Criteria for the Existence and Lower Bounds of Principal Eigenvalues of Time Periodic Nonlocal Dispersal Operators and Applications

The current paper is concerned with the spectral theory, in particular, the principal eigenvalue theory, of nonlocal dispersal operators with time periodic dependence, and its applications. Nonlocal and random dispersal operators are widely used to model diffusion systems in applied sciences and share many properties. There are also some essential differences between nonlocal and random dispersal operators, for example, a smooth random dispersal operator always has a principal eigenvalue, but a smooth nonlocal dispersal operator may not have a principal eigenvalue. In this paper, we first establish criteria for the existence of principal eigenvalues of time periodic nonlocal dispersal operators with Dirichlet type, Neumann type, or periodic type boundary conditions. It is shown that a time periodic nonlocal dispersal operator possesses a principal eigenvalue provided that the nonlocal dispersal distance is sufficiently small, or the time average of the underlying media satisfies some vanishing condition with respect to the space variable at a maximum point or is nearly globally homogeneous with respect to the space variable. Next we obtain lower bounds of the principal spectrum points of time periodic nonlocal dispersal operators in terms of the corresponding time averaged problems. Finally we discuss the applications of the established principal eigenvalue theory to time periodic Fisher or KPP type equations with nonlocal dispersal and prove that such equations are of monostable feature, that is, if the trivial solution is linearly unstable, then there is a unique time periodic positive solution which is globally asymptotically stable.     

Data-driven solutions and parameter discovery of the nonlocal mKdV equation via deep learning method

In this paper, we systematically study the integrability and data-driven solutions of the nonlocal mKdV equation. The infinite conservation laws of the nonlocal mKdV equation and the corresponding infinite conservation quantities are given through Riccti equation. The data-driven solutions of the zero boundary for the nonlocal mKdV equation are studied by using the multilayer physical information neural network algorithm, which include kink soliton, complex soliton, bright-bright soliton and the interaction between soliton and kink-type. For the data-driven solutions with nonzero boundary, we study kink, dark, anti-dark and rational solution. By means of image simulation, the relevant dynamic behavior and error analysis of these solutions are given. In addition, we discuss the inverse problem of the integrable nonlocal mKdV equation by applying the physics-informed neural network algorithm to discover the parameters of the nonlinear terms of the equation.

     

Age-Structured Population Dynamics with Nonlocal Diffusion

Random diffusive age-structured population models have been studied by many researchers. Though nonlocal diffusion processes are more applicable to many biological and physical problems compared with random diffusion processes, there are very few theoretical results on age-structured population models with nonlocal diffusion. In this paper our objective is to develop basic theory for age-structured population dynamics with nonlocal diffusion. In particular, we study the semigroup of linear operators associated to an age-structured model with nonlocal diffusion and use the spectral properties of its infinitesimal generator to determine the stability of the zero steady state. It is shown that (i) the structure of the semigroup for the age-structured model with nonlocal diffusion is essentially determined by that of the semigroups for the age-structured model without diffusion and the nonlocal operator when both birth and death rates are independent of spatial variables; (ii) the asymptotic behavior can be determined by the sign of spectral bound of the infinitesimal generator when both birth and death rates are dependent on spatial variables; (iii) the weak solution and comparison principle can be established when both birth and death rates are dependent on spatial variables and time; and (iv) the above results can be generalized to an age-size structured model. In addition, we compare our results with the age-structured model with Laplacian diffusion in the first two cases (i) and (ii).

     

Buckling analysis of functionally graded nanobeams under non-uniform temperature using stress-driven nonlocal elasticity

In this work, the size-dependent buckling of functionally graded(FG)Bernoulli-Euler beams under non-uniform temperature is analyzed based on the stressdriven nonlocal elasticity and nonlocal heat conduction. By utilizing the variational principle of virtual work, the governing equations and the associated standard boundary conditions are systematically extracted, and the thermal effect, equivalent to the induced thermal load, is explicitly assessed by using the nonlocal heat conduction law. The ...     

Pinning and Unpinning in Nonlocal Systems

We investigate pinning regions and unpinning asymptotics in nonlocal equations. We show that phenomena are related to but different from pinning in discrete and inhomogeneous media. We establish unpinning asymptotics using geometric singular perturbation theory in several examples. We also present numerical evidence for the dependence of unpinning asymptotics on regularity of the nonlocal convolution kernel.     

