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1

Problem with integral condition for evolution equation

Kalenyuk P. I., Kuduk G., Kohut I. V., Nytrebych Z. M.

Journal of Mathematics and Applications

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2015

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Vol. 38

71--76

EN

We propose a method of solving the problem with non-homogeneous integral condition for homogeneous evolution equation with abstract operator in a linear space H. For right-hand side of the integral condition which belongs to the special subspace H ⊆ L, in which the vectors are represented using Stieltjes integrals over a certain measure, the solution of the problem is represented in the form of Stieltjes integral over the same measure.

2

The effect of randomness on the stability of capillary gravity waves in the presence of air flowing over water

Majumder D. P., Dhar A. K.

International Journal of Applied Mechanics and Engineering

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2015

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Vol. 20, no. 4

835--855

EN

A nonlinear spectral transport equation for the narrow band Gaussian random surface wave trains is derived from a fourth order nonlinear evolution equation, which is a good starting point for the study of nonlinear water waves. The effect of randomness on the stability of deep water capillary gravity waves in the presence of air flowing over water is investigated. The stability is then considered for an initial homogenous wave spectrum having a simple normal form to small oblique long wave length perturbations for a range of spectral widths. An expression for the growth rate of instability is obtained; in which a higher order contribution comes from the fourth order term in the evolution equation, which is responsible for wave induced mean flow. This higher order contribution produces a decrease in the growth rate. The growth rate of instability is found to decrease with the increase of spectral width and the instability disappears if the spectral width increases beyond a certain critical value, which is not influenced by the fourth order term in the evolution equation.

3

Generalized method of Lie-algebraic discrete approximations for solving Cauchy problems with evolution equation

Kindybaliuk A.

Journal of Applied Mathematics and Computational Mechanics

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2014

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Vol. 13, nr 2

51--62

EN

We consider solving the Cauchy problem with an abstract linear evolution equation by means of the Generalized Method of Lie-algebraic discrete approximations. Discretization of the equation is performed by all variables in equation and leads to a factorial rate of convergence if Lagrange interpolation is used for building quasi representation of differential operator. The rank of a finite dimensional operator and approximation properties have been determined. Error estimations and the factorial rate of convergence have been proved.

4

Some theorems on fractional semilinear evolution equations

Tidke H. L.

Journal of Applied Analysis

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2012

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Vol. 18, nr 2

209-224

EN

In this paper, we investigate the existence, uniqueness and other properties of solutions of fractional semilinear evolution equations in Banach spaces. The results are obtained by using fractional calculus, the well-known Banach fixed point theorem coupled with Bielecki type norm and the integral inequality established by E. Hernandez.

5

Fourth-order evolution equations for a surface gravity wave packet in a two layer fluid

Senapati S., Debsarma S., Das K. P.

International Journal of Applied Mechanics and Engineering

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2010

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Vol. 15, no 4

1215-1225

EN

Fourth order nonlinear evolution equations are derived for a three dimensional surface gravity wave packet in the presence of long wave length an interfacial wave in a two layer fluid domain in which the lower fluid depth is infinite. For derivation of evolution equations, the multiple-scale method is used. Using these evolution equations, stability of uniform stokes wavetrain is investigated for different values of density ratio of the two fluids and for different values of the depth of the lighter fluid.

6

Entire graphs under a general flow

Mao J., Li G., Wu C.

Demonstratio Mathematica

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2009

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Vol. 42, nr 3

631-640

EN

An initial entire graph with bounded second fundamental form in Rn+1 over some hyperplane is evolving under a general flow defined in the paper. For an additionally suitable condition in the main theorem, we obtain gradient and curvature estimates, leading to long-time existence of the flow, and convergence to an entire graph in the limit.

7

On resonant interaction of capillary-gravity wave and internal wave

Debsarma S., Das K. P.

International Journal of Applied Mechanics and Engineering

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2008

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Vol. 13, no 3

653-668

EN

The fourth order nonlinear evolution equations are derived for a capillary-gravity wave packet for the case of resonant interaction with internal wave in the presence of a thin thermocline at a finite depth in deep water. These equations are used to make stability analysis of a uniform capillary-gravity wave train when resonance condition is satisfied. It is observed that for surface gravity waves the instability region expands with the decrease of thermocline depth. For surface capillary-gravity waves the growth rate of instability is much higher if the thermocline is formed at lower depth and for a fixed thermocline depth it increases with the increase of wave amplitude.

