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II-order perturbation methods new compuational metodology

Skrzypczyk J.

Architecture Civil Engineering Environment

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2008

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Vol. 1, no. 2

73-77

EN

The new numbers, called II-order perturbation numbers are introduced. Classical II-order perturbation problems can be solved in the new algebraic system as easily as usual problems of applied mathematics, theoretical physics and techniques. Static perturbation problems of a simple frame are discussed as well as dynamical vibration problems.

PL

W pracy przedstawiony jest nowy system algebraiczny, ze szczególnie zdefiniowanymi operacjami dodawania i mnożenia. Wprowadzone nowe liczby nazwano liczbami perturbacyjnymi II rzędu. W nowym systemie algebraicznym i zaproponowanej metodologii klasyczne zagadnienia perturbacyjne II-rzędu mogą być rozwiązywane tak łatwo, jak zwykłe zagadnienia matematyki stosowanej, fizyki teoretycznej i techniki. W przykładzie pokazano zastosowania do zagadnień perturbacyjnych w zadaniu dla prostej ramy, zarówno z punktu widzenia problemów statyki, jak i dynamiki.

2

Perturbation methods - new algebraic methodology with applications in mechanics

Skrzypczyk J.

Zeszyty Naukowe Katedry Mechaniki Stosowanej / Politechnika Śląska

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2005

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z. 29

413-418

EN

The aim of the paper is to present a new algebraic system with specifically defined addition and multiplication operations. The new numbers called perturbation numbers are introduced. It's proved that the new algebraic system (Re,+s,8E) is imbedded into the system of real numbers. Some additional properties as subtraction, inversion and division are presented too. Some elementary formulations of matrices and vectors of new numbers are introduced, as well as some basic properties of perturbation matrices, inverse matrices and solution properties of linear equations and eigenvalue problems in the new algebraic system of perturbation elements. Classical perturbation problems can be solved in the new algebraic system as easy as usual problems of applied mathematics, theoretical physics and techniques. Additional analytical transformations are not required.

3

Perturbation methods applied to the helical flow of a Casson fluid

Chiera C., Connell H. J., Shepherd J. J.

International Journal of Applied Mechanics and Engineering

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2005

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Vol. 10, no 1

39-51

EN

The helical flow of a Casson fluid between infinitely long coaxial cylinders is analyzed, when the inner cylinder has a given constant angular velocity, and a constant axial flow rate is imposed. Perturbation methods are applied in two circumstances of physical interest - that of low axial flow rates; and that of small intercylindrical gap width - to yield approximate expressions describing the fluid velocity field; and the Reiner-Riwlin equation, the fundamental relationship linking the angular velocity of the inner cylinder, the torque experienced there, and the given axial flow rate. The accuracy of these expressions is tested by comparison with solutions generated using numerical computation.

4

Approximate analytical solutions to thermo-poro-elastic equations

Yurusoy M., Pakdemirli M., Noyan O. F.

International Journal of Applied Mechanics and Engineering

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2004

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Vol. 9, no 1

181-187

EN

Thermo-poro-elastic equations describing the pressure and temperature diffusion from a deep layer of high temperature and pressure to a fluid saturated porous permeable layer are considered. The coupled partial differential system is transformed into an ordinary differential system via similarity transformations. Approximate analytical solutions to the ordinary differential system are found using perturbation methods and the criteria for validity of the solutions are established. Series solutions of the ordinary differential system are also given. Perturbation solutions and series solutions are contrasted with numerical solutions. When the criteria are met, perturbation solutions and numerical solutions are in good agreement. In contrast, the series solutions have a limited range of applicability.