Interdisciplinary Topics in Applied Mathematics, Modeling and Computational Science

In this chapter we obtain exact solutions of the Joseph-Egri equation with power law nonlinearity, which arises in various problems in many scientific applications. The Lie group analysis and simplest equation method are used to carry out the integration of this equation. The solutions obtained are travelling wave solutions. Moreover, the conservation laws for the Joseph-Egri equation with power law nonlinearity are constructed by using the multiplier method.

In this chapter, we consider the deconvolution modified Leray alpha (ML-α-deconvolution) model with fractional filter acting only in one variable

$\mathbb{A}_{3,\theta} = I + \alpha_3^{2\theta} {(-\partial_{3})^{2\theta}},$

where

$0 \le \theta \le 1$

controls the degree of smoothing in the filter. We study the global existence and uniqueness of solutions to the vertical ML-α-deconvolution model on a bounded product domain of the type

$D=\Omega \times (-\pi,\pi)$

, where

$\Omega$

is a smooth domain with homogeneous Dirichlet boundary conditions on the boundary

$\partial \Omega \times (-\pi,\pi)$

, and with periodic boundary conditions in the vertical variable. To present the model, we define the vertical

N

th Van Cittert deconvolution operator by

$D_{N,\theta} = \sum_{i = 0}^N ( I - \mathbb{A}_{3,\theta}^{-1})^i.$

The vertical ML-α-deconvolution model is then defined by replacing the nonlinear term in the Navier–Stokes equations

$({v} \cdot \nabla){v}$

by

$({v} \cdot \nabla)D_{N,\theta}(\overline{v})$

where

v

is the velocity and

$\overline{v}=\mathbb{A}_{3,\theta}^{-1}(v)$

is the smoothed velocity. We adapt the ideas from (H. Ali, Approximate Deconvolution Model in a bounded domain with a vertical regularization. J Math Anal Appl

408

, 355–363 (2013)) to prove that the vertical ML-α-deconvolution model which is derived by using

$\mathbb{A}_{3, \theta}$

, has a unique weak solution for any

$\theta> \frac{1}{2}$

.

In this chapter, we propose a modification of the Vogel Approximation Method (VAM) used to obtain near optimal solutions to linear transportation problems. This method, called Modified Vogel Method (MVM), consists of performing the row and column reduction of the cost matrix and then applying the classical Vogel method to the equivalent transportation problem with the reduced cost matrix. We prove that when no further reduction of a cost matrix is required, we do obtain an optimal solution, not an approximate one. We identify some cases when such a behavior occurs and provides rules that allow for fast new reductions and penalty calculations when needed. The method also allows us to make multiple assignments of variables. Numerical tests run on small tests show that the MVM over performs the original one in all instances while requiring comparable computing times. The tests also support the intuition that the new method provides optimal solutions almost all the time, making it a viable alternative to the classical transportation simplex.

This chapter deals with large-scale nonlinear delay stochastic systems where the system states are subject to impulsive effects and perturbed by some disturbance input having bounded energy. The interest is to develop a comparison principle and establish input-to-state stability (ISS) in the mean square (m.s.) using vector Lyapunov function and Razumikhin technique. Impulses are being viewed as perturbation to stable systems, and they have a stabilizing role to unstable systems.

We create pure strategy versions of Robert Axelrod’s well-known norms and metanorms games. Our findings show that the only evolutionarily stable strategy (ESS) in the norms game is one in which a player defects and is lenient. This result is derived using classic game theoretical tools, and we conclude that Axelrod’s original statement that the norms game always collapses holds. The metanorms game, however, has two evolutionarily stable strategies. The first is a repeat from the norms game, while the other is one in which a player follows the norm and punishes those who are lenient and those who defect.

This chapter presents an efficient computation for least squares conditioning or estimates of it. We propose performance results using new routines on top of the multicore-GPU library MAGMA. This set of routines is based on an efficient computation of the variance–covariance matrix for which, to our knowledge, there is no implementation in current public domain libraries LAPACK and ScaLAPACK.

