1. General information
Course:
DIFFERENTIAL EQUATIONS
Code:
38310
Type:
BASIC
ECTS credits:
6
Degree:
345 - UNDERGRADUATE DEGREE PROGRAMME IN CIVIL ENGINEERING
Academic year:
2022-23
Center:
603 - E.T.S. CIVIL ENGINEERS OF CR
Group(s):
20
Year:
2
Duration:
First semester
Main language:
Spanish
Second language:
English
Use of additional languages:
English Friendly:
Y
Web site:
Bilingual:
N
Lecturer:
GABRIEL FERNANDEZ CALVO
- Group(s):
20
Building/Office
Department
Phone number
Email
Office hours
Politecnico 2-D31
MATEMÁTICAS
6218
gabriel.fernandez@uclm.es
Will be provided at the beginning of the Course
Lecturer:
ROSA EVA PRUNEDA GONZALEZ
- Group(s):
20
Building/Office
Department
Phone number
Email
Office hours
Politecnico 2-D33
MATEMÁTICAS
3292
rosa.pruneda@uclm.es
Will be provided at the beginning of the Course
2. Pre-Requisites
Solving problems modeled by Ordinary and Partial Differential Equations is based on the concepts acquired in "Instrumentos Matemáticos I", "Instrumentos Matemáticos I" and "Herramientas Matemático-Informáticas para la Ingeniería".
3. Justification in the curriculum, relation to other subjects and to the profession
This course provides the necessary skills for solving engineering problems involvingdifferential equations. The different techniques and concepts studied have direct application in many areas of Civil Engineering and they will be useful in subjects as Technology of Materials,Geotechnics, Hydraulic Engineering and Hydrology or Maritime and Coastal Engineering; Economics; Transportation, etc. Applications such as beam deflection, consolidation equation or wave equation in maritime engineering will be addressed and modeled by differential equations.
4. Degree competences achieved in this course
| Course competences | |
|---|---|
| Code | Description |
| CE01 | Students can apply their knowledge in the practical solution of civil engineering problems, with capacity for the analysis and definition of the problem, the proposal of alternatives and their critical evaluation, choosing the optimal solution with technical arguments and with capacity of defense against third parties. |
| CE02 | Students have the ability to broaden their knowledge and solve problems in new or unfamiliar environments within broader (or multidisciplinary) contexts related to their area of study. Self-study ability, to undertake further studies with a high degree of autonomy |
| CE04 | Students have the 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 derivative equations; numerical methods; numerical algorithms; statistics and optimization. |
| CE06 | Students have a basic knowledge of the use and programming of computers, operating systems, databases and software with engineering application. |
| CG01 | Students achieve general knowledge of Information and Communication Technologies (ICT). |
5. Objectives or Learning Outcomes
| Course learning outcomes | |
|---|---|
| Description | |
| Students know how functions and data are approximated by means of power and Fourier series expansions and their applications. | |
| Students are able to express correctly both orally and in writing and, in particular, they can use the language of mathematics as a way of expressing accurately the quantities and operations in civil engineering. Students get used to teamwork and behave respectfully. | |
| Students use mathematical and computer tools to pose and solve civil engineering problems. | |
| Students can describe processes related to civil engineering subjects by means of ordinary and partial differential and equations, solve them and interpret their results. | |
| Students learn the most important approximations for numerical method resolution, use some statistical, data processing, mathematical calculation and visualization software packages at user level, develop algorithms and program using a high-level programming language, visualize functions, geometric shapes and data, design experiments, analyze data, and interpret results. | |
| Additional outcomes | |
| Not established. |
6. Units / Contents
-
Unit 1: INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS: Ordinary differential equations. Order and Degree. Linear differential equations. Notation. Definition of solution. Particular and general solutions. Initial value problems. Limit value problems. Classification of ordinary differential equations of the first order. Ordinary and differential form. Classification of first order ordinary differential equations.
