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Efficient and accurate spectral solvers for nonlocal models in any spatial dimension are presented. The approach we pursue is based on the Fourier multipliers of nonlocal Laplace operators introduced in a previous work. It is demonstrated that the Fourier multipliers, and the eigenvalues in particular, can be computed accurately and efficiently. This is achieved through utilizing the hypergeometric representation of the Fourier multipliers in which their computation in $n$ dimensions reduces to the computation of a 1D smooth function given in terms of $_2F_3$. We use this representation to develop spectral techniques to solve periodic nonlocal time-dependent problems. For linear problems, such as the nonlocal diffusion and nonlocal wave equations, we use the diagonalizability of the nonlocal operators to produce a semi-analytic approach. For nonlinear problems, we present a pseudo-spectral method and apply it to solve a Brusselator model with nonlocal diffusion. Accuracy and efficiency of the spectral solvers are compared against a finite-difference solver.

In this paper,we study the complicated dynamics of general Morris-Lecar model with the impact of Cl- fluctuations on firing patterns of this neuron model. After adding Cl- channel in the original Morris-Lecar model, the dynamics of the original model such as its bifurcations of equilibrium points would be changed and they occurred at different values compared to the primary model. We discover these qualitative changes in the point of dynamical systems and neuroscience. We will conduct the co-dimension two bifurcations analysis with respect to different control parameters to explore the complicated behaviors for this new neuron model.

In the current work, we study two infectious disease models and we use nonlinear optimization and optimal control theory which helps to find strategies towards transmission control and to forecast the international spread of the infectious diseases. The relationship between epidemiology, mathematical modeling and computational tools lets us to build and test theories on the development and fighting with a disease. This study is motivated by the study of epidemiological models applied to infectious diseases in an optimal control perspective. We use the numerical methods to display the solutions of the optimal control problems to find the effect of vaccination on these models. Finally, global sensitivity analysis LHS Monte Carlo method using Partial Rank Correlation Coefficient (PRCC) has been performed to investigate the key parameters in model equations. This present work will advance the understanding about the spread of infectious diseases and lead to novel conceptual understanding for spread of them.

In this paper, we investigate the complex dynamics of two-species Ricker-type discrete-time competitive model. We perform a local stability analysis for the fixed points and we will discuss about its persistence for boundary fixed points. This system inherits the dynamics of one-dimensional Ricker model such as cascade of period-doubling bifurcation, periodic windows and chaos. We explore the existence of chaos for the equilibrium points for a specific case of this system using Marotto theorem and proving the existence of snap-back repeller. We use several dynamical systems tools to demonstrate the qualitative behaviors of the system.

Discrete-time dynamical systems or difference equations have been increasingly used to model the biological and ecological systems for which there is time interval between each measurement. This modeling approach is done through using the iterative maps. Iterative maps are an essential part of nonlinear systems dynamics as they allow us to take the output of the previous state of the system and fit it back to the next iteration. In general, it is not easy to explicitly solve a system of difference equations. There are different methods of solving different types of difference equations. This book introduces concepts, theorems, and methods in discreet-time dynamical systems theory which are widely used in studying and analysis of local dynamics of biological systems and provides many traditional applications of the theory to different fields in biology. Our focus in this book is covering three important parts of discrete-time dynamical systems theory: Stability theory, Bifurcation theory and Chaos theory. Mathematically speaking, stability theory in the field of discrete-time dynamical systems deals with the stability of solutions of difference equations and of orbits of dynamical systems under small perturbations of initial conditions. In dynamical systems point of view, bifurcation theory addresses the changes in the qualitative behavior or topological structure of the solutions of a family of difference equations. Finally, chaos theory is a branch of dynamical systems which focuses on the study of chaotic states of a dynamical system which is often governed by deterministic laws and its solutions demonstrate irregular behavior and are highly sensitive to initial conditions. Therefore, this book is a blend of three important parts of discrete-time dynamical systems theory and their exciting applications to biology.