In this work we explore the use of spin torque nano-oscillators (STNOs) to produce a spintronics voltage oscillator in the microwave range. STNOs are quite small—on the order of 100 nm—and frequency agile. However, experimental results till date have produced power outputs that are too small for practical use. We attempt to increase power output by investigating the dynamics of a system of electrically-coupled STNOs. Transverse Lyapunov exponents are used to quantitatively measure the local stability of synchronized limit cycles. The synchronized solution is found to be stable for a large region of two-parameter space. However, a two-parameter bifurcation diagram reveals a competing out-of-phase solution, causing bistability.

Motivated by topics and issues critical to human health, and safety and efficacy of medical treatment practices, this communication investigates a nonlinear robust control approach of some uncertain biomedical nonlinear complex systems. The concept consists in setting the problem in the worst-case disturbances, which leads to the game theory in which controls and disturbances (which destabilize the dynamical behavior of the system) play antagonistic roles. The proposed strategy consists in controlling these instabilities by acting on certain parameters and data to maintain the system in a desired state (see

Stabilization, Optimal and Robust Control

, Springer, London (2008)). This approach is applied to two problems: first, controlling and regulating the blood glucose level in subjects with type 1 diabetes and predicting the dosages of insulin administered, and second, controlling and stabilizing the thermal distribution and damage during the treatment of cancer, in order to eradicate tumor while preserving the surrounding health tissues.

In this contribution, the model of heat and water transport in frozen porous media and fractured rock masses in conditions of freezing and thawing is analyzed. We present results concerning the existence of the numerical solution. Numerical scheme is based on semi-implicit discretization in time. The spacial discretization is carried out by the finite element method (FEM) and it is implemented in MATLAB. We also present an illustrative numerical example.

In this chapter we study a set-valued nonlinear Fredholm integral inclusion. We prove the existence of a solution and provide a numerical method based on the Steiner selection and Schauder bases to determine an approximated solution. We then discuss an inverse problem. Numerical results are also provided to show how the method works practically.

In this chapter a question of “how much over-consumption a renewable resource can tolerate” is addressed using a mathematical model, where a consumer population competes for the common resource, can contribute to resource restoration, and is subject to attacks of predators. The bifurcation analysis of the system shows that well-adapted predators can keep the system in a stable equilibrium even for “strong” prey over–consumption, when the initial system of resource–consumer goes to extinct. Thus, predators may extend the domain of total model system coexistence in niche.

A study on the influence of change in poling direction is carried for mechanical and electric strip yield model for a transversely isotropic piezoelectric plate cut along two equal collinear semipermeable cracks. Solution is obtained using Stroh formalism and complex variable technique. An illustrative numerical example is considered for a poled PZT-5H ceramic plate to show the effect of change in poling direction on energy release rate (ERR).

Using Fourier cosine integral transform technique, mode-III strip-induction-saturation model is proposed for a cracked transversely isotropic piezoelectromagnetic strip. Strip edges are subjected to combined anti-plane mechanical and in-plane electromagnetic loadings. Analytical closed-form expressions are derived for developed zones, field intensity factors, and energy release rates. Four impermeable/permeable electromagnetic crack-faces boundary conditions are considered. Results are plotted for

BaTiO

3

–

CoFe

2

O

4

ceramic to show the influence of electromagnetic fields on local energy release rate (LERR) and global energy release rate(GERR).

An adaptive parallel matrix transpose algorithm optimized for distributed multicore architectures running in a hybrid OpenMP/MPI configuration is presented. Significant boosts in speed are observed relative to the distributed transpose used in the state-of-the-art adaptive FFTW library. In some cases, a hybrid configuration allows one to reduce communication costs by reducing the number of message passing interface (MPI) nodes, and thereby increasing message sizes. This also allows for a more slab-like than pencil-like domain decomposition for multidimensional fast Fourier transforms (FFT), reducing the cost of, or even eliminating the need for, a second distributed transpose. Nonblocking all-to-all transfers enable user computation and communication to be overlapped.

Air Health Indicator (AHI) is a joint Health Canada/Environment Canada initiative. A component in the indicator is an estimate of the time-dependent population health risk due to short-term (acute) effects of air pollution. The standard approach for this risk estimation uses a generalized additive model (GAM) framework, which includes one or more air pollutants and one or more temperature terms as covariates, as well as a smooth function of time. In this risk-modeling framework, the temperature is not the primary focus, but is included to ensure that common structure between the mortality (response), the pollutant(s), and the temperature is not included in the risk attribution.

