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1

Control of chaos in the burke-shaw system of fractal-fractional order in the sense of Caputo-Fabrizio

Saber Sayed

Journal of Applied Mathematics and Computational Mechanics

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2024

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Vol. 23, nr 1

83-96

EN

For the Burke-Shaw system, we propose a fractal-fractional order in the sense of the Caputo-Fabrizio derivative. The proposed system is solved by utilizing the fractal-fractional derivative operator with an exponential decay kernel. Time-fractional Caputo-Fabrizio fractal fractional derivatives are applied to the Burke-Shaw-type nonlinear chaotic systems.Based on fixed point theory, it has been demonstrated that a fractal-fractional-order model under the Caputo-Fabrizio operator exists and is unique. Using a numerical power series method, we solve the fractional Burke-Shaw model. Using Newton’s interpolation polynomial, we solve the equation numerically by implementing a novel numerical scheme based on an efficient polynomial.

2

A Dai-Liao-type projection method for monotone nonlinear equations and signal processing

Ibrahim Abdulkarim Hassan, Kumam Poom, Abubakar Auwal Bala, Abdullahi Muhammad Sirajo, Mohammad Hassan

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

978--1013

EN

In this article, inspired by the projection technique of Solodov and Svaiter, we exploit the simple structure, low memory requirement, and good convergence properties of the mixed conjugate gradient method of Stanimirović et al. [New hybrid conjugate gradient and broyden-fletcher-goldfarbshanno conjugate gradient methods, J. Optim. Theory Appl. 178 (2018), no. 3, 860–884] for unconstrained optimization problems to solve convex constrained monotone nonlinear equations. The proposed method does not require Jacobian information. Under monotonicity and Lipschitz continuity assumptions, the global convergence properties of the proposed method are established. Computational experiments indicate that the proposed method is computationally efficient. Furthermore, the proposed method is applied to solve the ℓ1 -norm regularized problems to decode sparse signals and images in compressive sensing.

3

The New Extended KdV Equation for the Case of an Uneven Bottom

Karczewska A., Rozmej P.

Computational Methods in Science and Technology

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2018

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Vol. 24, No. 4

221--225

EN

The consistent derivation of the extended KdV equation for an uneven bottom for the case of α = O(β) and δ = O(β 2 ) is presented. This is the only one case when second order KdV type nonlinear wave equation can be derived for arbitrary bounded bottom function.

4

Modelowanie i symulacja cewki nieliniowej ze stratami w żelazie

Kolańska-Płuska J., Grochowicz B.

Prace Instytutu Elektrotechniki

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2016

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Z. 272

217--226

PL

Praca dotyczy badania dynamiki obwodu z cewką nieliniową z uwzględnieniem strat w żelazie. W pracy został przedstawiony model cewki oraz opis za pomocą zmiennych stanu. Przedstawiono również program do badania dynamiki cewki opracowany w środowisku C#, w którym do rozwiązania układu równań różniczkowych nieliniowych modelujących cewkę nieliniową ze stratami w żelazie zastosowano metodę niejawną RADAU IIA różnych rzędów.

EN

Solving rigid differential equations systems should be performed with the application of implicit or semi-explicit methods. As proven in the example given in this paper – the implicit RADAU IIA methods can be successfully applied for the purpose of solving the rigid non-linear systems. The high-order methods up to 11 cause that the application performs relatively large integration steps in pursuit of the steady state that is comparable with the explicit methods. Furthermore, the number of iterations and computation time for a sample non-linear equations formulated for a coil with iron losses is comparable to other methods such as, e.g., multistep Gear method. Also a sample test has proved that the implicit Runge-Kutta methods can be competitive for the purpose of research regarding electrical systems described with the rigid differential equations compared to the multistep Gear method.

5

A finite element method for extended KdV equations

Karczewska A., Rozmej P., Szczeciński M., Boguniewicz B.

International Journal of Applied Mathematics and Computer Science

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2016

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Vol. 26, no. 3

555--567

EN

The finite element method (FEM) is applied to obtain numerical solutions to a recently derived nonlinear equation for the shallow water wave problem. A weak formulation and the Petrov–Galerkin method are used. It is shown that the FEM gives a reasonable description of the wave dynamics of soliton waves governed by extended KdV equations. Some new results for several cases of bottom shapes are presented. The numerical scheme presented here is suitable for taking into account stochastic effects, which will be discussed in a subsequent paper.

6

Finite Element Method for Stochastic Extended KdV Equations

Karczewska A., Szczeciński M., Rozmej P., Boguniewicz B.