-
Unit 2: SEPARABLE DIFFERENTIAL EQUATIONS: General solution. Initial value problems. Homogeneous differential equations.
-
Unit 3: EXACT DIFFERENTIAL EQUATIONS: Definition. Resolution. Integration factors.
-
Unit 4: FIRST ORDER LINEAR DIFFERENTIAL EQUATIONS: Resolution. Applications.
-
Unit 5: HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS: Characteristic equation. Homogeneous equation resolution. Particular solution. Undetermined coefficients method. Variation of parameters.
-
Unit 6: LINEAR DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS: Introduction. Analytical functions. Ordinary points and singular points. Solutions by series of powers around an ordinary point. Method for homogeneous equations. Method for non-homogeneous equations.
-
Unit 7: LINEAR SYSTEMS WITH CONSTANT COEFFICIENTS: Introduction. Resolution of the initial value problem. Comparison of the solution methods. Reduction of a system of linear differential equations to a first-order system.
-
Unit 8: NUMERICAL METHODS FOR ODE: Introduction and motivation. Discretization of initial value ODE. Euler method. Heun method. Order of a numerical method. Runge-Kutta methods. Numerical resolution of EDO systems. Problems of the contour values: shooting method. Use of MATLAB to solving ODEs numerically.
-
Unit 9: STURM-LIOUVILLE PROBLEMS: Definition. Resolution. Fourier series.
-
Unit 10: PHYSICAL SYSTEMS AND PARTIAL DIFFERENTIAL EQUATIONS: Model. Resolution. Classification of partial differential equations. Second order problems. Reduction to canonical forms.
-
Unit 11: PARABOLIC PROBLEMS. DIFUSSION EQUATION: Diffusion problems: heat equation. Boundary conditions. Separation of variables. Resolution.
-
Unit 12: HYPERBOLIC PROBLEMS. WAVE EQUATION: The wave equation in one dimension. D'Alembert Solution. Boundary conditions associated with the wave equation. Finite string vibrating. Separation of variables.
-
Unit 13: ELLIPTIC PROBLEMS. LAPLACE EQUATION: Laplacian. Nature of problems with boundary conditions. Dirichlet problems.
-
Unit 14: NUMERICAL METHODS FOR PDE: Finite difference method applied to heat, wave and Laplace equations. Use of MATLAB to solve PDE numerically.
7. Activities, Units/Modules and Methodology
| Training Activity | Methodology | Related Competences (only degrees before RD 822/2021) | ECTS | Hours | As | Com | Description | |
| Class Attendance (theory) [ON-SITE] | Lectures | CE01 CE02 CE04 CE06 CG01 | 0.8 | 20 | N | N | Magistral lessons will be complemented with the resolution of exercises and the participation of the students in class. | |
| Group tutoring sessions [ON-SITE] | Problem solving and exercises | CE01 CE02 CG01 | 0.2 | 5 | N | N | Theoretical and practical student doubts will be solved in tutorials. | |
| Progress test [ON-SITE] | Problem solving and exercises | CE01 CE02 CG01 | 0.2 | 5 | Y | N | Recoverable | |
| Study and Exam Preparation [OFF-SITE] | Combination of methods | CE01 CE02 CE04 CE06 CG01 | 3.6 | 90 | N | N | ||
| Class Attendance (practical) [ON-SITE] | Project/Problem Based Learning (PBL) | CE01 CE02 CE04 CE06 CG01 | 0.6 | 15 | N | N | ||
| Computer room practice [ON-SITE] | Project/Problem Based Learning (PBL) | CE01 CE02 CE04 CE06 CG01 | 0.4 | 10 | Y | Y | Indispensable to pass the subject. Details on content, extension and requirements of the works or practices that have to be delivered in writing will be indicated in Campus Virtual at the beginning of the semester. The minimum score for those computer practices belonging to the part dedicated to Numerical Methods is 4.0 points out of 10. | |
| Final test [ON-SITE] | Assessment tests | CE01 CE02 CG01 | 0.2 | 5 | Y | Y | Recoverable. | |
| Total: | 6 | 150 | ||||||
| Total credits of in-class work: 2.4 | Total class time hours: 60 | |||||||
| Total credits of out of class work: 3.6 | Total hours of out of class work: 90 | |||||||
As: Assessable training activity Com: Training activity of compulsory overcoming (It will be essential to overcome both continuous and non-continuous assessment).