We examine the smooth function link that is commonly used when including temperature. We show that for a single lag of temperature, the traditional

J

-,

U

-, or

V

-shaped relationship between temperature and mortality is largely a function of low-frequency mortality structure and is thus accounted for by the smooth function of time typically included in risk models. We further compare and contrast the first two primary lags of temperature in the context of these findings, and demonstrate differences in their structure, advocating the inclusion of only the first (lag-0) parametric temperature series in the model.

Air Health Indicator (AHI) is a joint Health Canada/Environment Canada initiative that seeks to model the Canadian national population health risk due to short-term (acute) effects of air pollution. The commonly accepted model in the field uses cubic spline-based temporal smoothers embedded in generalized additive models (GAMs) to account for seasonal and long-term variations in the response. From a spectral point of view, it is natural to think of these smooth, long-term variations as low-frequency components, and the temporal smoother as a linear filter.

Examining the frequency response of the filters typically used, we show that the performance leaves much to be desired. Adapting the discrete prolate spheroidal sequences as filters, taking inspiration from their similar use in the multitaper method, we are able to significantly improve the frequency response of the smoother. We conclude with a discussion of the implications for controlling bias from the long timescale structure of parametric covariates, and suggest a prefiltering stage to such models.

In this chapter, we study three asset pricing models for valuing financial derivatives; namely, the constant elasticity of variance (CEV) model, the Bessel-K model, derived from the squared Bessel (SQB) process, and the unbounded Ornstein–Uhlenbeck (UOU) model, derived from the standard OU process. All three models are diffusion processes with linear drift and nonlinear diffusion coefficient functions. Specifically, the Bessel-K and UOU models are constructed based on a so-called diffusion canonical transformation methodology (Campolieti and Makarov, Int J Theor Appl Financ 10:1–38, 2007; Solvable Nonlinear Volatility Diffusion Models with Affine Drift, 2009; Math Finance 22:488–518, 2012). The models are calibrated to market prices of European options on the S&P500 index. It follows from the calibration analysis that the Bessel-K, UOU, and CEV models provide the best fit for pricing options that mature in 1 month, 3 months, and 1 year, respectively. The UOU model captures option data with a pronounced smile and hence it can be better calibrated to option data with short maturities. The CEV model provides a skewed local volatility and hence it works best for options with longer maturities. Moreover, we demonstrate that the CEV model is reasonably consistent through recalibration analysis on time series data in comparison with the Black–Scholes implied volatility.

This chapter considers the problem of borrowing and lending federal funds by a bank. The goal of the bank is to find the optimal borrowing/lending transaction policy while maintaining the reserve requirements. Within the model of [3] and [6] we describe the optimal net transaction amount using the Skorokhod formula developed in [8]. This formula provides a fast way of computing the optimal net transaction amount.

Speech communication devices and digital hearing aids must perform in the presence of high levels of ambient noise. Speech enhancement is a denoising process where Wiener-like filters are developed that require the estimation of the background noise spectrum from an additive combination of speech and noise. To follow statistical variations over time, the processes must be performed over short and overlapping frames of data resulting in time varying filters and spectra. We propose a novel algorithm to track the noise power of each frequency bin as it evolves over time. The proposed method uses an adaptation of the multitaper autoregressive spectral estimate. The resulting spectral components are smooth, low bias, and low variance and show superior tracking of the time-variation of the spectra.

This study focuses on the design and optimization of an electronics enclosure intended for operation in an outdoor commercial heating, ventilation, and air conditioning, HVAC, application. In particular the design was optimized for a high ambient environment without the aid of forced air cooling. As electronically controlled motor drive systems are increasing in use, designs need to operate in new challenging environments, reach higher power density, and enable higher levels of system integration. Computational fluid dynamics, CFD, was used for design, analysis, and optimization and correlated with test data. A design of experiments, DOE, was used to evaluate the sensitivity of the final design to the operating environment. A constrained optimization was performed to determine the optimal fin spacing, height, angle, and thickness of the enclosure geometry for thermal dissipation of the heat from the power electronics. Various fin topologies were also analyzed to evaluate the impact of increased surface area and enhanced thermal mixing effects. After a thorough review of the design space, general design recommendations are made and an optimized design reviewed.