Computational Methods in Science and Technology

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2016

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Vol. 22, No. 1

19--29

EN

The finite element method is applied to obtain numerical solutions to the recently derived nonlinear equation for shallow water wave problem for several cases of bottom shapes. Results for time evolution of KdV solitons and cnoidal waves under stochastic forces are presented. Though small effects originating from second order dynamics may be obscured by stochastic forces, the main waves, both cnoidal and solitary ones, remain very robust against any distortions.

7

A Note on the Singh Six-order Variant of Newton’s Method

Marciniak A., Wolf M.

Computational Methods in Science and Technology

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2015

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Vol. 21, No. 4

261--261

EN

In 2009 in this journal it was published the paper of M. K. Singh [1], in which the author presented a six-order variant of Newton’s method. Unfortunately, in this paper there were a number of printer errors and a serious error in the proof of theorem on the order of the method proposed. Therefore, we have opted for presenting the correct proof of this theorem.

8

Numeryczne modelowanie indukcyjnych nadprzewodnikowych ograniczników prądu zwarciowego

Łanczont M.

Przegląd Elektrotechniczny

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2014

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R. 90, nr 2

65--67

PL

Artykuł opisuje projekt obwodowego modelu numerycznego indukcyjnego nadprzewodnikowego ogranicznika prądu. Przedstawiono założenia modelu matematycznego, charakterystykę modelowanego obiektu oraz wyniki symulacji.

EN

This paper describes project the peripheral numerical model of induction superconducting fault current limiter. The assumptions of a mathematical model, characteristics of a modeled device and the simulation results where presented.

9

Stability and Resonant Behaviour of a Rotor in Slide Bearings with Bingham Fluid As a Lubricant

Perek A.

Machine Dynamics Research

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2012

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Vol. 36, No. 2

84-94

EN

The paper is concerned with a dynamic analysis of a rigid rotor supported in hydrodynamic bearings with rheological properties of the lubricant controlled by an external magnetic field. A fluid viscosity is an adaptive parameter, which controlled by the applied magnetic field, actively influence the dynamics of the rotor. Such an active lubricant can be identified with a magnetic fluid. Magnetic fluid as a suspension of sufficiently small ferro-particles can be treated as a homogenous substance, and used as a lubricating medium (Berkovsky et al., 1993, Miszczak and Wierzcholski). Due to magnetic interactions, it shows a non-Newtonian behaviour, which, with an appropriate synthesis of liquid, can be described by the Bingham model. In the following two effects of a magnetic fluid as a bearing lubricant are applied - increase of viscosity with the magnetic field and drop with the angular speed of the rotor. Both can be used to modify rotor dynamics to supply faster uplift to contactless rotation and desired resonant properties. Numerical simulations of nonlinear equations of the rotor transverse vibrations excited by unbalance forces show considerable influence of the magnetically controlled viscosity on the magnitude and position of the resonant zone. Magnetic field and shear rate- dependent effective lubricant viscosity influence the critical rotation speed of the rotor.

10

Analytical solution of forced-convective boundary-layer flow over a flat plate

Mirgolbabaei H., Barari A., Ibsen L. B., Esfahani M. G.

Archives of Civil and Mechanical Engineering

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2010

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Vol. 10, no 2

43-51

EN

In this letter, the problem of forced convection heat transfer over a horizontal flat plate is investigated by employing the Adomian Decomposition Method (ADM). The series solution of the nonlinear differential equations governing on the problem is developed. Comparison between results obtained and those of numerical solution shows excellent agreement, illustrating the effectiveness of the method. The solution obtained by ADM gives an explicit expression of temperature distribution and velocity distribution over a flat plate.

PL

W artykule przedstawiono zastosowanie metody dekompozycji Adomiana do wymuszonego, konwekcyjnie przepływu ciepła w poziomej, płaskiej płycie. Rozwiązania nieliniowych równań różniczkowych opisujących zagadnienie poszukiwana w postaci szeregów Adomiana. Z porównania otrzymanych wyników z wynikami innych metod numerycznych wynika doskonała ich zgodność, która potwierdza skuteczność zastosowanej metody. Otrzymane rozwiązanie pozwoliło jednoznacznie wyznaczyć rozkład i prędkości mian temperatury w analizowanej płycie.

11

Matematiceskoe modelirovanie nelinejnoj dinamiki i massoperenosa

Holpanov L.