A Line-force loading on the surface of a nonlocal elastic half-infinite medium

Transmission of a concentrated force into a half-infinite nonlocal elastic medium is examined. A nonlocal concentrated force boundary condition which is different from that of Nowinski is derived from the nonlocal principle. The nonlocal stress field is obtained analytically. The result eliminates the stress singularity which occurs in the classical stresses and also the nonlocal stresses derived by Nowinski. A theoretical cut-off strength P_c is discussed.     

Pressure transient response of stochastically heterogeneous fractured reservoirs

A stochastic model for flow through inhomogeneous fractured reservoirs of double porosity, based on Barenblattet al.'s continuum approach, is presented. The fractured formation is conceptualized as an interconnected fracture network surrounding porous blocks, and amenable to the continuum approach. The block permeability is negligible in comparison to that of the fractures, and therefore the reservoir permeability is represented by the permeability of the fracture network. The fractured reservoir inhomogeneity is attributed to the fracture network, while the blocks are considered homogeneous. The mathematical model is represented by a coupled system of partial differential random equations, and a general solution for the average and for the correlation moments of the fracture pressure are obtained by the Neumann expansion (or Adomian decomposition). The solution for pressure is represented by an infinite series and an approximate solution for radial flow, is obtained by retaining the first two terms of the series. The purpose of this investigation is to get an insight on the pressure behavior in inhomogeneous fractured reservoirs and not to obtain type curves for determination of reservoir properties, which owing to the nonuniqueness of the solution, is impossible. For the present analysis we assumed an ideal reservoir with cylindrical symmetric inhomogeneity around the well. Fractured rock reservoirs being practically inhomogeneous, it is of interest to compare the pressure behavior of such reservoirs, with Warren and Roots's solution for (ideal) homogeneous reservoirs, used as a routine for determining the fractured reservoir characteristic parameters and, using the results of well tests. The comparison of the results show that inhomogeneous and homogeneous reservoirs exhibit a similar pressure behavior. While the behavior is identical, the same drawdown or a build-up pressure curve may be fitted by different characteristic dimensionless parameters and, when attributed to an inhomogeneous or a homogeneous reservoir. It is concluded that the ambiguity in determining the fractured reservoir and, makes questionable the usefulness of determination of these parameters. Computations were also carried out to determine the correlation between the fracture pressure at the well and the fracture pressure at different points.     

Two notes on nonlocal field theory

In this paper, two fundamental problems completely unsolved in nonlocal field theory are studied. The first is the dependence of nonlocal residuals. By studying this problem, a theorem concerning the relationship between the residuals of nonlocal body force and nonlocal moment of momentum is given and proven. The other problem is how to give the stress boundary conditions in the linear theory of nonlocal elasticity. The stress boundary conditions obtained in this paper can not only answer why the nonlocal stress solution satisfying the boundary conditionst _ji ^(s) n _j ¦O ₂ =p _i (p _i is a constant) on the surface of crack does not exist but also give a model of the molecular cohesive stress on the crack tip.     

Local discontinuous Galerkin method for a nonlocal viscous conservation laws

The purpose of this article is to investigate high‐order numerical approximations of scalar conservation laws with nonlocal viscous term. The viscous term is given in the form of convolution in space variable. With the help of the characteristic of viscous term, we design a semidiscrete local discontinuous Galerkin (LDG) method to solve the nonlocal model. We prove stability and convergence of semidiscrete LDG method in L² norm. The theoretical analysis reveals that the present numerical scheme is stable with optimal convergence order for the linear case, and it is stable with sub‐optimal convergence order for nonlinear case. To demonstrate the validity and accuracy of our scheme, we test the Burgers equation with two typical nonlocal fractional viscous terms. The numerical results show the convergence order accuracy in space for both linear and nonlinear cases. Some numerical simulations are provided to show the robustness and effectiveness of the present numerical scheme.     

Nonlocal versus local elastoplastic behaviour of heterogeneous materials

The scale transition methods have been developed for many years in order to obtain the overall behavior of polycrystalline materials from their microscopic behavior and their microstructure. Nevertheless, some basic aspects are absent from such formalisms. The most significant one seems to be the heterogeneization by plastic straining which involves nonlocality of hardening. In this article, a nonlocal theory based upon crystalline plasticity is developed from which a nonlocal constitutive equation at the grain level is derived. With regard to the polycrystal, in order to deduce the behavior of a local equivalent homogeneous medium, an integral equation is proposed and solved for nonlocal inhomogeneous materials by the self-consistent approximation. This scheme is developed in case of a two-phase nonlocal material representing the dislocation cell structure induced during plastic straining. Numerical simulations based on a simplified model show significant effects on the intragranular heterogeneization.     