8

A higher order nonlinear evolution equation for much broader bandwidth gravity waves in deep water

Debsarma S., Das K. P.

International Journal of Applied Mechanics and Engineering

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2007

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Vol. 12, no 2

557-563

EN

A higher order nonlinear evolution equation for gravity waves in deep water is derived from Zakharov's integral equation which is valid for a much broader bandwidth gravity waves than considered previously. The instability regions in the perturbed wave-number space for a uniform Stokes wave obtained from this equation is shown to fit nicely those obtained by McLean et al. [Phys. Rev. Lett. 46, 817-820(1981)] by exact numerical method.

9

Non-isothermal phase-field models and evolution equation

Morro A.

Archives of Mechanics

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2006

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Vol. 58, nr 3

257-271

EN

Phase transitions between two phases are modelled as space regions where a phase field, or order parameter, changes smoothly. The literature shows a seeming contradiction in that some papers lead to the use of the reduced chemical potential through the temperature, others do not. The paper has a threefold purpose. First, to revise the arguments of known approaches and possibly generalize the associated schemes. Secondly, to show that a further approach is possible which involves the phase field as an internal variable. Thirdly, to contrast the various schemes and the corresponding results. It follows that differences arise because different fields enter the models and different forms are considered for the balance of energy and the second law of thermodynamics.

10

On the existence of global solutions of evolution equations

Medved M.

Demonstratio Mathematica

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2004

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Vol. 37, nr 4

871--882

EN

In this paper a sufficient condition for the existence of global solutions of evolution equations is proved. In the proof a modification of the Bihari type integral inequality to the case of a weakly singular nonlinear integral inequality is used. An application to a reaction-diffusion problem is given.

11

Fourth order nonlinear evolution equation for capillary gravity waves in deep water in the presence of a thin thermocline including the effect of wind

Debsarma S., Das K. P.

International Journal of Applied Mechanics and Engineering

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2003

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Vol. 8, no 2

177-193

EN

A fourth order non-linear evolution equation is derived for a capillary-gravity wave packet in deep water in the presence of a thin thermocline including the effect of wind and viscous dissipation in water. In deriving this equation it has been assumed that the wind induced basic current in water is exponential and the effect of shear in air flow and viscous dissipation in water is accounted for by including a term in the evolution equation. The nonlinear evolution equation is used to study the stability of a uniform capillary-gravity wave train. Expressions for the maximum growth rate of instability and wave number at marginal stability are obtained. From results shown graphically it is found that the inclusion of wind effect increases the growth rate of instability irrespective of the presence of a thin thermocline. For waves with a small wave number, a thin thermocline has a stabilizing influence both in the presence and in the absence of wind input and the maximum growth rate of instability decreases with the increase of thermocline depth. But for waves with a large wave number a thin thermocline has no influence.

12

The Asymptotical Stability of a Dynamic System With Structural Damping

Hou X.

International Journal of Applied Mathematics and Computer Science

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2003

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Vol. 13, no 2

131-138

EN

A dynamic system with structural damping described by partial differential equations is investigated. The system is first converted to an abstract evolution equation in an appropriate Hilbert space, and the spectral and semigroup properties of the system operator are discussed. Finally, the well-posedness and the asymptotical stability of the system are obtained by means of a semigroup of linear operators.

13

Periodic solutions for evolution inclusions with time-dependent subdifferentials

Matzakos N., Papageorgiu N. S.

Demonstratio Mathematica

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2002

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Vol. 35, nr 3

521-538

EN

In this paper we examine a periodic evolution equation driven by a time-dependent subdifferential and with a multivalued forcing term. Using a fixed point theorem for pseudo-acyclic multifunctions we prove the existence of periodic trajectories. This approach requires a study of the structure of the solution set of the Cauchy problem, which is also conducted in this paper.

