System Of Nonlinear Integro-differential Equations Research Articles

The existence and convergence of the periodic solution of the system of complementary equations - nonlinear differential of the second class type Volterra

In this paper we study the existence and approximation of the periodic solutions for system of nonlinear integro-differential equations of Volterra type by Lebesque integrable function, By using weaker conditions of the function F(t,s,x,x) to be measurable in t and bounded by Lebesque integrable function.

Periodic solutions of a system of nonlinear integro-differential equations with pulse action

We establish a condition for the existence and uniqueness of a periodic solution of a system of nonlinear integro-differential equations with pulse action. The solution is represented as the limit of periodic iterations. We give estimates for the rate of convergence and for the exact solution of the system.

The numerical solution of the self-consistent evolution of plasma equilibria

An advanced mathematical model is developed for a self-consistent description of the evolution of free boundary toroidal plasmas. New numerical algorithms for the solution of the appropriate non-linear system of integro-differential equations are presented. An unusual feature of the system is that time derivatives enter only one of the equations. The model and algorithms are implemented in the SCoPE code. Examples of numerical modelling for tokamak plasmas are given.

Large dynamic displacements and deformations of a massive extensible thread in a moving medium

In a Lagrangian system of coordinates, non-linear differential equations are obtained that describe the deformation of a massive extensible thread in a moving medium. Three specific problems with specified numerical parameters are solved by a numerical method developed by the author for solving systems of non-linear Integro-differential equations.

Solving the problem of biological species living together by Adomian decomposition method

In this paper, we use Adomian decomposition method for solving the system of nonlinear integro-differential equations derived from considering biological species living together.

Self-induced acoustic transparency in the long-short-wave resonance mode

The effect of self-induced acoustic transparency for transverse-longitudinal pulses propagating along an external magnetic field in a system of resonance paramagnetic impurities with the effective spin S=1/2 is theoretically investigated. In this case, the short-wave transverse component of the pulse causes quantum transitions, and the longitudinal long-wave component dynamically shifts the frequency of those transitions. When the speeds of the longitudinal and transverse acoustic waves in the crystal matrix are close to each other, both components interact in the mode of the long-short-wave resonance, which is described by a system of nonlinear integro-differential equations. It is shown that this interaction results, in particular, in the modulation of the carrier frequency of the circular-polarized component of the pulse. More precisely, the frequency in the neighborhood of the signal’s maximum is less than in the vicinity of its edges. Solutions in the form of traveling 2π-pulses are analyzed analytically and numerically. It is shown that there exist solutions that include a longitudinal component and cannot be reduced to well-known transverse solitons of the sinus-Gordon equation.

A Conformal Lubricated Contact of Cylindrical Surfaces Involved in a Non-Steady Motion

A non-steady problem for a conformal elastohydrodynamically lubricated (EHL) contact of two infinite cylindrical surfaces with parallel axes is considered. It takes into account the elasticity of cylinders, lubricant viscosity, contact surface velocities, and the applied load. The problem is solved based on the “modified” formulation proposed by Kudish et al. (2000) which is free of such defects as discontinuity of its solution and independence of the solution from some of the initial data. The problem is reduced to a system of nonlinear integro-differential equations. The additional conditions include initial and boundary conditions and Newton’s second law applied to the internal cylinder motion. The main emphasis of the paper is three fold: the analysis of the transient dynamics of the system under constant external conditions and due to abrupt changes in applied load, and the system behavior in the case of a bump/dent presence on the shaft surface. The numerical solutions exhibit damped oscillatory behavior while approaching a steady state. It is observed that in a transient motion the radial displacement of the shaft center may vary by no more than 2.5 percent while the maximum pressure may vary by as much as 350 percent. Moreover, the variations of pressure are greater for stiffer materials.

On a Sociologically Structured Human Community Dynamics Model

A deterministic model for pair formation in an age-sex and religion-structured nonlimited human community is analyzed. The model consists of a system of nonlinear integro-differential equations and describes the dynamics of interacting religions which tolerate both uniconfessional pairs and those with different religions. The existence and uniqueness theorem is proved.

Heat and mass transfer in annular liquid jets: III. Combustion within the volume enclosed by the jet

The nonlinear dynamics of and heat and mass transfer processes in annular liquid jets are analyzed by means of a nonlinear system of integrodifferential equations which account for the liquid motion and the gases enclosed by the jet. Both linear and sinusoidal heat and mass addition sources are considered to take place homogeneously within the volume enclosed by the jet's inner interface in an attempt to simulate the combustion of hazardous wastes or materials within this volume. It is shown that the liquid's temperature at the jet's inner interface increases rapidly with linear heat addition, but drops also quickly to its initial value once heat addition is ended, whereas the pressure coefficient and the volume enclosed by the jet increase until they reach a maximum value and then decrease in an oscillatory manner towards their steady values. For the case of sinusoidal heat addition, it is shown that the pressure coefficient and interfacial concentration, temperature and heat and mass fluxes oscillate in a sinusoidal manner with the same frequency as that of the sinusoidal heat source. It is also shown that mass transfer phenomena are much slower than heat transfer ones. For the case of linear mass addition, it is shown that the temperature of the gases enclosed by the jet first decreases because of dilution and then it increases until it reaches a constant value that corresponds to the same temperature for the gases and the flowing liquid. The pressure of the gases enclosed by the jet first increases because of mass addition and then slowly decreases because of mass absorption by the jet.

The problem of evolution of toroidal plasma equilibria

This paper is devoted to an advanced mathematical model for a self-consistent description of the evolution of free boundary toroidal plasmas, with a description of numerical algorithms for the solution of the appropriate nonlinear system of integro-differential equations, and discussion of some results from the model.

