Difference between revisions of "Mathematical Methods in Mechanics"

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The areas of study are integral equations and calculus of variations.
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'''The course consists of two parts.''' The first part is devoted to '''tensor calculus''' and the second one is devoted to '''nonlinear dynamics'''.
  
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Integral equations area involves:
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'''Tensor Calculus'''
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classification of integral equations, methods of solving Volterra integral equations of both the first and the second kind,
 
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some approaches to solutions of Fredholm integral equations (degenerative kernels, Fredholm resolvent, continuous kernels),
 
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theory of Fredholm integral equations with symmetric kernels,
 
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Hilbert-Schmidt theorem, application to Sturm-Liouville problem, green function
 
  
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Calculus of variations area involves:
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This part of course provides students with invariant formulations of theory of tensor calculus. Concepts and techniques of tensor algebra, theory of tensor functions and tensor analysis are introduced. The symmetries of tensors and tensor functions are considered. The attention is drawn to application of tensor algebra and calculus in various fields of science. Students who have completed the course should be aware of application of techniques of tensor calculus to solve a wide range of problems in different area of mechanics.
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functionals and proximity of functions, Lagrange’s lemma. Necessary condition for an extremum. Increment of a functional and its variation. Transversality conditions. Functionals of several dependent variables. Functionals dependent on higher derivatives. Functionals dependent on functions of several independent variables. Conditional extremum. Isoperimetric problem. Sturm-Liouville systems as variational problems. Lagrange problem. Direct methods for solving variational problems (Ritz’s, Galerkin’s and Kantorovich’s methods)
 
  
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'''Nonlinear Dynamics'''
  
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back to [[IMDP| International MSc program "Advanced Dynamics of Discrete and Continuum Systems"]]
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The course introduces students to the current state of nonlinear wave mechanics problems. Students learn how to solve these problems and apply methods of mathematical and numerical modeling by using Wolfram Mathematica. Students who have completed the course should be aware of up-to-date state of mechanics of deformable media, theoretical and applied mechanics. They should know how to formulate the major mechanics problems. Students have to be able to apply this knowledge to solve nonlinear dynamic problems of mechanics and conduct experimental studies using software packages. Students have to use methods of mathematical modelling for the analysis and solution of problems of natural science.
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back to [[IMDP| International MSc program "Mechanics and Mathematical Modeling"]]

Latest revision as of 14:12, 10 April 2019

The course consists of two parts. The first part is devoted to tensor calculus and the second one is devoted to nonlinear dynamics.

Tensor Calculus

This part of course provides students with invariant formulations of theory of tensor calculus. Concepts and techniques of tensor algebra, theory of tensor functions and tensor analysis are introduced. The symmetries of tensors and tensor functions are considered. The attention is drawn to application of tensor algebra and calculus in various fields of science. Students who have completed the course should be aware of application of techniques of tensor calculus to solve a wide range of problems in different area of mechanics.

Nonlinear Dynamics

The course introduces students to the current state of nonlinear wave mechanics problems. Students learn how to solve these problems and apply methods of mathematical and numerical modeling by using Wolfram Mathematica. Students who have completed the course should be aware of up-to-date state of mechanics of deformable media, theoretical and applied mechanics. They should know how to formulate the major mechanics problems. Students have to be able to apply this knowledge to solve nonlinear dynamic problems of mechanics and conduct experimental studies using software packages. Students have to use methods of mathematical modelling for the analysis and solution of problems of natural science.

back to International MSc program "Mechanics and Mathematical Modeling"

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