5441336452

Codc: BSD001

MATHEMATICS

Field of study: Chil Engineering						Responsible Person: prof. dr hab. inż. Eligiusz Mieloszyk
Postgraduatc studies (MSc - coursc)
Full-timc studies						Department of Differential Eąuations and Applications of Mathematics
Year of study: I / Seinester: 1						Language: English
Hours in semester	lec	tut	proj	lab	sem	ECTS Points: 5
	30	30				Assessment: exam

Topics:

Partial differential equations. Classification of partial differential eąuations. Discriminant of a partia] differential eąuation of the second order with rcal coefficients. Elliptic. parabolic and hyperbolic eąuations. Applications of partial differential eąuations. Selected methods of solving of partial differential eąuations.

Elements of calculus of variations. A definition of a functional. a definition of an extreine of a functional. a fundamental leimna of calculus of variations. Euler eąuation. necessary condition of existence of an extreme of a functional. Jacobi eąuation. Jacobi condition. Sufficient conditions of existence of an extreme of a functional.

Tensor calculus. Similar matrixes. Base of vector space. A matrix of transition in base to base. A linear operation and its matrix. A inatrix of operation undcr transition of base. Detennination of an eigemalue and an eigemector of a linear operation. A tensor of tank one and a tensor of tank two. An inertia tensor. An eigem alue and an eigemector of an inertia tensor. Imariants of transition of base of an inertia tensor. Tensor ąuadric. canonical form of tensor ąuadric and its application.

Orthogonal siąuences and series. Fourier series. Trigonometrical Fourier series. Diriclilet conditions. Trigonometrical Fourier series for even and uneven functions. An application of Fourier series to solving of partial differential eąuations.

Operator methods. Laplace and Fourier transform. Fundamental properties of Laplace and Fourier transform. Comolution of functions. Borel theorem. An application of operator methods to solving partial differential eąuations.

Objectives:

•    knowledge of fonnulating standard initial-value problem.

•    boundary value problem

•    mastering fundamentals of tensor calculus.

Rccommcndcd literaturę:

1.    BatemanH.: Tables ofintegral Transforms. McGraw-Hill Book Company.

2.    Evans L.C.: Partial Differential Eąuations AMS.

3.    Gelfand I.M., Fomin S.W.: Rachunek wariacyjny. PWN.

4.    Krasnov M.I., Makarenko G.I., Kiselev A.I.: Problems and exercises in the calculus of variations. Mir Publishers.

5.    McConnel A.J.: Application of tensor analysis. Dover Publications Inc.

6.    Mieloszyk E.: Nieklasyczny rachunek operatorów w zastosowaniu do uogólnionych układów dynamicznych. Wyd. PAN.

7.    Thomson W.T.: Theory ofYibrations. Unwin Hyinan