Fourier multiplier analysis is developed for nonlocal peridynamic-type Laplace operators, which are defined for scalar fields in $\mathbb{R}^n$. The Fourier multipliers are given through an integral representation. We show that the integral representation of the Fourier multipliers is recognized explicitly through a unified and general formula in terms of the hypergeometric function $_2F_3$ in any spatial dimension $n$. Asymptotic analysis of $_2F_3$ is utilized to identify the asymptotic behavior of the Fourier multipliers $m(\nu)$ as $\|\nu\|\rightarrow \infty$. We show that the multipliers are bounded when the peridynamic Laplacian has an integrable kernel, and diverge when the kernel is singular. The bounds and decay rates are presented explicitly in terms of the dimension $n$, the integral kernel, and the peridynamic Laplacian nonlocality. The asymptotic analysis is applied in the periodic setting to prove a regularity result for the peridynamic Poisson equation and, moreover, show that its solution converges to the solution of the classical Poisson equation.

A mathematical model can be defined as an abstract model which uses mathematical language to describe the behavior and evolution of a system. Mathematical models are used widely in the many different sciences and engineering disciplines (such as physics, biology, chemistry and engineering). Mathematical models may have many different forms, including continuous time and discrete time dynamical systems (using differential equations an difference equations respectively), statistical models, partial differential equations, or game theoretic models. Mathematical modeling has an important role in the discovering the problems which occur in our daily life. Mathematical and computational models has been frequently used to help interpret experimental data. Models also can help to describe our beliefs about how different phenomenon around the world functions. In mathematical modeling, we try to transfer those beliefs and pictures into the language of mathematics. This transformation is very beneficial. First, Mathematics is an exact and delicate language. Second, we can easily formulate ideas and also determine the basic assumptions. The governed rules in Mathematics help us to manipulate the problem. Strongly speaking, in Mathematical modeling, we are using the results which have been already proved by mathematicians over hundreds of years. Computers play an important role to perform numerical simulations and calculations. Although, many of the systems in the real world are too complicated to model but we can solve this problem by identifying the most important parts of the system and then we include them in the model, the rest will be excluded. Afterward, computer simulations can be applied to handle the model equations and desired manipulations.

Closed form expression for the buckling load of generally anisotropic laminated composite simply supported thin plates is derived. The Rayleigh-Ritz displacement field approximation based on the energy approach introduced an upper bound solution compared to the FE results. Therefore, the critical stability matrix is used to obtain an accurate buckling formula. The effective axial, coupling and flexural stiffness coefficients of the anisotropic layup is determined from the generalized constitutive relationship using dimensional reduction by static condensation of the 6 × 6 composite stiffness matrix. The resulting explicit formula has an additional term, which is a function of the effective coupling and axial stiffness. This formula reduces down to Euler buckling formula once the effective coupling stiffness term vanishes for isotropic and certain classes of laminated composites. The closed form results are verified against finite element Eigenvalue solutions for a wide range of anisotropic laminated layups yielding high accuracy. Comparisons with a limited number of experiments are also performed showing good correspondence. A brief parametric study is then conducted to examine the effect of ply orientations and material properties including hybrid carbon/glass fiber composites. Relevance of the numerical and closed form results is discussed for all these cases.

A nonlocal vector calculus was introduced in Du et al. (Math Model Meth Appl Sci 23:493–540, 2013) that has proved useful for the analysis of the peridynamics model of nonlocal mechanics and nonlocal diffusion models. A formulation is developed that provides a more general setting for the nonlocal vector calculus that is independent of particular nonlocal models. It is shown that general nonlocal calculus operators are integral operators with specific integral kernels. General nonlocal calculus properties are developed, including nonlocal integration by parts formula and Green’s identities. The nonlocal vector calculus introduced in Du et al. (Math Model Meth Appl Sci 23:493–540, 2013) is shown to be recoverable from the general formulation as a special example. This special nonlocal vector calculus is used to reformulate the peridynamics equation of motion in terms of the nonlocal gradient operator and its adjoint. A new example of nonlocal vector calculus operators is introduced, which shows the potential use of the general formulation for general nonlocal models.