This study focuses on the design and optimization of an electronics enclosure intended for use in a centrifugal fan being driven by an electric motor for the residential heating, ventilating, and air conditioning(HVAC) market. Typically in these systems the motor is mounted directly in the airstream of the centrifugal fan, but in this case the Regal Beloit’s axial motor technology allows for the minimization of this obstruction to the airflow. In the system analyzed the axial motor is mounted in the center of the centrifugal fan and the electronics used to drive the system is enclosed and mounted to the axial motor. This enclosure has been optimized for system airflow efficiency and thermal management of the electronics. A sensitivity analysis was also performed to understand the optimized design’s performance under various application environments. Computational fluid dynamics (CFD) was used as a test platform and tool for optimization. The CFD analysis was driven by goal optimization software to explore the design space and lead to an optimized design for overall efficiency. Results were validated to test data and test visualization methods. This presentation will cover the design requirements and details of the application, the optimization and CFD techniques used, and the criteria used for CFD model validation.

We present here an agent-based model (ABM) of adoption of new products including: dynamic consumer preferences and product demands, a 2D characteristics space where products are placed, global and local (nearest neighbours) social influence. The ABM model is built from a continuous time model of the market (Cojocaru et al., Environ Model Softw, 2013), driven by agents’ heterogeneity and their local connections. We find that consumer populations where a large fraction of population is sensitive to product popularity displays higher adoption levels of a new product, especially when local social connections are taken into account.

In this work, we study a modified Novikov equation using group methods. A complete group classification is carried out. Then from the point symmetry generators, we find the one-parameter group of local diffeomorfisms which preserves the equation. From the Lie symmetry generators, we also obtain exact solutions to the considered equation. It is also proved that only one nontrivial conservation law can be established using Ibragimov’s recent developments.

This presents an application of stochastic resonance in a data-driven nonlinear bistable system, in which inhibitory and excitatory electrophysiological neuronal activity in the prefrontal cortex (PFC) is quantified in a control and a putative rodent model of schizophrenia brains. An empirical mode decomposition protocol was applied for processing and analyzing the spike data. Within the different experimental conditions, we extracted different asymmetric shapes of bistable model potentials using the Fokker–Planck equation (FPE). Our analyses in control brains suggest that neuronal firing, along with noise (e.g., synaptic activity) before and after amphetamine administration provide asymmetries with phase transition in the bistable model allowing bidirectional information flow. Such transitions appear to be impaired in the disease model.

In this chapter, we generalize our earlier model for the spread of ectoparasites and diseases transmitted by them by including disease-induced mortality. The qualitative behavior of the system is similar to that of the original model: three reproduction numbers determine which of the four possible equilibria is globally asymptotically stable. We conclude that a moderate mortality decreases the size of the population, while a high mortality leads to the eradication of the infection. The main tools used for the proofs include persistence theory, Lyapunov–LaSalle theory and Dulac’s criteria.

Shape memory alloys (SMAs) exhibit hysteresis behaviors upon stress- and temperature-induced loadings. In this chapter, we focus on numerical simulations of microstructure evolution of cubic-to-tetragonal martensitic phase transformations in SMAs in 3D settings under the dynamic loading conditions. A phase-field (PF) model has been developed to capture coupled dynamic thermo-mechanical behavior of such SMA structures and the system of governing equations have been solved numerically using the isogeometric analysis. Temperature induced reverse and forward transformations have been applied to a cubic SMA specimen, starting with evolved accommodated martensitic microstructure. We have observed that during the forward transformation, the martensitic variants nucleate abruptly. The transient microstructures are aligned along

$\left[ 110 \right]$

planes, which is in accordance with the crystallographic theory and experimental results.

Hydrocephalus is a serious neurological disorder characterized by abnormalities in the circulation of cerebrospinal fluid (CSF) within the brain. Unfortunately, the response of the patients who have been treated for hydrocephalus continues to be poor and thus better therapy protocols are desperately needed. Mathematical models of CSF dynamics and CSF–brain interactions could play important roles in the design of improved, patient-specific treatments. To capture some of brain’s dynamics during the evolution of hydrocephalus we propose a new mathematical model using Hamilton’s principle. We assume the existence of current healthy healing and abnormal inflammation states and investigate the relationship between these states using volumetric data of healthy and untreated hydrocephalic mice.

