Nonlocal Integral Term Research Articles

Splitting-particle methods for structured population models: Convergence and applications

We propose a new numerical scheme designed for a wide class of structured population models based on the idea of operator splitting and particle approximations. This scheme is related to the Escalator Boxcar Train (EBT) method commonly used in biology, which is in essence an analogue of particle methods used in physics. Our method exploits the split-up technique, thanks to which the transport step and the nonlocal integral terms in the equation can be separately considered. The order of convergence of the proposed method is obtained in the natural space of finite non-negative Radon measures equipped with the flat metric. This convergence is studied even adding reconstruction and approximation steps in the particle simulation to keep the number of approximation particles under control. We validate our scheme in several test cases showing the theoretical convergence error. Finally, we use the scheme in situations in which the EBT method does not apply showing the flexibility of this new method to cope with the different terms in general structured population models.

Coupled fractional nonlinear differential equations and exact Jacobian elliptic solutions for exciton–phonon dynamics

Abstract An improved quantum model for exciton–phonon dynamics in an α-helix is investigated taking into account the interspine coupling and the influence of power-law long-range exciton–exciton interactions. Having constructed the model Hamiltonian, we derive the lattice equations and employ the Fourier transforms to go in continuum space showing that the long-range interactions (LRI) lead to a nonlocal integral term in the equations of motion. Indeed, the non-locality originating from the LRI results in the dynamic equations with space derivatives of fractional order. New theoretical frameworks are derived, such that: fractional generalization of coupled Zakharov equations, coupled nonlinear fractional Schrodinger equations, coupled fractional Ginzburg–Landau equations, coupled Hilbert–Zakharov equations, coupled nonlinear Hilbert–Ginzburg–Landau equations, coupled nonlinear Schrodinger equations and coupled nonlinear Hilbert–Schrodinger equations. Through the F-expansion method, we derive a set of exact Jacobian solutions of coupled nonlinear Schrodinger equations. These solutions include Jacobian periodic solutions as well as bright and dark soliton which are important in the process of energy transport in the molecule. We also discuss of the impact of LRI on the energy transport in the molecule.

Modified reduced Ostrovsky equation: Integrability and breaking

The modified reduced Ostrovsky equation is a reduction of the modified Korteweg-de Vries equation, in which the usual linear dispersive term with a third-order derivative is replaced by a linear nonlocal integral term, representing the effect of background rotation. Here we study the case when the cubic nonlinear term has the same polarity as the rotation term. This equation is integrable provided certain slope constraints are satisfied. We demonstrate, through theoretical analysis and numerical simulations, that when this constraint is not satisfied at the initial time, wave breaking inevitably occurs.

The Reduced Ostrovsky Equation: Integrability and Breaking

The reduced Ostrovsky equation is a modification of the Korteweg‐de Vries equation, in which the usual linear dispersive term with a third‐order derivative is replaced by a linear nonlocal integral term, which represents the effect of background rotation. This equation is integrable provided a certain curvature constraint is satisfied. We demonstrate, through theoretical analysis and numerical simulations, that when this curvature constraint is not satisfied at the initial time, then wave breaking inevitably occurs.

Existence of periodic and solitary waves for a nonlinear Schrödinger equation with nonlocal integral term of convolution type

We prove the existence of periodic solutions and solitons in the nonlinear Schrödinger equation with a nonlocal integral term of convolution type. By separating phase and amplitude, the problem is reduced to an integro-differential formulation that can be written as a fixed point problem for a suitable operator on a Banach space. Then a fixed point theorem due to Krasnoselskii can be applied.

Bernstein estimates: weakly coupled systems and integral equations

In this paper we extend the classical Bernstein estimates for systems of weakly coupled fully non-linear elliptic equations as well as scalar elliptic equations with non-local integral terms and singular kernels.

Continuum-level modelling of cellular adhesion and matrix production in aggregates

Key regulators in tissue-engineering processes such as cell culture and cellular organisation are the cell–cell and cell–matrix interactions. As mathematical models are increasingly applied to investigate biological phenomena in the biomedical field, it is important, for some applications, that these models incorporate an adequate description of cell adhesion. This study describes the development of a continuum model that represents a cell-in-gel culture system used in bone-tissue engineering, namely that of a cell aggregate embedded in a hydrogel. Cell adhesion is modelled through the use of non-local (integral) terms in the partial differential equations. The simulation results demonstrate that the effects of cell–cell and cell–matrix adhesion are particularly important for the survival and growth of the cell population and the production of extracellular matrix by the cells, concurring with experimental observations in the literature.

A matrix formulation to the wave equation with non-local boundary condition

Recently, various sequential numerical schemes have been proposed for the solution of non-classical hyperbolic initial value problems which involve non-local integral terms over the spatial domain. In this paper, we focus on the wave equation with the non-local boundary condition. Two matrix formulation techniques based on the shifted standard and shifted Chebyshev bases are proposed for the numerical solution. Several numerical examples and also some comparisons with another methods will be investigated to confirm the efficiency of this procedure.