8. Evaluation criteria and Grading System
| Evaluation System | Continuous assessment | Non-continuous evaluation * | Description |
| Progress Tests | 40.00% | 0.00% | |
| Final test | 60.00% | 100.00% | |
| Total: | 100.00% | 100.00% |
According to art. 4 of the UCLM Student Evaluation Regulations, it must be provided to students who cannot regularly attend face-to-face training activities the passing of the subject, having the right (art. 12.2) to be globally graded, in 2 annual calls per subject , an ordinary and an extraordinary one (evaluating 100% of the competences).
Evaluation criteria for the final exam:
-
Continuous assessment:Topics 1 to 7 (partial 1) and 9 to 13 (partial 2) from the Analytical Methods part are evaluated by the average of 2 partial exams (minimum score 4 out of 10) (60%) and progress tests (40%). In the exams, a minimum grade of 4 out of 10 points is required. If a partial exam and/or progress test is passed with minimum score of 4 out of 10, the contents of that partial or progress tests are released for the Ordinary and the Extraordinary call. Progress tests can be retrieved in the Ordinary call by assigning the punctuation obtained in the exam.
Topics 8 and 14 (Numerical Methods) are evaluated exclusively through a MANDATORY practice each (60%) and a final test (40%), which will take place on the dates of the ordinary/extraordinary exams.
The final grade of the Course consists of the marks from both the Analytical Methods (80%) and the Numerical Methods (20%). Indispensable to pass the subject. The minimum score for those computer practices belonging to the part dedicated to Numerical Methods is 4.0 points out of 10. The marks of the computer practices done by the student during the previous academic year will be kept for the next academic year as long as those marks were at least 4.0 points out of 10. -
Non-continuous evaluation:The student will have to do a global exam that will include all the course and competences content. To pass the course, the student must obtain at least a 5 out of 10, which will constitute 100% of his/her grade.
Unless stated otherwise, continuous evaluation criteria will be applied to all students.
Anyone choosing non-continuous assessment must notify it to the lecturer within the class period of the subject. The option is only available if the student's participation in evaluation activities (from the continuous assessment) has not reached 50% of the total Course evaluation.
For the retake exam, the assessment type used for the final exam will remain valid.
Specifications for the resit/retake exam:
Same criteria that apply in final exam.
Specifications for the second resit / retake exam:
The student will have to do a global exam that will include all the course and competences content. To pass the course, the student must obtain at least a 5 out of 10, which will constitute 100% of his/her grade.
