System Of Nonlinear Integro-differential Equations Research Articles

Numerical solutions of systems of nonlinear integro-differential equations by Homotopy-perturbation method

In this study, the numerical solutions of a system of two nonlinear integro-differential equations, which describes biological species living together, are derived employing the well-known Homotopy-perturbation method. The approximate solutions are in excellent agreement with those obtained by the Adomian decomposition method. Furthermore, we present an analytical approximate solution for a more general form of the system of nonlinear integro-differential equations. The numerical result indicates that the proposed method is straightforward to implement, efficient and accurate for solving nonlinear integro-differential equations.

A dynamic neural field model of mesoscopic cortical activity captured with voltage-sensitive dye imaging.

A neural field model is presented that captures the essential non-linear characteristics of activity dynamics across several millimeters of visual cortex in response to local flashed and moving stimuli. We account for physiological data obtained by voltage-sensitive dye (VSD) imaging which reports mesoscopic population activity at high spatio-temporal resolution. Stimulation included a single flashed square, a single flashed bar, the line-motion paradigm – for which psychophysical studies showed that flashing a square briefly before a bar produces sensation of illusory motion within the bar – and moving squares controls. We consider a two-layer neural field (NF) model describing an excitatory and an inhibitory layer of neurons as a coupled system of non-linear integro-differential equations. Under the assumption that the aggregated activity of both layers is reflected by VSD imaging, our phenomenological model quantitatively accounts for the observed spatio-temporal activity patterns. Moreover, the model generalizes to novel similar stimuli as it matches activity evoked by moving squares of different speeds. Our results indicate that feedback from higher brain areas is not required to produce motion patterns in the case of the illusory line-motion paradigm. Physiological interpretation of the model suggests that a considerable fraction of the VSD signal may be due to inhibitory activity, supporting the notion that balanced intra-layer cortical interactions between inhibitory and excitatory populations play a major role in shaping dynamic stimulus representations in the early visual cortex.

Modelling aggregation–fragmentation phenomena from kinetic to macroscopic scales

This paper deals with the modelling of aggregation and/or fragmentation physical phenomena for large systems of interacting living entities in the framework of the mathematical kinetic theory for active particles. After introducing various mathematical structures in terms of systems of nonlinear integro-differential equations with quadratic type nonlinearities and variable number of equations, the relative qualitative analysis of the initial value problem is presented. Finally, the paper deals with the derivation of macroscopic equations based on the underlying description at the microscopic scale delivered by the kinetic theory models.

A new homotopy perturbation method for system of nonlinear integro-differential equations

In this paper, a new homotopy perturbation method (NHPM) is introduced to obtain exact solutions of system of nonlinear integro-differential equations. Theoretical considerations are discussed. Two examples are given to demonstrate the efficiency of NHPM to the classical HPM and variational iteration methods.

Construction of a nontrivial solution of a system of nonlinear integro-differential equations

The paper studies a system of nonlinear, vector integro-differential equations with sum and difference kernel and possessing the trivial solution. A nontrivial solution of the system in the Sobolev spaceW ∞,2 ×n (0,+∞) is constructed.

Boundary value problems for systems of nonlinear integro-differential equations with deviating arguments

By establishing a comparison result and using the method of upper and lower solutions and the monotone iterative technique, we investigate the systems of nonlinear mixed integro-differential equations with deviating arguments and mixed boundary value conditions. Sufficient conditions are established for the existence of extremal system of solutions of the given problem.

Large time asymptotic and numerical solution of a nonlinear diffusion model with memory

Large time behavior of solutions and finite difference approximation of a nonlinear system of integro-differential equations associated with the penetration of a magnetic field into a substance is studied. Two initial-boundary value problems are investigated: the first with homogeneous conditions on whole boundary and the second with nonhomogeneous boundary data on one side of lateral boundary. The rates of convergence are also given. Mathematical results presented show that there is a difference between stabilization rates of solutions with homogeneous and nonhomogeneous boundary conditions. The convergence of the corresponding finite difference scheme is also proved. The decay of the numerical solution is compared with the analytical results.

Nonlinear vibrations and dynamic stability of a viscoelastic circular cylindrical shell with shear strain and inertia of rotation taken into account

We consider nonlinear vibration and dynamic stability problems for a viscoelastic circular cylindrical shell according to the refined Timoshenko theory, which takes into account the shear strain and the inertia of rotation, in a geometrically nonlinear setting. The problem data are reduced to systems of nonlinear integro-differential equations with singular relaxation kernels, which can be solved by the Bubnov-Galerkin method combined with a numerical method based on quadrature formulas. We study the numerical convergence of the Bubnov-Galerkin method. We analyze the shell dynamic behavior in a wide range of physical-mechanical and geometric parameters. We demonstrate the influence of the viscoelastic properties of the material on the nonlinear vibrations and dynamic stability of a circular cylindrical shell. We also compare the results obtained according to different theories.

Existence theorems for systems of nonlinear integro-differential equations

The existence of solutions for many systems of integro-differential equations discovered and generalized in the process of applying the Galerkin method for some initial-boundary value problems will be investigated in this paper.

Bounded and periodic solutions of nonlinear integro-differential equations with infinite delay

By using the concept of integrable dichotomy, the fixed point theory, functional analysis methods and some new technique of analysis, we obtain new criteria for the existence and uniqueness of bounded and peri- odic solutions of general and periodic systems of nonlinear integro-differential equations with infinite delay.

Existence of a periodic solutions for certain system of nonlinear integro-differential equations

In this paper we investigate the existence and approximation of the periodic solutions for certain systems of nonlinear integro-differential equations, by using the method of successive periodic approximation of ordinary differential equations which is given by A. M. Samoilenko. Also these investigation lead us to the improving the extending the above method.