Multi-type Entire Solutions in a Nonlocal Dispersal Epidemic Model

This paper deals with entire solutions of a nonlocal dispersal epidemic model. Unlike local (random) dispersal problems, a nonlocal dispersal operator is not compact and the solutions of nonlocal dispersal system studied here lack regularity in suitable spaces, which affects the uniform convergence of the solution sequences and the technique details in constructing the entire solutions. In the monostable case, some new types of entire solutions are constructed by combining leftward and rightward traveling fronts with different speeds and a spatially independent solution. In the bistable case, the existence of many different entire solutions with merging fronts are proved by constructing different sub- and super-solutions. Various qualitative features of the entire solutions are also investigated. A key idea is to characterize the asymptotic behaviors of the traveling wave solutions at infinite in terms of appropriate sub- and super-solutions. Finally, we also obtain the smoothness of the entire solutions in space, i.e., the solutions established in our paper are global Lipschitz continuous in space.     

Accurate and straightforward symplectic approach for fracture analysis of fractional viscoelastic media

An accurate and straightforward symplectic method is presented for the fracture analysis of fractional two-dimensional(2D) viscoelastic media. The fractional Kelvin-Zener constitutive model is used to describe the time-dependent behavior of viscoelastic materials. Within the framework of symplectic elasticity, the governing equations in the Hamiltonian form for the frequency domain(s-domain) can be directly and rigorously calculated. In the s-domain, the analytical solutions of the displacement ...     

Radial Flow in a Bounded Randomly Heterogeneous Aquifer

Flow to wells in nonuniform geologic formations is of central interest to hydrogeologists and petroleum engineers. There are, however, very few mathematical analyses of such flow. We present analytical expressions for leading statistical moments of vertically averaged hydraulic head and flux under steady-state flow to a well that pumps water from a bounded, randomly heterogeneous aquifer. Like in the widely used Thiem equation, we prescribe a constant pumping rate deterministically at the well and a constant head at a circular outer boundary of radius L. We model the natural logarithm Y = lnT of aquifer transmissivity T as a statistically homogeneous random field with a Gaussian spatial correlation function. Our solution is based on exact nonlocal moment equations for multidimensional steady state flow in bounded, randomly heterogeneous porous media. Perturbation of these nonlocal equations leads to a system of local recursive moment equations that we solve analytically to second order in the standard deviation of Y. In contrast to most stochastic analyses of flow, which require that log transmissivity be multivariate Gaussian, our solution is free of any distributional assumptions. It yields expected values of head and flux, and the variance–covariance of these quantities, as functions of distance from the well. It also yields an apparent transmissivity, T _a, defined as the negative ratio between expected flux and head gradient at any radial distance. The solution is supported by numerical Monte Carlo simulations, which demonstrate that it is applicable to strongly heterogeneous aquifers, characterized by large values of log transmissivity variance. The two-dimensional nature of our solution renders it useful for relatively thin aquifers in which vertical heterogeneity tends to be of minor concern relative to that in the horizontal plane. It also applies to thicker aquifers when information about their vertical heterogeneity is lacking, as is commonly the case when measurements of head and flow rate are done in wells that penetrate much of the aquifer thickness. Potential uses include the analysis of pumping tests and tracer test conducted in such wells, the statistical delineation of their respective capture zones, and the analysis of contaminant transport toward fully penetrating wells.     

General hoyle–youngdahl and love solutions in the linear inhomogeneous theory of elasticity

The general Hoyle–Youngdahl and Love solutions in the three-dimensional theory of inhomogeneous linear elastic materials are proposed. Following a brief historical outline of various general solutions existing in the classical linear elasticity of homogeneous isotropic media, key steps of the derivation of the Hoyle–Youngdahl and Love solutions are presented. The procedure is then generalized to the case of inhomogeneous elastic materials with elastic constants depending on the z-coordinate. The significance of the solutions and their relevance to modeling of functionally graded materials is discussed in brief     