MATHEMATICAL MODELLING OF CANCER INVASION: THE IMPORTANCE OF CELL–CELL ADHESION AND CELL–MATRIX ADHESION

The process of invasion of tissue by cancer cells is crucial for metastasis — the formation of secondary tumours — which is the main cause of mortality in patients with cancer. In the invasion process itself, adhesion, both cell–cell and cell–matrix, plays an extremely important role. In this paper, a mathematical model of cancer cell invasion of the extracellular matrix is developed by incorporating cell–cell adhesion as well as cell–matrix adhesion into the model. Considering the interactions between cancer cells, extracellular matrix and matrix degrading enzymes, the model consists of a system of reaction–diffusion partial integro–differential equations, with nonlocal (integral) terms describing the adhesive interactions between cancer cells and the host tissue, i.e. cell–cell adhesion and cell–matrix adhesion. Having formulated the model, we prove the existence and uniqueness of global in time classical solutions which are uniformly bounded. Then, using computational simulations, we investigate the effects of the relative importance of cell–cell adhesion and cell–matrix adhesion on the invasion process. In particular, we examine the roles of cell–cell adhesion and cell–matrix adhesion in generating heterogeneous spatio-temporal solutions. Finally, in the discussion section, concluding remarks are made and open problems are indicated.

Nonlinear development of two-layer Couette–Poiseuille flow in the presence of surfactant

The two-dimensional nonlinear evolution of the interface between two superposed layers of viscous fluid moving in a channel in the presence of an insoluble surfactant is examined. A pair of coupled weakly nonlinear equations is derived describing the interfacial and surfactant dynamics when one of the two fluid layers is very thin in comparison to the other. In contrast to previous work, the dynamics in the thin film are coupled to the dynamics in the thicker layer through a nonlocal integral term. For asymptotically small Reynolds number, the flow in the thicker layer is governed by the Stokes equations. A linearized analysis confirms the linear instability identified by previous workers and it is proven that the film flow is linearly unstable if the undisturbed surfactant concentration exceeds a threshold value. Numerical simulations of the weakly nonlinear equations reveal the existence of finite amplitude traveling-wave solutions. For order one Reynolds number, the flow in the thicker layer is governed by the linearized Navier–Stokes equations. In this case the weakly nonlinear film dynamics are more complex and include the possibility of periodic traveling-waves and chaotic flow.

Attractor for the nonlinear Schrödinger equation with a nonlocal nonlinear term

In this paper, we consider the long-time behavior of solutions of the dissipative 1D nonlinear Schrodinger (NLS) equation with nonlocal integral term and with periodic boundary conditions. We prove the existence of the global attractor $ \mathcal{A} $ for the nonlocal equation in the strong topology of H 1(?). We also prove that the global attractor is regular, i.e., $ \mathcal{A} \subset {H^2}\left( \Omega \right) $ , assuming that f(x) is of class C 2. Furthermore, we estimate the number of the determining modes for this equation.

Type II functional response for continuous, physiologically structured models

The goal of this work is to formulate a general Holling-type functional, or behavioral, response for continuous physiologically structured populations, where both the predator and the prey have physiological densities and certain rules apply to their interactions. The physiological variable can be, for example, a development stage, weight, age, or a characteristic length. The model leads to a Fredholm integral equation for the functional response, and, when inserted into population balance laws, it produces a coupled system of partial differential–integral equations for the two species, with a nonlocal integral term that arises from rules of interaction in the functional response. The general model is, typically, analytically intractable, but specialization to a structured prey–unstructured predator model leads to some analytic results that reveal interesting and unexpected dynamics caused by the presence of size-dependent handling times in the functional response. In this case, steady-states are shown to exist over long times, similar to the stable age-structure solutions for the McKendick–von Foerster model with exponential growth rates determined by the Euler–Lotka equation. But, for type II responses, there are early transient oscillations in the number of births that bifurcate in a few generations into either the decaying or growing steady-state. The bifurcation parameter is the initial level of prey. This special case is applied to a problem of the biological control of a structured pest population (e.g., aphids) by a predator (e.g., lady beetles).

MATHEMATICAL MODELLING OF CANCER INVASION OF TISSUE: THE ROLE AND EFFECT OF NONLOCAL INTERACTIONS

In this paper we consider a mathematical model of cancer cell invasion of tissue (extracellular matrix). Two crucial components of tissue invasion are (i) cancer cell proliferation, and (ii) over-expression and secretion of proteolytic enzymes by the cancer cells. The proteolytic enzymes are responsible for the degradation of the tissue, enabling the proliferating cancer cells to actively invade and migrate into the degraded tissue. Our model focuses on the role of nonlocal kinetic terms modelling competition for space and degradation. The model consists of a system of reaction-diffusion-taxis partial differential equations, with nonlocal (integral) terms describing the interactions between cancer cells and the host tissue. First of all we prove results concerning the local existence, uniqueness and regularity of solutions. We then prove global existence. Using Green's functions, we transform our original nonlocal equations into a coupled system of parabolic and elliptic equations and we undertake a numerical analysis of this equivalent system, presenting computational simulation results from our model showing the effect of the nonlocal terms (travelling waves we observed have the shape closely linked to the nonlocal terms). Finally, in the discussion section, concluding remarks are made and open problems are indicated.