Influence of a surface irregularity on the rigidity of a lubricated contact

A technique is presented for numerical solution of the problem of the elastohydrodynamic point contact characteristic of antifriction bearings and tooth gears, taking into account that on the surface in the contact area there is an irregularity in the form of a hill or a dimple. Distributions of pressure and thickness of the lubricant film in the contact area are obtained. The influence of the type of irregularity on the elastic characteristics and rigidity of the elastohydrodynamic point contact is shown. The presented results may be used in calculations of concentrated lubricated contacts. The mathematical model of the elastohydrodynamic contact is described by a system of nonlinear integrodifferential equations with a free boundary (the outflow boundary of the contact area). The computational algorithm is based on the Newton method.

The periodic solution for some systems of nonlinear integro-differential equations of Volterra type.

The periodic solution for some systems of nonlinear integro-differential equations of Volterra type.

Complete integrable particle methods and the recurrence of initial states for a nonlinear shallow-water wave equation

We propose an algorithm for an asymptotic model of shallow-water wave dynamics in a periodic domain. The algorithm is based on the Hamiltonian structure of the equation and corresponds to a completely integrable particle lattice. In particular, “periodic particles” are introduced in the algorithm for waves travelling through the domain. Each periodic particle in this method travels along a characteristic curve of the shallow-water wave model, determined by solving a system of nonlinear integro-differential equations. We introduce a fast summation algorithm to reduce the computational cost from O(N2) to O(N), where N is the number of particles. With the aim of providing a test of the algorithms, we scale the shallow-water wave equation to make it asymptotically equivalent to the KdV equation in the form studied by Zabusky and Kruskal in their seminal 1965 paper, thereby also testing the equivalence of the two models derived under similar asymptotic approximations of shallow-water wave dynamics. By using the fast summation algorithm and the asymptotic scaling analysis, we further test this equivalence by investigating the interaction of solitons and recurrence of initial states for the shallow-water wave equation in periodic domains. Finally, to illustrate the hyperbolic nature of the dynamics of the shallow-water wave model, we introduce a particle algorithm and its integral counterpart for the initial-boundary value problem with homogeneous boundary conditions on finite intervals.

Dynamic stability of viscoelastic circular cylindrical shells taking into account shear deformation and rotatory inertia

The present work discusses the problem of dynamic stability of a viscoelastic circular cylindrical shell, according to revised Timoshenko theory, with an account of shear deformation and rotatory inertia in the geometrically nonlinear statement. Proceeding by Bubnov-Galerkin method in combination with a numerical method based on the quadrature formula the problem is reduced to a solution of a system of nonlinear integro-differential equations with singular kernel of relaxation. For a wide range of variation of physical mechanical and geometrical parameters, the dynamic behavior of the shell is studied. The influence of viscoelastic properties of the material on the dynamical stability of the circular cylindrical shell is shown. Results obtained using different theories are compared.

He’s homotopy perturbation method for systems of integro-differential equations

In this article, the homotopy perturbation method [He JH. Homotopy perturbation technique. Comput Meth Appl Mech Eng 1999;178:257–62; He JH. A coupling method of homotopy technique and perturbation technique for nonlinear problems. Int J Non-Linear Mech 2000;35(1):37–43; He JH. Comparison of homotopy perturbation method and homotopy analysis method. Appl Math Comput 2004;156:527–39; He JH. Homotopy perturbation method: a new nonlinear analytical technique. Appl Math Comput 2003;135:73–79; He JH. The homotopy perturbation method for nonlinear oscillators with discontinuities. Appl Math Comput 2004;151:287–92; He JH. Application of homotopy perturbation method to nonlinear wave equations Chaos, Solitons & Fractals 2005;26:695–700] is applied to solve linear and nonlinear systems of integro-differential equations. Some nonlinear examples are presented to illustrate the ability of the method for such system. Examples for linear system are so easy that has been ignored.

The existence and approximation of the periodic solution of a system of nonlinear differential-differential equations of the first order with independent variable delay with boundary conditions

In this paper we study the existence and approximation of the solutions for some systems of nonlinear integro-differential equations with having retarded arguments and boundary conditions. The numerical–analytic method has been used to study the periodic solutions of ordinary differential equations which were introduced by A. M. Samoilenko.

A numerical method for a nonlinear system of integro-differential equations arising in thermoviscoelasticity

This paper deals with the numerical solution of nonlinear integro-differential equations modelling a one-dimensional quasi-static contact problem in thermoviscoelasticity. A finite element approximation is proposed and analysed and some numerical results are given.

Asymptotics of a solution of a nonlinear system of diffusion of a magnetic field into a substance

We study the asymptotic behavior at large time of a solution to a system of nonlinear integro-differential equations which arises in mathematical modeling of diffusion of a magnetic field into a substance. We establish the corresponding stabilization rate.

An Asymptotic Framework for the Analysis of Hydraulic Fractures: The Impermeable Case

This paper presents a novel asymptotic framework to obtain detailed solutions describing the propagation of hydraulic fractures in an elastic material. The problem consists of a system of nonlinear integro-differential equations and a free boundary problem. This combination of local and nonlocal effects leads to transitions on a small scale near the crack tip, which control the behavior across the whole fracture profile. These transitions depend upon the dominant physical process(es) and are identified by simultaneously scaling the associated parameters with the distance from the tip. A smooth analytic solution incorporating several physical processes in the crucial tip region can be constructed using this new framework. In order to clarify the exposition of the new methodology, this paper is confined to considering the impermeable case in which only the two physical processes of viscous dissipation and structure energy release compete.