The convergence of a peridynamic model for solid mechanics inside heterogeneous media in the limit of vanishing nonlocality is analyzed. It is shown that the operator of linear peridynamics for an isotropic heterogeneous medium converges to the corresponding operator of linear elasticity when the material properties are sufficiently regular. On the other hand, when the material properties are discontinuous, i.e., when material interfaces are present, it is shown that the operator of linear peridynamics diverges, in the limit of vanishing nonlocality, at material interfaces. Nonlocal interface conditions, whose local limit implies the classical interface conditions of elasticity, are then developed and discussed. A peridynamics material interface model is introduced which generalizes the classical interface model of elasticity. The model consists of a new peridynamics operator along with nonlocal interface conditions. The new peridynamics interface model converges to the classical interface model of linear elasticity.

A methodology is presented for bounding the higher L p norms, 2 ⩽ p ⩽ ∞ , of the local strain inside random media. We present optimal lower bounds that are given in terms of the applied loading and volume fractions for random two phase composites. These bounds provide a means to measure load transfer across length scales relating the excursions of the local fields to applied loads. These results deliver tight upper bounds on the macroscopic strength domains for statistically defined heterogeneous media.

A methodology is presented for investigating the dynamics of heterogeneous media using the nonlocal continuum model given by the peridynamic formulation. The approach presented here provides the ability to model the macroscopic dynamics while at the same time resolving the dynamics at the length scales of the microstructure. Central to the methodology is a novel two-scale evolution equation. The rescaled solution of this equation is shown to provide a strong approximation to the actual deformation inside the peridynamic material. The two scale evolution can be split into a microscopic component tracking the dynamics at the length scale of the heterogeneities and a macroscopic component tracking the volume averaged (homogenized) dynamics. The interplay between the microscopic and macroscopic dynamics is given by a coupled system of evolution equations. The equations show that the forces generated by the homogenized deformation inside the medium are related to the homogenized deformation through a history dependent constitutive relation.

A methodology is presented for bounding the higher $L^p$ norms, $2\leq p\leq\infty$, of the local stress inside random media. We present optimal lower bounds that are given in terms of the applied loading and volume fractions for random two phase composites. These bounds provide a means to measure load transfer across length scales relating the excursions of the local fields to applied loads. These results deliver tight upper bounds on the macroscopic strength domains for statistically defined heterogeneous media.

A method is presented for approximating the effective conductivity of composite media with thin interphase regions, which is exact to first order in the interphase thickness. The approximations are computationally efficient in the sense the fields need to be computed only in a reference composite in which the interphases have been replaced by perfect interfaces. The results apply whether any two phases of the composite are separated by a single interphase or multiple interphases, whether the conductivities of the composite phases are isotropic or anisotropic, and whether the thickness of an interphase is uniform or varies as a function of position. It is assumed that the conductivities of the interphase materials have intermediate values as opposed to very high or very low conductivities.

During the Arctic melt season, the sea ice surface undergoes a remarkable transformation from vast expanses of snow covered ice to complex mosaics of ice and melt ponds. Sea ice albedo, a key parameter in climate modeling, is determined by the complex evolution of melt pond configurations. In fact, ice–albedo feedback has played a major role in the recent declines of the summer Arctic sea ice pack. However, understanding melt pond evolution remains a significant challenge to improving climate projections. By analyzing area–perimeter data from hundreds of thousands of melt ponds, we find here an unexpected separation of scales, where pond fractal dimension D transitions from 1 to 2 around a critical length scale of 100 m2 in area. Pond complexity increases rapidly through the transition as smaller ponds coalesce to form large connected regions, and reaches a maximum for ponds larger than 1000 m2, whose boundaries resemble space-filling curves, with D ≈ 2. These universal features of Arctic melt pond evolution are similar to phase transitions in statistical physics. The results impact sea ice albedo, the transmitted radiation fields under melting sea ice, the heat balance of sea ice and the upper ocean, and biological productivity such as under ice phytoplankton blooms.