equal to 5 over 10 in the written part or in the computer lab part do not have to take the passed part in the extraordinary call, unless they resign by writting to the lecturer of the course.

Class and computer practical notes of the subject available on eGela.

Basic bibliography

- "Ecuaciones Diferenciales con aplicaciones y notas históricas". F. SIMMONS, Ed McGraw-Hill, 1993.



- "Fundamentos de Ecuaciones Diferenciales". R.K. NAGLE y E.B. SAFF. - Ed Addison-Wesley iberoamericana, 1992.



- "Ecuaciones Diferenciales y Problemas de valores en la frontera". W.E. BOYCE y R.C. DIPRIMA. Ed Limusa, 1991.



- "Análisis Numérico" 7ª ed. Burden, Richard L. & Faires, J. Douglas International Thomson. 2002.



- "Variable compleja y aplicaciones" Churchill y Brown - McGraw-Hill - 1993



- "MATLAB: Una introducción con ejemplos prácticos".A.Gilat, Ed. Reverté, Barcelona 2006.



- "MATLAB: An Introduction with Applications". A.Gilat, Ed. John Wiley & Sons, 2004.



- "Applied Numerical Methods with MATLAB for engineers and scientists".



S.C.Chapra. McGraw-Hill Higher Education, 2008.

In-depth bibliography

- "Differential Equation Analysis in Biomedical Science and Engineering: Ordinary Differential Equation Applications with R". John Wiley & Sons. Schiesser, W. E. (2014).

- "Ecuaciones Diferenciales. Un enfoque de modelado". G. LEDDER W.H. Ed. Mc Graw-Hill, 2006.

- "Numerical mathematics and computing" 6th Cheney, E.W. & Kincaid, David ed. Brooks Cole. 2007.

- "Métodos numéricos para ingenieros" .Chapra, Steven C. & Canale, Raymond P.5ª ed. McGraw-Hill. 2007.

- Strang G., "Introduction to Applied Mathematics", Wellesley-Cambridge Press, 1986.

- "Scientific Computing with MATLAB and Octave". A.Quarteroni, F.Saleri. Springer, 2006.

Web addresses

egela.ehu.es
http://www.ehu.es/
http://www.ingenierosbilbao.com/
https://en.wikipedia.org/wiki/MATLAB
https://en.wikibooks.org/wiki/MATLAB_Programming
https://es.mathworks.com/help/matlab/index.html
https://es.mathworks.com/help/matlab/getting-started-with-matlab.html