Czasopismo Techniczne. Mechanika

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2008

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R. 105, z. 2-M

95-112

EN

In the message the methods of the decision of nonlinear dynamics and mass transfer are presented. The methods are illustrated on examples of a wave liquid films flow, mass transfer in a wave films, and mass transfer involving the inlet portion in different regimes. The laws of occurrence of self-organizing and chaos are submitted.

12

Propagation and interaction of hyperbolic plane waves in nonlinear elastic solids

Domański Wł.

Prace Instytutu Podstawowych Problemów Techniki PAN

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2006

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nr 4

3-169

EN

Our aim is to create a mathematical theory of nonlinear plane waves which could have applications in many branches of nonlinear material science. Using asymptotic expansions we derive simplified models to complicated, dynamical, nonlinear phenomena (including wave interactions), described originally by large systems of nonlinear partial differential equations. We are particularly interested in applications to elastic solids. The models which we derive are weakly nonlinear. Weak nonlinearity means that we are interested in small amplitude solutions. This is more refined than linearized models. Asymptotic methods are very useful when we deal with complicated systems of PDEs. Weakly nonlinear geometric optic (WNGO) is suitable for nonlinear hyberbolic equations modelling wave type phenomena. The method works for small amplitude (weakly nonlinear) and high frequency waves. WNGO is based on the introduction of the additional "fast" independet variable and the use of a multiple scale analysis. It provides transport equations as the simplified asymptotic models for the evolution of wave profiles. The classical WNGO expansion results in the decoupled (in the nonresonant case) inviscid Burgers equations as the canonical asymptotic evolution equations for strictly hyberbolic and genuinely nonlinear waves. However in real models of continuum mechanics or physics it is quite common that the wave speeds may coincide ank often they are not monotonic in certain directions fo the assumptions of strict hyberbolicity and genuine nonlinearity are violated. This happent e.g. in nonlinear elastodynamics, the model which we are studying in the second part of this work. In order to take into consideration the loss of strict hyperbolicity and loss of genuine nonlinearity we modify theclassical WNGO expansion and derive new asymptotic evolution equations. The new reults include: 1. Derivation of a general structure of nonlinear plane waves' elastodynamics in terms of the strain energy function for an arbitrary direction of propagation, regardless of the anisotropy type. 2. Presentation of explicit nonlinear plane waves' elasktodynamics equations for a cubic crystal in the case - an arbitrary direction of propagation and a geometrically nonlinear but physically linear model. - an arbitrary direction of propagation in a cube face and both geometrically and physically nonlinear model, - three selected derections of pure mode propagation: (1,0,0), (1,1,0) and (1,1,1) with the inclusion of hihger order terms. 3. Derivation of the simlified evolution equations for nonlinear elastic waves in the isotropic and the cubic crystal cases. 4. Establishing a new model - the complex Burgers equation as the evolution equation describing interaction of a pair of transverse elastic waves propagatin along the three-fold symmetry axis in a cubic crystal. 5. Derivation of general analytical formulas for all interaction coefficients of nonlinear elastic plane waves propagating in any direction in an arbitrary hyperelastic medium. 6. Calculations of all interaction coefficients analytically for all considered models. 7. New formulations and interpretation of a null condition with the use of self-interaction coefficients. We focused in particular on cases where a local loss of strict hyperbolicity and/or genuine nonlinearity occurs. We showed that this happens for transverse or quasi-transverse elastic waves and implies the presence of a cubic nonlinearity in the evolution equations for decoupled (quasi)-transverse waves. However, we found cases when (quasi)-transverse waves do interact at a quadratically nonlinear level for certain directions in anisotropic media. We showed that this happens e.g. for a cubic crystal for a (1,1,1) direction. This fact is in contradiction to the isotropic case where it is known that shear elastic waves cannot interact at a quadratically nonlinear level see (75). We proved that the coupled systems with quadratic nonlinearity (complex Burgers equations) are new asymptotic models for interacting pairs of (quasi)-transverse waves propagating in the (1,1,1) direction in a cubic crystal. This is for the first time where complex Burgers equations appear in the context of nonlinear elasticity.