Business intelligence (BI) tools and techniques, when applied to a data store of bibliographical references, can provide a researcher with valuable information and metrics. In contrast to specialized research platforms that provide a number of analysis tools, such as the Web of Knowledge™ (WOK) or PubMed™, the techniques discussed in this chapter provide a more generalized approach that can be used with most bibliographical data sets as well as with a number of different analysis tools. As a point of reference, the system utilizes the WOK’s Web of Science (WOS) database schema, chosen because it provides a comprehensive number of bibliographical information fields. This chapter will discuss how to transform WOK formatted data into an online analytical processing (OLAP) cube as well as provide a few examples of using this technology to analyze bibliographical information.

Past few years have witnessed a considerable level of research activity in the field of exceptional orthogonal polynomials (EOPs), which are new complete orthogonal polynomial systems, and these are first observed as a result of the development of a direct approach to exact or quasi-exact solvability for spectral problems in quantum mechanics that would go beyond the classical Lie algebraic formulations. We have discovered new EOP families associated to such kind of above systems in the framework of supersymmetric quantum mechanics. We have studied thoroughly some fundamental properties of those EOP families. We also have been able to prove completeness of few such EOP categories in weighted Hilbert space, associated with solutions of certain conditionally exactly solvable potentials obtained via unbroken as well as broken supersymmetry. Some important key properties of such polynomials, e.g, recurrence relation, Rodrigues formula, ladder operators, differential equations, etc., have been obtained.

A new numerical method is developed for the solution of the Dirac equation for 3D axisymmetric geometries using cylindrical coordinates. It is based on a split-step scheme in coordinate space, which can be parallelized very efficiently. A new technique to circumvent the coordinate singularity at

r

 = 0 using Poisson’s integral solution of the wave equation for the radial operator is used. The general strategy is to interpolate the solution using cubic Hermite polynomials and to integrate exactly the Poisson solution. The result of this procedure gives a nonstandard finite difference scheme on a time staggered grid. The numerical method is then utilized to evaluate the ground state of an electron bound in a Coulomb potential.

In this chapter, we study a generalized Fisher equation based on the theory of symmetry reductions in partial differential equations. Optimal systems and reduced equations are obtained. We derive some travelling wave solutions by applying the

(G'/G)

-expansion method to one of these reduced equation.

The aim of this work is to perform some numerical experiments for the resolution of a strongly coupled parabolic–elliptic system that describes the heating induction–conduction industrial process of a steel workpiece, whose unknowns are the electric potential, the magnetic vector potential, and the temperature. In order to make the numerical simulations lighter, and taking into account the different time scales between the potentials and the temperature, a new system of nonlinear partial differential equations (PDEs) has been constructed that describes the heating process in the harmonic regime.

The markets in which manufacturers and service firms compete are increasingly influenced by intense foreign competition, rapid technological change, and shorter product life cycles. In this new scenario, flexibility may be one of the most important capabilities needed for firms to achieve competitive advantage. The possible behaviors of the companyto the problems it faces are called levers of flexibilities. In a supply chain, the flexibility of one entity is highly dependent on the flexibility of upstream entities. It is a natural area for metrics. A metric is a standard of measurement of performance and gives the basis on which to evaluate the performance of processes in the supply chain. Thus, the purpose of the study is to determine and evaluate the supply chain flexibility levers in order to calculate the benefit of preferring a flexibility lever to another one. The analytic network process (ANP) technique is used for prioritizing evaluated flexibility levers. We are handling the automotive sector for the study.

In the face of increasing extinction rates, it is vital to have estimates of relative and absolute species abundance and their relationship to important factors. For species that live in the oceans or large lakes, this can be a difficult task. Here, we present a method for estimating absolute abundance from a single binary acoustic time series. The dependence in the series is exploited to allow the estimation of abundance when some animals remain hidden, and in the face of uncertainty about the range over which sounds carry. Simulations show that the method works well, even when some assumptions are violated. The method is illustrated using data on sperm whales in the Sargasso Sea.

