The final qualification of the subject consists of three parts:

Qualification of internships through ICT (CP): between 0 and 2.5 points
Problem Solving Qualification (CR): between 0 and 2.5 points
Mixed test qualification (CE): between 0 and 5 points.

The final qualification will be the sum of three parts: Final_Note= CP + CR + CE, if the qualification of the mixed test (CE) is greater than 1.3 (over 5 points). In other case, the final qualification will be the mark obtained on the mixed test, CE.

The qualifications of practices through ICT (CR) and problem solving (CP) will be retained in the second opportunity of the evaluation.

Students who do not show up for the final mixed test will be considered as "Not presented".

Observations on o “Students with recognition of part-time dedication and academic exemption from attendance exemption”: As specific personalized attention measures for o “Students with recognition of part-time dedication and academic exemption from attendance exemption” for or study da matter, the continuous assessment of the practices through ICT and the resolution of problems will be carried out through online partial tests.

Learning outcomes	Study programme competences
Identify the different types of differential equations and problems associated with them. Especially those originating in nanoscience and nanotechnology	A3 A7	B2 B4 B6 B7 B8 B9	C3 C9
Know and acquire fluency in the techniques to obtain analytical and numerical solutions of models based on ordinary differential equations	A3 A7	B2 B4 B6 B7 B8 B9 B12	C7 C8 C9
Know and acquire fluency in the techniques to obtain analytical and numerical solutions of models based on partial differential equations	A3	B2 B5 B10 B11	C3 C7 C8 C9
Have criteria to choose the most efficient analytical and numerical techniques for models of real problems, especially those related to nanoscience and nanotechnology.	A3	B2 B4 B5 B6 B7 B8 B9 B10 B11 B12	C3 C7 C8 C9
Manage software tools that implement the methodologies studied and know how to analyze the results	A3 A7	B2 B4 B5 B6 B7 B9 B10 B12	C3 C9

Topic	Sub-topic
Unit 1: First order ordinary differential equations	- Initial value problem - Analytic resolution - Mathematical models - Numerical resolution: Explicit Euler, Implicit Euler, Heun, Runge-Kutta. - Aplications
Unit 2: Systems of differential equations	- Systems of differential equations - Analytic resoluiton - Estability - Mathematical models - Numerical schemes: Explicit Euler, Implicit Euler, Heun, Runge-Kutta. - Applications
Unit 3: Second order ordinary differential equations	- Initial value problem. - Analytic resolution. Laplace transform, Fourier transform. - Mathematical models. - Numerical resoltion - Aplications - Contour problems - Analytic resolution. - Numerical resolution. Finite difference method. - Sturm-Liouville problems. Numerical approximation of eigenvalues and eigenfunctions - Aplications
Tema 4: Ecuacións en derivadas parciais.	- Ecuación de transporte. Resolución analítica mediante o método de características. Resolución numérica mediante el esquema de Godunov. - Ecuación do calor 1D. Resolución analítica mediante separación de variables. Resolución numérica por diferencias finitas. - Ecuación de ondas 1D. Resolución analítica mediante separación de variables. Resolución numérica por diferencias finitas. - Ecuación de Laplace e Poisson. Resolución analítica mediante separación de variables. Resolución numérica por diferencias finitas - Ecuación de calor 2D. Resolución analítica mediante separación de variables. Resolución numérica por diferencias finitas. - Ecuación de Schrödinger. Resolución analítica mediante separación de variables.. Resolución numérica por diferencias finitas. - Aplicacións

Methodologies / tests	Competencies	Ordinary class hours	Student’s personal work hours	Total hours
Guest lecture / keynote speech	A3 B2 B4 B5 B6 B7 B11 C8	28	56	84
ICT practicals	A3 A7 B2 B4 B10 C3 C7 C9	12	25	37
Problem solving	A7 B8 B12	8	16	24
Mixed objective/subjective test	B7 B9 C9	3	0	3

Personalized attention		2	0	2

(*)The information in the planning table is for guidance only and does not take into account the heterogeneity of the students.

Methodologies	Description
Guest lecture / keynote speech	Exhibition of the contents specified in the program of the subject, for which audiovisual media (tablet) will be used.
ICT practicals	Interactive practices in which relevant problems in the field of Science and Engineering will be solved, using Python programming language.
Problem solving	Sessions where relevant problems in the field of Science and Engineering will be presented, which will be solved both analytically and numerically. The student must be able to reach the solution of any problem using pencil and paper or alternatively using computer tools (using Python), and compare the results.
Mixed objective/subjective test	Development of issues and problems of the subject.

Methodologies	Competencies	Description	Qualification
Mixed objective/subjective test	B7 B9 C9	Test that includes the resolution of questions and problems of the subject (by hand and/or Python)	50
Problem solving	A7 B8 B12	Rosolution of practical problems	25
ICT practicals	A3 A7 B2 B4 B10 C3 C7 C9	Resolution of practical problems using the Python programming language	25

Assessment comments
The final qualification of the subject consists of three parts: Qualification of internships through ICT (CP): between 0 and 2.5 points Problem Solving Qualification (CR): between 0 and 2.5 points Mixed test qualification (CE): between 0 and 5 points. The final qualification will be the sum of three parts: Final_Note= CP + CR + CE, if the qualification of the mixed test (CE) is greater than 1.3 (over 5 points). In other case, the final qualification will be the mark obtained on the mixed test, CE. The qualifications of practices through ICT (CR) and problem solving (CP) will be retained in the second opportunity of the evaluation. Students who do not show up for the final mixed test will be considered as "Not presented". Observations on o “Students with recognition of part-time dedication and academic exemption from attendance exemption”: As specific personalized attention measures for o “Students with recognition of part-time dedication and academic exemption from attendance exemption” for or study da matter, the continuous assessment of the practices through ICT and the resolution of problems will be carried out through online partial tests.

Basic	Richard G. Rice, Duong D. Do (2012). Applied Mathematics And Modeling For Chemical Engineers (2º ed). John Wiley & Sons Wei-Chau Xie (2014). Differential Equations for Engineers (2º ed). Cambridge University Press Stephen Lynch (2018). Dynamical Systems with Applications using Python. Springer Dennis G. Zill (2018). Ecuaciones diferenciales con problemas con valores en la frontera (9ª ed). Cengage C. Henry Edwards, David E. Penney (2017). Ecuaciones diferenciales y problemas con valores en la frontera. Cómputo y modelado (4ª ed). Pearson Education William E. Boyce, Richard C. DiPrima, Douglas B. Meade (2017). Elementary Differential Equations and Boundary Value Problems, (11ª Ed). Willey

Complementary	George F. Simmons (2016). Differential Equations with Applications and Historical Notes. Chapman and Hall/ William E. Boyce, Richard C. DiPrima, Douglas B. Meade (2017). Elementary Differential Equations and Boundary Value Problems, Student Solutions Manual, (11ª Ed). Wiley Steven C. Chapra , Raymond P. Canale (2015). Métodos Nméricos para Ingenieros (7ª ed). McGraw-Hill J. C. Butcher (2016). Numerical Methods for Ordinary Differential Equations, (3ª ed). Wiley Svein LingeHans Petter Langtangen (2017). Programming for Computations - Python A Gentle Introduction to Numerical Simulations with Python. Springer

Other comments
Estudio diario dos contidos tratados na aula, complementándoos coa bibliografía recomendada. Para axudar a conseguir unha contorna inmediata sustentable e cumprir co punto 6 da "Declaración Ambiental da Facultade de Ciencias (2020)", os traballos documentais que se realicen nesta materia:  Solicitaranse maioritariamente en formato virtual e soporte informático.