14

The effect of randomness on stability of surface gravity waves from fourth order nonlinear evolution equation

Dhar A. K., Das K. P.

International Journal of Applied Mechanics and Engineering

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2001

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Vol. 6, no 1

11-34

EN

A fourth order nonlinear evolution equation, which is a good starting point for the study of nonlinear water waves of wave-steepness up to 0.25, is used here to investigate the effect of randomness on stability of deep-water surface gravity waves in the presence of wind blowing over water. A spectral transport equation for narrow band Gaussian surface wave is derived. With the use of this transport equation stability analysis is made for an initial homogeneous wave spectrum having a very simple normal form to small oblique long wave length perturbations for a range of spectral widths. An expression for the growth rate of instability is obtained, in which higher order contribution comes from only one of the fourth order terms in the evolution equation, which is responsible for wave-induced mean flow. This higher order contribution in this expression for growth rate of instability produces a decrease in the growth rate. The growth rate of instability is found to decrease with the increase of spectral width and ultimately the instability disappears if the spectral width increases beyond a certain critical value, which is not influenced by the fourth order terms in the evolution equation.

15

On interaction between damage growth and material stiffness in 3D structures

Mika P.

Journal of Theoretical and Applied Mechanics

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1999

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nr 4

755-778

EN

Deterioration of material properties; such as, rupture toughness, strength, and rigidity as well as lifetime reductions are modelled by a symmetric second order damage tensor introduced into the constitutive description, when employing the theory of tensor function representations. The growth of damage and state failure are described by a one parameter model of damage evolution and a three parameter failure criterion. The constitutive description for different modes of failure frontpropagation is of particular interest in this study. The comparision between solutions obtained by means of two different physical models; i.e., the linear elastic and linear elastic with damages ones, respectively, has been presented. In both cases the stress redistribution due to geometrrical changes of the structure induced by the crack propagation has been considered. Numerical calculations have been made employing the ABAQUS programme, Runge--Kutta procedures for integrationof the evolution equations, standard methods of matrix division in LU decomposition and Gauss method for solving the set of linear equations.

PL

Procesy degradacji własności materiału, takich jak wytrzymałość, sztywność i redukcja czasu twrałości są modelowane przez symetryczny tensor uszkodzeń drugiego rzędu wprowadzony do opisu konstytutywnego z zastosowaniem teorii reprezentacji funkcji tensorowych. Wzrost uszkodzeń oraz stan zniszczenia są opisywane przez jednoparametrowe równanie evolucji oraz trójparametrowe kryterium zniszczenia. W pracy przedstawiono różne formy propagacji frontu zniszczenia w odniesieniu do opisu konstytutywnego. W tym kontekście dokonano porównania rozwiązań otrzymanych z zastosowaniem dwu odmiennych modeli fizycznych: liniowo sprężystego i liniowo sprężystego z uszkodzeniami. W obydwu przypadkach rozważana jest redystrybucja naprężeń spowodowana zmianami geometrycznymi konstrukcji wywołanymi propagacją zarysowań. W obliczeniach numerycznych odwołano się do programu ABAQUS, procedur Rungego--Kutty w całkowaniu równania ewolucji oraz standardowych metod rozkładu macierzy na górno-- i dolno--trójkątną, a w zakresie rozwiązywania układu równań algebraicznych -- do metody Gaussa.

16

A fourth order evolution equation for capillary-gravity waves including the effects of wind input and shear in the water current

Dhar A. K., Das K. P.

Applied Mechanics and Engineering

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1999

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Vol. 4, no 1

5-24

EN

A fourth order nonlinear evolution equation is derived for capillary gravity waves in deep water including the effect of a surface drift current in the water and shear in the air flow. From this evolution equation instability conditions are derived for a uniform capillary-gravity wave train. Graphs are plotted showing the maximum growth rate of instability and instability regions for weakly damped (linearly) and weakly growing (linearly) waves for some different values of friction velocity of the flow. From these graphs it is found that the effect of the wind input and shear in water current is to produce a decrease in the growth rate for weakly damped (linearly) waves and to produce an increase in growth rate for weakly growing (linearly) waves. The shear in the water current and the wind input are found to produce a shrinkage in the instability regions.