The increasing usage of mobile electrical units demands higher energy efficiencies for these devices. Self-sustaining units that harvest various forms of ambient energy can help significantly with their regular battery replacements. In this chapter two hybrid energy harvesting units are proposed that employ piezoelectric, magnetostrictive, and electromagnetic technologies to capture ambient vibrational energy. The first harvester is made of piezoelectric and magnetostrictive materials while the second harvester is composed of a piezoelectric layer and a magnet. Both proposed harvesters employ a spiral piezoelectric layer in order to reduce the compliance of the piezoelectric unit. The advantages of the first design is that it allows for more efficient harvesting over a wider range of frequencies than traditional harvesting units while the second design reduces the natural frequency of the system that results in better energy harvesting at low frequencies.

In this chapter, we update a mathematical model based on the Lotka-Volterra predator–prey model to describe the elk–coyote–wolf interactions after the reintroduction of wolf in Yellowstone in 1995. A Markov Chain Monte Carlo algorithm is applied to calibrate the model parameters based on data compiled since wolves were released in the park. Our model predictions match the published experimental data very well. The objective of this study is to predict the impact of wolf reintroduction into the Yellowstone National Park on elk and coyote population.

We describe a numerical technique for discovering forward invariant sets for discrete-time nonlinear dynamical systems. Given a region of interest in the state- space, our technique uses simulation traces originating at states within this region to construct candidate Lyapunov functions, which are in turn used to obtain candidate forward invariant sets. To vet a candidate invariant set, our technique samples a finite number of states from the set and tests them. We derive sufficient conditions on the sample density that formally guarantee that the candidate invariant set is indeed forward invariant. Finally, we present a numerical example illustrating the efficacy of the technique.

We have used molecular dynamics simulation to investigate hydrophilic–hydrophobic interfaces between calcium chloride (CaCl

2

) aqueous solutions and normal hexane. The results demonstrate the increasing impact of salt concentration on the liquid–liquid interfacial tension, hence, negative adsorption of CaCl

2

according to Gibbs adsorption isotherm. Moreover, we calculated the density profiles of hexane, water, and the counter ions. The results reveal an electrical double layer near the interface and the less affinity of calcium cations toward the interface than that of chloride anions. Orientation of water molecules at the studied concentrations may result in developing a positively charged interface and, consequently, accumulation of anions close to the charged interface. Our calculations show that the interfacial width decreases by increasing salt concentration. Therefore, consistent with the calculated interfacial tension (IFT) data, aqueous salt solutions are less miscible in normal hexane at higher salt concentrations.

This study presents experimental results of gaining the accuracy of 18.4 % when the pairwise comparisons method was used instead of the direct method for area estimation of random images. Random images were produced by deblurring the Gaussian blur applied to randomly generated polygons. Participants were asked to estimate the areas of five random images by using an online questionnaire. Images have been compared to a provided unit of measure and in pairs. Our intensive Internet searches could not find another Monte Carlo experimentation for 2D case conducted in the past.

This chapter shows the controllability of second order impulsive differential systems in Banach spaces. Sufficient conditions for the controllability are obtained by using the theory of strong continuous cosine families and the contraction mapping principle. Particularly, the compactness of the cosine family of operators is not needed in this chapter.

We present the results of the symmetric and one-sided smoothness-increasing accuracy-conserving (SIAC) filter applied to a discontinuous Galerkin (DG) approximation for two examples of nonlinear hyperbolic conservation laws. The traditional symmetric SIAC filter relies on having a translation invariant mesh, periodic boundary conditions, and linear equations. However, for practical applications that are modeled by nonlinear hyperbolic equations, this is not feasible. Instead we must concentrate on a filter that allows error reduction for nonuniform/unstructured meshes and nonperiodic boundary conditions for nonlinear hyperbolic equations. This proceedings is an introductory exploration into the feasibility of these requirements for efficient filtering of nonlinear equations.

Differential algebraic equations (DAEs) appear frequently in applications involving equation based modeling, from robotics to chemical engineering. A common way of making a DAE amenable to numerical solution is by reducing the index to obtain a corresponding ordinary differential equations (ODE) and using an ODE solution method. The signature matrix method developed by Pryce does not rely on an index reduction step and instead solves the DAE directly via Taylor series. The chapter draws comparisons between these two different approaches and shows the signature matrix method is in some sense equivalent to the dummy derivative index reduction method developed by Mattsson and Söderlind. The ideas are illustrated via a DAE from Campbell and Griepentrog that models a robot arm. The authors acknowledge G. Tan and N. Nedialkov at McMaster University, Hamilton, Canada for their support in this chapter and the talk that accompanied it at AMMCS-2013.

