Universitatea Babeş-Bolyai Cluj-Napoca
Facultatea de Matematică şi Informatică
Ciclul de studii: Masterat
Domeniul: Informatica
Programul de studii: Modelare si simulare - în limba engleză
Limba de predare: Engleză

SYLLABUS

I. General data
Code	Subject	
MME1008	Mathematical Modeling in Real Sciences	

Semester	Hours: C+S+L	Category	Status	
1	2+1+0	speciality	compulsory	

II. Full status faculty members
Name and	Scientific	Didactic	Chair	Type of activity	
surname	title	title		C	S	L	
SERBAN Marcel Adrian	Ph.D.	Lect.	Applied Mathematics	*			

Associated faculty members
Name and	Scientific	Institution	Type of	Type of activity	
surname	title		position	C	S	L	

III. Course objectives
The course contains mathematical aspects in real sciences modeling. There are studied models from mechanics, biology and medicine using differential equations. The objectives are the knowledge and simulation of some models oriented on software products. Development and construction of  procedures for the study of these models.	

IV. Course contents
1. Mathematical modeling
2. Autonomous differential systems
3. Classical mechanics
4. Population dynamics models
5. Epidemiological models	

V. Bibliography
1. I.A.Rus, C. Iancu, Mathematical modeling, Transilvania Press, 2000.
2. J.D. Logan, A first course in differential equations, Springer, 2001.
3. W.E. Boyce, R.C. DiPrima, Elementary differential equations and boundary value problems, John Wiley & Sons, Inc., 2001
4. H.R. Thieme, Mathematics in population biology, Princeton University Press, 2003.
5. J.D. Murray, Mathematical biology, Springer, 2001.	

VI. Thematic of didactic activities per weeks
Lecture 1: Aspects in mathematical modeling, clasification of mathematical models
 [1, ch.2]

Lecture 2: First order differential equations: equations and solutions, geometrical interpretation, solvable first order differential equations, boundary value problems
Seminar 1: Exercises for solvable first order differential equations and boundary value problems.
[2, ch. 1]

Lecture 3: The dynamical system of scalar autonomous differential equations: the flow of autonomous first order differential equation, equilibrium points, phaze portrait, stability of equilibrium points.
 [2, ch. 1]

Lecture 4: Second order differential equations: reduction of order, linear differential equations, boundary value problems.
Seminar 2:: the constructions of the flow, the study of stability equilibrium points 
[2, ch. 3]

Lecture 5: The dynamical systems of planar autonomous differential equations system: the flow of planar autonomous differential equations system, equilibrium points, phaze portrait, stability of equilibrium points.
[2, ch. 6]

Lecture 6: Modeling with first order differential equations: Temperature problem, compound interest problem, the problem of escape velocity, radioactive dezintegration and radiocarbon dating problem.
Seminar 3: Exercises: the constructions of the flow, the study of stability equilibrium points. 
[3, ch. 2]

Lecture 7: Modeling with first order differential equations: One specie population models, the Malthus model, the logistic Verhulst’s model.
Seminar 4: Problems regarding one specie population models, simulation of dynamics using MAPLE
[4, ch.3, ch.7]

Lecture 8-9: Modeling with second order differential equations: Newton’s law, mechanical and electrical vibrations, forced vibrations, the falling body problems, Archimedes’ principle and the buoyancy problem.
[3, ch. 2]

Lecture 10:Modeling with systems of differential equations: Two species population models, the prey-predator, competition and mutualism models.
Seminar 5: Test paper. Exercises regarding Lecture 8-10 models, simulation of dynamics using MAPLE.
[5, ch.3]

Lecture 11: Modeling with systems of differential equations: Population growth under basic stage structure.
[4, ch.11]

Lecture 12: Modeling with systems of differential equations: Epidemiological models, SIS, SIR, SEIR models.
Seminar 6: Exercises regarding Lecture 11-12 models, simulation of dynamics using MAPLE.
[4, ch.18, ch. 21]

Lecture 13: Chaos theory: definition of chaos, chaotic continuous-time models, the Loretnz model
Seminar 7: simulation of  chaotic continuous-time models using MAPLE
[3. ch.9]

Lecture 14: Remarks and conclusions in mathematical modeling.	

VII. Didactic methods used
Lectures, interactive presentations, exercises, individual study, visualization using MAPLE.	

VIII. Assessment
Final grade consists from:
Test paper during the semester : 30% 
Final written exam : 70%

IX. Additional references
1. D.K. Arrowsmith, Dynamical systems, Differential equations, maps and chaotic behaviour, Chapmann and Hall, 1992.
2. Lynch S. Dynamical systems with applications using MAPLE, Birkhauser, 2001.
3. Nino Boccara, Modeling complex systems, Springer, 2004.	

	Dean,							Chair head,
Prof. Leon Ţâmbulea, Ph.D.				Prof. Octavian Agratini, Ph.D.

	Date,							Course responsible,
       06.11.2009						Lect. SERBAN Marcel Adrian, Ph.D.