Mathematical modelling of cancer cell invasion of tissue: Local and non-local models and the effect of adhesion

The ability to invade tissue is one of the hallmarks of cancer. Cancer cells achieve this through the secretion of matrix degrading enzymes, cell proliferation, loss of cell–cell adhesion, enhanced cell–matrix adhesion and active migration. Invasion of tissue by the cancer cells is one of the key components in the metastatic cascade, whereby cancer cells spread to distant parts of the host and initiate the growth of secondary tumours (metastases). A better understanding of the complex processes involved in cancer invasion may ultimately lead to treatments being developed which can localise cancer and prevent metastasis. In this paper we formulate a novel continuum model of cancer cell invasion of tissue which explicitly incorporates the important biological processes of cell–cell and cell–matrix adhesion. This is achieved using non-local (integral) terms in a system of partial differential equations where the cells use a so-called “sensing radius” R to detect their environment. We show that in the limit as R → 0 the non-local model converges to a related system of reaction–diffusion–taxis equations. A numerical exploration of this model using computational simulations shows that it can form the basis for future models incorporating more details of the invasion process.

The one-dimensional heat equation subject to a boundary integral specification

Various processes in the natural sciences and engineering lead to the nonclassical parabolic initial boundary value problems which involve nonlocal integral terms over the spatial domain. The integral term may appear in the boundary conditions. It is the reason for which such problems gained much attention in recent years, not only in engineering but also in the mathematics community. In this paper the problem of solving the one-dimensional parabolic partial differential equation subject to given initial and nonlocal boundary conditions is considered. Several approaches for the numerical solution of this boundary value problem which have been considered in the literature, are reported. New finite difference techniques are proposed for the numerical solution of the one-dimensional heat equation subject to the specification of mass. Numerical examples are given at the end of this paper to compare the efficiency of the new techniques. Some specific applications in various engineering models are introduced.

On electro-kinetic fluids: One dimensional configurations

Electro-kinetic fluids can be modeled by hydrodynamic systems describing the coupling between fluids and electric charges. The system consists of a momentum equation together with transport equations of charges. In the dynamics, the special coupling between the Lorentz force in the velocity equation and the material transport in the charge equation gives an energy dissipation law. In stationary situations, the system reduces to a Poisson-Boltzmann type of equation. In particular, under the no flux boundary conditions, the conservation of the total charge densities gives nonlocal integral terms in the equation. In this paper, we analyze the qualitative properties of solutions to such an equation, especially when the Debye constant $\epsilon$ approaches zero. Explicit properties can be derived for the one dimensional case while some may be generalized to higher dimensions. We also present some numerical simulation results of the system.

Existence of weak solutions to a system of nonlinear partial differential equations modelling ice streams

This paper deals with the mathematical analysis of a nonlinear system of three differential equations of mixed type. It describes the generation of fast ice streams in ice sheets flowing along soft and deformable beds. The system involves a nonlinear parabolic PDE with a multivalued term in order to deal properly with a free boundary which is naturally associated to the problem of determining the basal water flux in a drainage system. The other two equations in the system are an ODE with a nonlocal (integral) term for the ice thickness, which accounts for mass conservation and a first order PDE describing the ice velocity of the system. We first consider an iterative decoupling procedure to the system equations to obtain the existence and uniqueness of solutions for the uncoupled problems. Then we prove the convergence of the iterative decoupling scheme to a bounded weak solution for the original system.

Motion of weakly curved vortices and Hertzian radiation potential

A correspondence linking the electromagnetic theory with dipole sources to the far-field of vortices is developed in which the vector potential and the phase gradient correspond. This approach is applied to study the self-induced accelerated motion of the φ string. Acceleration dependence on the shape of the string is obtained by an asymptotic matching procedure between the core solution and the outer field. The non-locally induced field generated by the oscillating string is analyzed for several modes of oscillation. Vortex motion is analyzed and a rotational instability is found for non-planar vortex shapes due to a non-local integral along the vortex. Additional effects due to the relativistic nature of the field include the Doppler effect. For waves on the vortex with light velocity the non-local integral term vanishes, thus the motion law is purely local. The relevance of the model to magnetic fluxons is considered.

Solving parabolic integro-differential equations by an explicit integration method

The subject of this note is the numerical integration in time of nonclassical parabolic initial-boundary value problems which involve nonlocal integral terms over the spatial domain. The integral terms may appear in the boundary conditions and/or in the governing partial differential equation itself. These terms generally complicate the application of standard methods. Our purpose here is to draw attention to an explicit method, the Runge-Kutta-Chebyshev method, whose application remains straightforward in many cases of practical interest. We discuss two numerical examples to illustrate this. A comparison is made, respectively, with the implicit BDF code DASSL and with the Keller-box scheme.