In this chapter we study a coupled system of nonlinear partial differential equations (PDEs), namely, the Klein–Gordon–Zakharov equations. The travelling wave hypothesis approach along with the simplest equation methods are utilized to obtain exact solutions of this system.

We study the physical and collision properties of the combined KdV–mKdV solitons given by the Gardner equation which possess solitary wave solution characterized by

sech

function. A collision of the two solitary waves produces 2-soliton solution. We make a physical form of the 2-soliton solution where the fast soliton moves with speed

c

1

and the slow soliton moves with speed

c

2

. In the collision described by the 2-soliton solution, the solitary waves preserve their shapes and speeds, but get a shift in position where the fast soliton overtakes the slow soliton if their speeds have same direction, and two solitons cross head-on if their speeds have opposite direction. For a collision there exist three different types of interactions which depend on the relative ratio

$c_1/c_2$

of speeds and the relative orientation of the two solitary waves.

In this chapter, we consider a generalized coupled Boussinesq system of KdV–KdV type, which belongs to the class of Boussinesq systems modeling two-way propagation of long waves of small amplitude on the surface of an ideal fluid. We obtain conservation laws for this system using Noether theorem. Since this system does not have a Lagrangian, we increase the order of the partial differential equations by using the transformations

$u={U_{x}}$

,

$v={V_{x}}$

and convert the Boussinesq system to a fourth-order system in

U

,

V

variables, which has a Lagrangian. Consequently, we find infinitely many nonlocal conserved quantities for our original Boussinesq system of KdV–KdV type.

In this chapter,

$(G'/G)$

-expansion method is employed to derive new exact solutions of a coupled Boussinesq equation. Three types of solutions are obtained, namely, hyperbolic function solutions, trigonometric function solutions and rational solutions. These solutions are travelling wave solutions.

In this chapter we briefly review recent advances in Error Control B-spline Gaussian Collocation software for the numerical solution of 1D parabolic partial differential equations (PDEs). BACOL and BACOLR, two packages of this type, developed over the last decade, have been shown to be efficient, reliable, and robust, especially for problems having solutions with sharp moving layers and for stringent tolerances. These packages use high order methods in time and space and feature

adaptive control of high order estimates of the temporal and spatial errors

. The spatial error estimates require the computation of a second collocation solution, which introduces a substantial computational overhead. In order to address this issue, a new software package, called BACOLI, has recently been developed (through a substantial modification of BACOL) in which the computation of the second collocation solution is replaced by the computation of a high order interpolant. Numerical results have shown that BACOLI computes spatial error estimates that are generally of comparable quality to those computed by BACOL and that the new code is generally substantially more efficient than BACOL.

Better methods are needed to statistically downscale climate variability to agricultural ecosystem impact scales and to reduce uncertainty in regional climate model (RCM) predictions. We present a multivariate, multisite model for downscaling climate to the 10 km scale for agricultural areas across Canada. Scenario data was obtained from NARCCAP (North American Regional Climate Change Assessment Program). This method employs variable-selection for a multivariate set of regional climate model predictors, and may offer a rapid (automated) and reliable (cross-validated) way to generate high-resolution climate surfaces for use in agricultural decision-making. We provide selected results that show the model can significantly reduce bias in mean precipitation.

We develop constructive methods for solving periodic boundary value problems (PBVPs) associated with a nonlinear first order scalar differential equation in a unified setting. The method of generalized quasilinearization which we employ yields rapid convergence of monotone iterates to the solution of the PBVP. The monotone iterates in our approach are solutions of linear initial value problems (IVPs) as opposed to the linear PBVPs which appear in conventional methods. We provide graphical and numerical illustrations of our results.

In this chapter we derive fast, recursive, and numerically stable radix-2 algorithms for discrete sine transformations (DST) having sparse and orthogonal factors. These real radix-2 stable algorithms are completely recursive, fast, and based on the simple orthogonal factors. Comparing to the known bulky and mostly unstable DST algorithms, our algorithms are easy to implement and use only permutations, scaling by constants, butterfly operations, and plane rotations/rotation-reflections.