Nonrelativistic wave equation for a system of interacting particles

It is shown that the Schr\"{o}dinger nonrelativistic equation of a system of interacting particles is not a rigorously nonrelativistic equation since it is based on the implicit assumption of finiteness of the interaction propagation velocity. For a system of interacting particles, a fully nonrelativistic nonlinear system of integro-differential equations is proposed. In the case where the size of the system of particles is of the same order as the Compton wavelength associated with particles, certain essential differences are shown to exist as compared with traditional consequences of the nonrelativistic Schr\"{o}dinger equation.

A mixed problem for a nonstationary system of nonlinear integro-differential equations

A mixed problem for a nonstationary system of nonlinear integro-differential equations

Bifurcation analysis for a nonlinear system of integro-differential equations modelling tumor-immune cells competition

A bifurcation analysis is developed for the initial value problem for a nonlinear system of integro-differential equations modelling the competition between tumor cells and immune system. It is shown that there exists a critical value of a bifurcation parameter separating situations where the immune system controls the neoplastic growth from those where tumor growth is not contrasted. This result refers to conjecture that the action of cytokine signals can modify the outputs of the competition.

A Fast Multilevel Algorithm for the Solution of Nonlinear Systems of Conductive-Radiative Heat Transfer Equations in Two Space Dimensions

In this paper we describe a fast multilevel algorithm for the solution of a system of nonlinear integro-differential equations that model steady-state combined conductive-radiative heat transfer in two space dimensions. This extends our previous work in one space dimension. We formulate the equations as a compact fixed point problem with the temperature as the unknown. The fixed point map requires both a Poisson solve and a transport solve for its evaluation. As a solver for both the transport problem and the full system we apply the Atkinson--Brakhage algorithm, using Newton-GMRES as the solver on the coarse mesh. We compare our solver choices with Newton-GMRES. Under modest stability and convergence assumptions on the transport solver, we prove convergence of the multilevel method for the complete system.

Modelling and simulating the effect of cytokines on the immune response to tumor cells

In this paper, we present a model for the competition between tumor cells and the immune system. We formulate the problem as an initial value problem for a nonlinear system of Integro-differential equations. We present simulations of the immune response to the onset of tumor cells. Furthermore, the analysis is aimed to investigate the tumor immune system interactions for activated versus depressed immune systems and for the action of cytokine signals.

Size distributions of misrejoining DNA fragments in irradiated cells.

When ionizing radiation strikes a cell it induces DNA double strand breaks (DSBs). Subsequently, some of the DSBs misrejoin and thus cause alterations in the size distribution of the DNA fragments. We derive a system of non-linear integro-differential equations describing the misrejoining interactions of five classes of DNA fragments, including rings and various types of linear fragments. The fragment classes are represented by density functions; the shape of a density function determines the probability that a fragment has a particular size and the amplitude (integral) equals the expected number of such fragments per cell. The equations are solved: analytically for exponentially distributed initial fragment sizes (corresponding to high doses) and numerically for arbitrary initial conditions. Computed final fragment size distributions are applied to situations representative of flow karyotypes and pulsed-field gel assays. For human flow karyotypes, the model can be used to obtain misrejoining estimates at doses too high for conventional methods of data analysis. For pulsed-field gel assays in which human chromosomes are digested with restriction endonucleases to form 'cut-somes' (restriction fragments), the model provides a means of misrejoining estimation when the cut-some sizes are non-random. The model suggests that if the cut-some size distribution for unirradiated cells is completely random, misrejoining of radiation-induced DSBs will not be detectable in the final size distribution.

Internal stresses, rheology, and structure formation in drying of a concentrated disperse system

A mathematical model is suggested for the stress strain state and structure formation in drying of a concentrated disperse system. The interrelated process of deformation of the medium and motion of the liquid in a saturated region is described by equations of filtration consolidation. The rheological relationship of a saturated medium is determined on the basis of the micromechanics of the relative motion of the particles. A system of nonlinear integrodifferential equations that describe the process is obtained and numerical analysis of the problem is carried out under different drying conditions.

Microscopic Behavior of the Classical Electron in the Absence of External Forces

A system of nonlinear integro-differential equations is derived for the motion of the classical electron with a rigid and spherically symmetric 3D gaussian distribution of charge. The equations are analyzed for stability around the state of rest and of uniform rectilinear motion with velocity small with respect to the velocity of light. The extremely high-frequency and radiationless micro-oscillations that the electron executes when disturbed from the equilibrium states show the inconsistency of the Abraham-Lorentz equation and of all concepts associated with this equation, like the notion that the electron may have a mass of electromagnetic origin.

A Fast Multilevel Algorithm for the Solution of Nonlinear Systems of Conductive-Radiative Heat Transfer Equations

In this paper we describe and analyze a fast multilevel algorithm for the solution of a system of nonlinear integro-differential equations that model steady-state combined conductive-radiative heat transfer. This system of equations for radiative intensity and temperature can be formulated as a compact fixed point problem in temperature alone with a fixed point map that requires both a solution of the linear transport equation and the linear heat equation for its evaluation. We obtain an efficient evaluation of the fixed point map by coupling a finite element diffusion solver with a fast transport solver developed by the second author. As a solver we apply a modification of the Atkinson--Brakhage method, with Newton--GMRES as the coarse mesh solver, to the full nonlinear system. We compare our discretization/solver pair with Newton--GMRES and the classical Atkinson--Brakhage algorithm.

Time-periodic solutions to a system of nonlinear integrodifferential equations

Time-periodic solutions to a system of nonlinear integrodifferential equations