9. Assignments, course calendar and important dates
| Not related to the syllabus/contents | |
|---|---|
| Hours | hours |
| Progress test [PRESENCIAL][Problem solving and exercises] | 5 |
| Study and Exam Preparation [AUTÓNOMA][Combination of methods] | 8 |
| Final test [PRESENCIAL][Assessment tests] | 5 |
| Unit 1 (de 14): INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS: Ordinary differential equations. Order and Degree. Linear differential equations. Notation. Definition of solution. Particular and general solutions. Initial value problems. Limit value problems. Classification of ordinary differential equations of the first order. Ordinary and differential form. Classification of first order ordinary differential equations. | |
|---|---|
| Activities | Hours |
| Class Attendance (theory) [PRESENCIAL][Lectures] | 1 |
| Study and Exam Preparation [AUTÓNOMA][Combination of methods] | 1 |
| Unit 2 (de 14): SEPARABLE DIFFERENTIAL EQUATIONS: General solution. Initial value problems. Homogeneous differential equations. | |
|---|---|
| Activities | Hours |
| Class Attendance (theory) [PRESENCIAL][Lectures] | 1 |
| Study and Exam Preparation [AUTÓNOMA][Combination of methods] | 6 |
| Unit 3 (de 14): EXACT DIFFERENTIAL EQUATIONS: Definition. Resolution. Integration factors. | |
|---|---|
| Activities | Hours |
| Class Attendance (theory) [PRESENCIAL][Lectures] | 1 |
| Group tutoring sessions [PRESENCIAL][Problem solving and exercises] | .5 |
| Study and Exam Preparation [AUTÓNOMA][Combination of methods] | 5 |
| Class Attendance (practical) [PRESENCIAL][Project/Problem Based Learning (PBL)] | 1 |
| Unit 4 (de 14): FIRST ORDER LINEAR DIFFERENTIAL EQUATIONS: Resolution. Applications. | |
|---|---|
| Activities | Hours |
| Class Attendance (theory) [PRESENCIAL][Lectures] | 2 |
| Group tutoring sessions [PRESENCIAL][Problem solving and exercises] | .25 |
| Study and Exam Preparation [AUTÓNOMA][Combination of methods] | 6 |
| Class Attendance (practical) [PRESENCIAL][Project/Problem Based Learning (PBL)] | 1 |
| Unit 5 (de 14): HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS: Characteristic equation. Homogeneous equation resolution. Particular solution. Undetermined coefficients method. Variation of parameters. | |
|---|---|
| Activities | Hours |
| Class Attendance (theory) [PRESENCIAL][Lectures] | 2 |
| Group tutoring sessions [PRESENCIAL][Problem solving and exercises] | .25 |
| Study and Exam Preparation [AUTÓNOMA][Combination of methods] | 7 |
| Class Attendance (practical) [PRESENCIAL][Project/Problem Based Learning (PBL)] | 3 |
| Computer room practice [PRESENCIAL][Project/Problem Based Learning (PBL)] | 2 |
| Unit 6 (de 14): LINEAR DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS: Introduction. Analytical functions. Ordinary points and singular points. Solutions by series of powers around an ordinary point. Method for homogeneous equations. Method for non-homogeneous equations. | |
|---|---|
| Activities | Hours |
| Class Attendance (theory) [PRESENCIAL][Lectures] | 1 |
| Group tutoring sessions [PRESENCIAL][Problem solving and exercises] | .5 |
| Study and Exam Preparation [AUTÓNOMA][Combination of methods] | 5 |
| Class Attendance (practical) [PRESENCIAL][Project/Problem Based Learning (PBL)] | 1 |
| Unit 7 (de 14): LINEAR SYSTEMS WITH CONSTANT COEFFICIENTS: Introduction. Resolution of the initial value problem. Comparison of the solution methods. Reduction of a system of linear differential equations to a first-order system. | |
|---|---|
| Activities | Hours |
| Class Attendance (theory) [PRESENCIAL][Lectures] | 2 |
| Group tutoring sessions [PRESENCIAL][Problem solving and exercises] | .5 |
| Study and Exam Preparation [AUTÓNOMA][Combination of methods] | 7 |
| Class Attendance (practical) [PRESENCIAL][Project/Problem Based Learning (PBL)] | 3 |