Two-to-one internal resonance in microscanners

To realize large scanning angles, torsional microscanners are normally excited at their natural frequencies. Usually, a bias DC voltage is also applied to scan around a desired nonzero tilt angle. As a result, a deep understanding of the mirror’s response to a DC-shifted primary resonance excitation is imperative. Along these lines, we use the method of multiple scales to obtain a second-order nonlinear approximate analytical solution of the mirror steady-state response. We show that the response of the mirror exhibits a softening-type behavior that increases as the magnitude of the DC component increases. For a given mirror, we can also identify a DC voltage range wherein the mirror exhibits a two-to-one internal resonance between the first two modes; that is, ω ₂≈2ω ₁. To analyze the mirror behavior within that range, we first treat the case where the excitation frequency is near the first-mode frequency; that is, Ω≈ω ₁. Then we treat the case where the excitation frequency is near the second-mode frequency; that is, Ω≈ω ₂. We analyze the stability of the response and compare the analytical results to numerical solutions obtained via long-time integration of the equations of motion. We show that, due to the internal resonance, the mirror exhibits complex dynamic behavior characterized by aperiodic responses to primary resonance excitations. This behavior results in undesirable oscillations that are detrimental to the mirror performance, namely bringing the target point in and out of focus and resulting in distorted images.

     

Finite element solutions for nonhomogeneous nonlocal elastic problems

A finite element procedure for analysing nonhomogeneous nonlocal elastic 2D problems is presented and discussed. The procedure grounds on a variationally consistent approach known, in the relevant literature, as Nonlocal Finite Element Method. The latter is recast making use of a recently theorized phenomenological strain-difference-based nonhomogeneous nonlocal elastic model. The peculiarities of the numerical procedure together with the pertinent nonlocal operators are expounded and discussed. Two simple numerical 2D examples close the paper.     

Flow and Transport Properties of Deforming Porous Media. II. Electrical Conductivity

In Part I of this series, we presented a new theoretical approach for computing the effective permeability of porous media that are under deformation by a hydrostatic pressure P. Beginning with the initial pore-size distribution (PSD) of a porous medium before deformation and given the Young’s modulus and Poisson’s ratio of its grains, the model used an extension of the Hertz–Mindlin theory of contact between grains to compute the new PSD that results from applying the pressure P to the medium and utilized the updated PSD in the effective-medium approximation (EMA) to estimate the effective permeability. In the present paper, we extend the theory in order to compute the electrical conductivity of the same porous media that are saturated by brine. We account for the possible contribution of surface conduction, in order to estimate the electrical conductivity of brine-saturated porous media. We then utilize the theory to update the PSD and, hence, the pore-conductance distribution, which is then used in the EMA to predict the pressure dependence of the electrical conductivity. Comparison between the predictions and experimental data for twenty-six sandstones indicates agreement between the two that ranges from excellent to good.

     

A nonlocal constitutive model generated by matrix functions for polyatomic periodic linear chains

We establish a discrete lattice dynamics model and its continuum limits for nonlocal constitutive behavior of polyatomic cyclically closed linear chains being formed by periodically repeated unit cells (molecules), each consisting of \({n \geq 1}\) atoms which all are of different species, e.g., distinguished by their masses. Nonlocality is introduced by elastic potentials which are quadratic forms of finite differences of orders \({m \in \mathbf{N}}\) of the displacement field leading by application of Hamilton’s variational principle to nondiagonal and hence nonlocal Laplacian matrices. These Laplacian matrices are obtained as matrix power functions of even orders 2m of the local discrete Laplacian of the next neighbor Born-von-Karman linear chain. The present paper is a generalization of a recent model that we proposed for the monoatomic chain. We analyze the vibrational dispersion relation and continuum limits of our nonlocal approach. “Anomalous” dispersion relation characteristics due to strong nonlocality which cannot be captured by classical lattice models is found and discussed. The requirement of finiteness of the elastic energies and total masses in the continuum limits requires a certain scaling behavior of the material constants. In this way, we deduce rigorously the continuum limit kernels of the Laplacian matrices of our nonlocal lattice model. The approach guarantees that these kernels correspond to physically admissible, elastically stable chains. The present approach has the potential to be extended to 2D and 3D lattices.     

Colloid Transport in Porous Media: A Review of Classical Mechanisms and Emerging Topics

To celebrate the tenth anniversary of InterPore, we present an interdisciplinary review of colloid transport through porous media. This review aims to explore both classical colloid transport and topics that fall outside that purview and thus offer transformative insights into the physics governing transport behavior. First, we discuss the unique colloid characteristics relative to molecules and larger particles. Then, the classical advection–dispersion–filtration models (both conceptual and mathematical) of colloid transport are introduced as well as anomalous transport behaviors. Next, the forces of interaction between colloids and porous media surfaces are discussed. Fourth, applications that are interested in maximizing the transport of colloids through porous media are considered. Then the concept of motile, active biocolloids is introduced, and finally, colloid swarming as a newly recognized mode of transport is summarized.