Universitatea Babeş-Bolyai Cluj-Napoca
Facultatea de Matematică şi Informatică
Ciclul de studii: Licenţă
Domeniul: Calculatoare şi tehnologia informaţiei
Programul de studii: Ingineria informatiei - in limba engleza
Limba de predare: Engleză

SYLLABUS

I. General data
Code	Subject	
MME0012	Equations of Mathematical Physics	

Semester	Hours: C+S+L+P	Category	Status	
5	2+1+1+0	speciality	compulsory	

II. Full status faculty members
Name and	Scientific	Didactic	Department	Type of activity	
surname	title	Title		C	S	L	
ILEA Veronica Ana	Ph.D.	Lect.	Mathematics	*			

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

III. Course objectives
The lecture presents classical and modern methods for functionals’ minimizations and classical and modern theory of partial differential equations and helps learning basic concepts of partial differential equations theory and develops capacity of theoretical analysis of some equations from physics.	

IV. Course contents
Calculus of variations. Introduction. Examples. Variation of first and second order of a functional. Variational problems for different functional.
Elements of functional analysis. Fourier series.
Partial differential equations. Introduction. Examples from physics. Clasifications.
The canonical form of partial differential equations. 
Separation of variables method. 	

V. Bibliography
1 Lawrence C. Evans, Partial Differential Equations, AMS,Providence, Rhode Island, 1998.
2. Gerald B. Folland, Introduction to Partial Differential Equations, second edition, Princeton University Press, Princeton, New Jersey, 1995.
3. Radu Precup, Lectii de ecuatii cu derivate partiale, Presa Universitara Clujeana, Cluj-Napoca, 2004.
4. Irinel Radimir, Horiana Ovesea, Matematici speciale, Editura Albastra, Cluj-Napoca, 2001.
5. Damian Trif, http://math.ubbcluj.ro/~dtrif
6. Frank Morgan, Real Analysis and Applications, AMS, 2005.
7. Mike Mesterton-Gibbons, A Primer on the Calculus of Variations and Optimal Control Theory, AMS, 2009.
8. Hans Sagan, Boundary and Eigenvalue Problems in Mathematical Physics, Dover Publications, New York, 1989. 	

VI. Thematic of didactic activities per weeks
Calculus of variations. Utility. Examples. Problems that lead to minimization of a functional (Problem of Dido, brahistocron’s problem).
Problems that lead to minimization of a functional (problem of equilibrium shape of a hard wire; problem with variable border). Formulation of variational problem. Fundamental lemmas in calculus of variations. Applications to variational calculations.
Calculus of variations. Variation of first and second order of a functional. Examples. 
Necessary conditions of extreme. Euler’s theorem and equation. Particular cases. Applications.
Calculus of variations. Generalization of Euler’s equation. Variational problems for different functional (functional that depend on multiple functions; functionals that depend on superior order derivation). 
Extreme conditioning problems. Problems with variable border. Applications. Test paper.
Notion of partial differential equation. Partial differential equations of first order. Notations. Applications.
Partial differential equations. Examples from physics (equation oscillatory processes; diffusion equation; equation of stationary processes). Clasifications of differential equation of second order..Applications.
The canonical form of partial differential equations of second order with two independent variables. 
The canonical form of partial differential equations. Problems that are attached to partial differential equation of second order (heat equation, equation vibrant chord; Cauchy problem for equation chord vibrant, mixed problem for equation vibrant chord).
Problems that are attached to partial differential equation of second order (Poisson's equation, Dirichlet problem for Poisson's equation, Neumann problem for Poisson's equation). Applications.
Separation of variables method. Mixed problem for equation vibrant chord. Utility. 
Special functions applied in variable separation method (Legendre’s polinom; recurrence formula; diferential equation of Legendre’s polinom). Applications.
Separation of variables method. Dimensional heat equation. Laplace equation (rectangular area, circular area). Applications.	

VII. Didactic methods used

VIII. Assessment
Final grade consists of:
   A seminar paper. Share 15% 
   The laboratory work. Share 15%
   Evaluation topics and homework during this semester. Share  10%
   Automatically score. Share 10%
   Final paper. Share 50%	

IX. Additional references

	Dean,							Department head,

	Date,							Course responsible,
								Lect. ILEA Veronica Ana, Ph.D.