Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Master
SUBJECT
| MME1008 | Mathematical Modeling in Real Sciences |
| Teaching Staff in Charge |
Lect. SERBAN Marcel Adrian, Ph.D., mserban math.ubbcluj.ro |
| Aims |
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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. |
| Content |
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Lecture 1: Aspects in mathematical modeling, clasification of mathematical models
Seminar 1: Simple 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 2: 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. Seminar 3: Exercises: the constructions of the flow, the study of stability equilibrium points. [2, ch. 1] Lecture 4: Second order differential equations: reduction of order, linear differential equations, boundary value problems. Seminar 4: Exercises for solvable second order differential equations and boundary value problems. [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. Seminar 5: Exercises: the constructions of the flow, the study of stability 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 6: Exercises regarding Lecture 6 problems. [3, ch. 2] Lecture 7: Modeling with first order differential equations: One specie population models, the Malthus model, the logistic Verhulst’s model, the Allee effect. Seminar 7: 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. Seminar 8-9: Exercises regarding Lecture 8-9 problems, simulation of dynamics using MAPLE. [3, ch. 2] Lecture 10:Modeling with systems of differential equations: Two species population models, the prey-predator, competition and mutualism models. Seminar 10: Exercises regarding Lecture 10 models, simulation of dynamics using MAPLE. [5, ch.3] Lecture 11: Modeling with systems of differential equations: Population growth under basic stage structure. Seminar 11: Exercises regarding Lecture 11 models, simulation of dynamics using MAPLE. [4, ch.11] Lecture 12: Modeling with systems of differential equations: Epidemiological models, SIS, SIR, SEIR models. Seminar 12: Exercises regarding Lecture 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 13: simulation of chaotic continuous-time models using MAPLE [3. ch.9] Lecture 14: Remarks and conclusions in mathematical modeling. |
| References |
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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. |
| Assessment |
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Final grade consists from:
Test paper during the semester : 30% Final written exam : 70% |
| Links: | Syllabus for all subjects Romanian version for this subject Rtf format for this subject |