PL

Praca poświęcona jest analizie propagacji i oddziaływania fal w nieliniowych materiałach sprężystych przy pomocy metody słabo nieliniowej optyki geometrycnej. Rozważono zarówno materiały izotropowe jak i kryształy o symetrii kubicznej. Celem pracy jest wyprowadzenie równań ewolucyjnych na amplitudy propagujących się fal oraz znalezienie analitycznych wzrów na współczynniki oddziaływania między falami. Praca składa się z dwóch części. W części pierwszej wprowadzono aparat matematyczny użyty w centralnej części drugiej. Rozdział pierwszy to wstęp, w którym przedstawione są motywacje, cele pracy, użyte metody i główne rezultaty. Rozdział drugi to przegląd zagadnień dotyczących hiperbolicznych praw zachowania. Nowe rezultaty zawarte są głównie w paragrafie 2.1.1. W rozdziale trzecim zaprezentowana jest metoda nieliniowej optyki geometrycznej. Nowe wyniki dotyczą lokalnej struktury układów 2 x 2 dla krotnych pierwiastków oraz modyfikacji asymptotyki w przypadku lokalnej utraty własności istotnej nieliniowości. W ramach rozdziału czwartego dokonano przeglądu nieliniowych równań cząstkowych, które mogą służyć jako kanoniczne asymptotyczne modele dla skomplikowanych dynamicznych zagadnień nieliniowych. Oprócz modeli hiperbolicznych zaprezentowano również modele dyssypatywne i dyspersyjne, w tym z nieliniową dyspersją. Część druga stanowi główną część rozprawy i zawiera zastosowania metod zaprezentowanych w części pierwszej do materiałów sprężystych. W rozdziale piątym rozważone są równania nieliniowej elastodynamiki (przy założeniu zarówno geometrycznej, jak i fizycznej nieliniowości). W tym rozdziale godny uwagi jest zwłaszcza paragraf 5.5, gdzie przedstawiono ogólną ideę wyprowadzenia równań nieliniowej hipersprężystości fal płaskich. Pokazano także oryginalny temat sprowadzając zagadnienie na wartości i wektory własne do badania macierzy 3 x 3. Warto podkreślić, iż cała analiza przeprowadzona jest dla dowolnego ośrodka i dowolnego kierunku propagacji płaskiej fali. W rozdziale szóstym omówiono związki konstytutywne dla nieliniowego ośrodka izotropowego. Rozdział siódmy zawiera dyskusję różnych postaci tzw. "null condition" - warunku zerowania. Warunek ten zapewnia gładkość rozwiązań nieliniowych dynamicznych równań sprężystości w ośrodku izotropowym przy małych danych początkowych, a co za tym idzie zapobiega powstawaniu fal uderzeniowych oraz udaremnia eksplozję rozwiązań. Oryginalnym wkładem autora jest zaproponowanie kilku alternatywnych postaci tego warunku bez użycia niezmienników oraz interpretacja warunku zerowania poprzez współczynniki samooddziaływania fal. W rozdziale ósmym wyprowadzono ogólne wzory na współczynniki oddziaływania falowego, abstrahując od rodzaju anizotropii materiału i kierunku propagacji fali płaskiej. Podano w ten sposób przepis na wyznaczanie współczynników w zależności od pochodnych funkcji energii. Korzystając z tego przepisu wyznaczono wszystkie współczynniki dla materiałów izotropowych podkreślając różnice jakie wprowadza fizyczna nieliniowość. W rozdziale dziewiątym wyprowadzono asymptotyczne równania ewolucyjne dla materiału izotropowego uwypuklając różnice między otrzymanymi nieliniowymi równaniami dla fal podłużnych i fal poprzecznych. W rozdziale 10, 11 i 12 znalazły się zasktosowania dla ośrodka anizotropowego - rezultaty te stanowią oryginalny wkład autora w dziedzinie propagacji i oddziaływania fal płaskich w krysztale o symetrii kubicznej. Rozważono trzy wybrane kierunki propagacji i wyprowadzono dla nich równania fal płaskich uwzględniając efekty wyższego rzędu, uzyskano równania asymptotyczne dla tych fal i obliczono analitycznie wszystkie współczynniki oddziaływania międzyfalowego, wykazując, że zależą one od stałych materiałowych. Dodatkowo uzyskano także równania fal płaskich dla dowolnego kierunku propagacji przy założeniu fizycznej liniowości oraz w ogólnym przypadku dla dowolnego kierunku leżącego w płaszczyźnie ściany bocznej kryształu. Rezultatem zasługującym na szczególne podkreślenie jest udowodnienie, że fale poprzeczne mogą ze sobą oddziaływać na poziomie kwadratowej nieliniowości w przeciwieństwie do tego, co zostało pokazane dla ośrodka izotropowego (75). W rozprawie wykazano, iż na to miejsce dla fal poprzecznych propagujących się w krysztale o symetrii kubicznej wzdłuż osi trzykrotnej symetrii (1,1,1). Wyprowadzono nowe równania ewolucyjne opisujące oddziaływanie tych fal (zespolone równania Burgersa).