| Computer room practice [PRESENCIAL][Project/Problem Based Learning (PBL)] | 1 |
| Unit 8 (de 14): NUMERICAL METHODS FOR ODE: Introduction and motivation. Discretization of initial value ODE. Euler method. Heun method. Order of a numerical method. Runge-Kutta methods. Numerical resolution of EDO systems. Problems of the contour values: shooting method. Use of MATLAB to solving ODEs numerically. | |
|---|---|
| Activities | Hours |
| Class Attendance (theory) [PRESENCIAL][Lectures] | 1 |
| Group tutoring sessions [PRESENCIAL][Problem solving and exercises] | .5 |
| Study and Exam Preparation [AUTÓNOMA][Combination of methods] | 6 |
| Class Attendance (practical) [PRESENCIAL][Project/Problem Based Learning (PBL)] | 1 |
| Computer room practice [PRESENCIAL][Project/Problem Based Learning (PBL)] | 3 |
| Unit 9 (de 14): STURM-LIOUVILLE PROBLEMS: Definition. Resolution. Fourier series. | |
|---|---|
| Activities | Hours |
| Class Attendance (theory) [PRESENCIAL][Lectures] | 1 |
| Study and Exam Preparation [AUTÓNOMA][Combination of methods] | 5 |
| Class Attendance (practical) [PRESENCIAL][Project/Problem Based Learning (PBL)] | 1 |
| Unit 10 (de 14): PHYSICAL SYSTEMS AND PARTIAL DIFFERENTIAL EQUATIONS: Model. Resolution. Classification of partial differential equations. Second order problems. Reduction to canonical forms. | |
|---|---|
| Activities | Hours |
| Class Attendance (theory) [PRESENCIAL][Lectures] | 1 |
| Group tutoring sessions [PRESENCIAL][Problem solving and exercises] | .5 |
| Study and Exam Preparation [AUTÓNOMA][Combination of methods] | 1 |
| Class Attendance (practical) [PRESENCIAL][Project/Problem Based Learning (PBL)] | 1 |
| Unit 11 (de 14): PARABOLIC PROBLEMS. DIFUSSION EQUATION: Diffusion problems: heat equation. Boundary conditions. Separation of variables. Resolution. | |
|---|---|
| Activities | Hours |
| Class Attendance (theory) [PRESENCIAL][Lectures] | 2 |
| Group tutoring sessions [PRESENCIAL][Problem solving and exercises] | .5 |
| Study and Exam Preparation [AUTÓNOMA][Combination of methods] | 3 |
| Class Attendance (practical) [PRESENCIAL][Project/Problem Based Learning (PBL)] | 2 |
| Unit 12 (de 14): HYPERBOLIC PROBLEMS. WAVE EQUATION: The wave equation in one dimension. D'Alembert Solution. Boundary conditions associated with the wave equation. Finite string vibrating. Separation of variables. | |
|---|---|
| Activities | Hours |
| Class Attendance (theory) [PRESENCIAL][Lectures] | 2 |
| Group tutoring sessions [PRESENCIAL][Problem solving and exercises] | .5 |
| Study and Exam Preparation [AUTÓNOMA][Combination of methods] | 1 |
| Class Attendance (practical) [PRESENCIAL][Project/Problem Based Learning (PBL)] | 1 |
| Computer room practice [PRESENCIAL][Project/Problem Based Learning (PBL)] | 1 |
| Unit 13 (de 14): ELLIPTIC PROBLEMS. LAPLACE EQUATION: Laplacian. Nature of problems with boundary conditions. Dirichlet problems. | |
|---|---|
| Activities | Hours |
| Class Attendance (theory) [PRESENCIAL][Lectures] | 2 |
| Group tutoring sessions [PRESENCIAL][Problem solving and exercises] | .5 |
| Study and Exam Preparation [AUTÓNOMA][Combination of methods] | 1 |
| Class Attendance (practical) [PRESENCIAL][Project/Problem Based Learning (PBL)] | 3 |
| Unit 14 (de 14): NUMERICAL METHODS FOR PDE: Finite difference method applied to heat, wave and Laplace equations. Use of MATLAB to solve PDE numerically. | |
|---|---|
| Activities | Hours |
| Class Attendance (theory) [PRESENCIAL][Lectures] | 1 |
| Group tutoring sessions [PRESENCIAL][Problem solving and exercises] | .5 |
| Study and Exam Preparation [AUTÓNOMA][Combination of methods] | 8 |
| Class Attendance (practical) [PRESENCIAL][Project/Problem Based Learning (PBL)] | 2 |
| Computer room practice [PRESENCIAL][Project/Problem Based Learning (PBL)] | 3 |
| Global activity | |
|---|---|
| Activities | hours |
10. Bibliography and Sources