13

Stabilization of nonlinear systems: Radial optimal control approach

Sangelaji Z., Banks S.P.

Systems Science

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2003

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Vol. 29, nr 4

29-41

EN

This paper represents the optimal control of nonlinear systems based upon the associated angular approach. In the latter a general class of nonlinear systems is converted to two associated systems: a nonlinear equation on a sphere (spherical), and a radial differential system. By decoupling the two subsystems and considering only the radial system, a finite-horizon radial optimal control is designed which minimizes the radial cost function. Successive approximation technique is then introduced in which the equations are replaced by a sequence of linear, time-varying approximations. The resulting optimal control is then applied to the original angular system. This control forces the original angular system to the origin.

14

Nonlinear equations of acoustics - numerical solutions and their visualization

Wójcik J., Lewin P. A., Nowicki A., Filipczyński L., Kujawska T., Radulescu E.

Hydroacoustics

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2003

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Vol. 6

137--142

EN

The novel effective numeric solver of the nonlinear scalar wave equation describing the acoustic wave propagation in the attenuating media was derived. The solver was developed for the PC environment. The standard computation data include al! stationary and dynamic characteristics of the radiated ultrasonic pressure field, especially its 4D (space/time) visualization. The results obtained with the solver can be used as the supporting tools (tool) in designing and developments of the multielement linear and phase array transducers applied in ultrasonography.

15

Solving the nonlinear problem of gas-lubricated hybrid circular bearings. Part II: unsteady case

Nedelcu S., Postelnicu A.

International Journal of Applied Mechanics and Engineering

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2002

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Vol. 7, no 2

465-478

EN

In the paper, a dynamic analysis of gas-lubricated hybrid circular bearings is made. The mathematical model is the Reynolds equation in unsteady regimes along with the boundary conditions for a multiple connected domain. Within the hypothesis of a periodic relative motion of bearing surfaces, the method of small perturbations is used. The equations of the model are solved numerically using a difference finite method and finally, the curves of variation of the critical mass versus the eccentricity are obtained.

16

Allosteric systems on biological membrane analysis by time minimum optimal control principle

Hirayama H., Okita Y.

Systems Science

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2000

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Vol. 26, nr 4

121-145

EN

We calculated temporal changes in concentrations of species in biological allosteric system. The allosteric nature was described by co-operativity and concerted actions of identical subunits composing the system. The co-operativity was characterized by accelerated bindings of substrates to the subunits and their structural changes. The concerted mechanism provokes all or non-conformational change of an entire molecule from its inactive to active sate at one time. As the number of bound substrate increases, the concerted conformational transition of all the subunits strongly directs to active state. These two properties save the time to complete activation of the system and full substrate binding. The allosteric actions are frequently observed in the critical emergency conditions such as oxygen binding, immune defense reaction and ionic channel gating on excitable membrane of the bio-signal transmission. Hence, the temporal actions of allosteric systems can be interpreted as the shortest time controlled system. We thus, proposed the time minimum optimal control principle as an organizing strategy for activating and binding processes of the allosteric systems. We presented fifteen non-linear differential equations for describing the temporal changes in concentrations of the species composing a two activators-two substrates allosteric system. Another set of fifteen non-linear differential equations for co-state variables were obtained by partial differentiation of the Hamiltonian of the system based on the minimum optimal control theory. The computed temporal concentrations of species oscillated. A reduction of an allosteric parameter shifted these temporal changes synchronously. When the amounts of substrate and activator were set as time invariant, the oscillations disappeared and the concentration of the species approached a steady level. These computed results are qualitatively consistent with actual biophysical phenomena. The present mathematical description of transient changes in concentrations of species in biological allosteric system will be available for evaluating the effective and economical performance of the allosteric system acting under the critical emergency circumstances.

17

Solving the nolinear problem of gas-lubricated hybrid circular bearings. Part I: Steady case

Nedelcu S., Postelnicu A.

Applied Mechanics and Engineering

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2000

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Vol. 5, no 3

629-653

EN

In hybrid bearings, the carrying effect is produced by supplying under pressure and by the relative motion of the bearing surfaces. Because the pressure distribution in the bearing is the solution to a nonlinear partial differential equation of second order, the two causes cannot be studied separately and solving the problem is a difficult task. The mathematical model considered by us is the Reynolds equation for compressible fluids in a multiple connected domain. The boundary conditions on the inner boundaries are derived from the flow-rate continuity through the supplying orifices and are expressed as nonlinear integral-differential equations, which are solved numerically.