For a given vector

$\textbf{x}$

, we also analyze error bounds of computing

$\textbf{y}=S\textbf{x}$

for the presented DST algorithms:

S

. Finally a classification of these real radix-2 DST algorithms enables us to establish the excellent forward and backward stability based on the sparse and orthogonal factors.

We consider essentially nonlinear autonomous and nonautonomous dynamic systems described by ordinary differential equations. In such systems, for the same parameters of the system and forcing, different stable and unstable periodic solutions of different periods can exist. In addition, along with the ordered movements, the existence of a strange attractor is known. In such circumstances, the search for periodic solutions and their stability analysis is not a trivial problem. In order to find periodic solutions of the dynamical systems, we offer an interactive computer algorithm based on finding the initial conditions corresponding to the periodic solutions with the possibility of interactive intervention and operational control of the computing process. We demonstrate the algorithm and various numerical examples of finding new and complex stable and unstable periodic solutions in strongly nonlinear dynamical systems with one and two degrees of freedom. We also consider the mutual influence of oscillations in multidimensional nonlinear dynamic systems.

We consider the simulation of the Black–Derman–Toy (BDT) model with log-normally distributed rates in the spot measure, in discrete time and with a continuous state variable. We note an explosive behavior in the Eurodollar futures convexity adjustment at a critical value of the volatility, which depends on maturity, rate tenor, and simulation time step size. In the limit of a very small time step, this singularity appears for any volatility, and reproduces the Hogan–Weintraub singularity, which is generic for short-rate interest rate models with lognormally distributed rates. The singular behavior arises from a region in the state space which is usually truncated off in finite difference and tree methods, or is very poorly sampled in Monte Carlo methods, and thus is not observed under usual simulation methods.

The authors have written two codes to solve differential algebraic equations (DAEs) by structural analysis (SA). The first is written in C++ (

Daets

) and deals with the solution of DAE initial value problems, using SA. Upon seeing how informative the SA could be the authors wrote

Daesa

(in Matlab) to do only the structural analysis. These codes rely on exploiting the block triagular form (BTF) of a DAE, this chapter explains how.

A general methodology to analyze the solution of a many-objective optimization problem (MOOP) to landscape design is presented. The landscape design problem consists on assigning different types of land use to specific areas identified as

candidate sites

to be changed. Some local and global ecological criteria are considered. In order to gain some clarity during the analysis of the solutions that conform to an optimal set some clustering strategies are usually utilized. In this chapter, we use a simple strategy called favour ranking to place similar solutions together. Then, the solutions are visualized using a state-of-the-art technique.

Habitat fragmentation and loss is a key issue for land-use planning and environmental policy implementation. Greenlands Systems have been proposed as one solution to this issue. This chapter discusses aspects of implementation of an evolutionary multiobjective optimization (EMO) methodology for a Greenlands System design. The application of landscape ecology principles via EMO, combined with analysis of the Pareto front of nondominated solutions and the measure of favor of these solutions provides a methodology to address the deterioration of ecological function in urban fringe areas and insights into the steps that can be taken to promote sustainable peri-urban landscapes. The results of particular landscape metrics using a real-world data set of a small study area bring into relief issues concerning the interplay between the mathematization of landscape ecology principles of design and the resulting set of estimates of the Pareto optimal solutions.

The generalized dispersion model is used to study the dispersion process in unsteady flow in a tube with wall absorption by modeling the flowing fluid as Casson fluid. According to this model, the entire dispersion process is expressed in terms of three transport coefficients viz., the absorption, convection, and dispersion coefficients. This study brings out the effects of pulsatility, yield stress and wall absorption on these three transport coefficients. It is observed that the convection and the dispersion coefficients are dependent on absorption parameter, yield stress, pressure fluctuating component, and frequency parameter whereas the absorption coefficient depends only on wall absorption parameter. This study can be used to understand dispersion process in blood flows.

Cutaneous leishmaniasis (CL) represents a serious public health problem in Algeria. In the aim to understand the transmission dynamics of CL in the human–

Phlebotomus

papatasi

–rodent cycle, and to improve the preventive strategies set up in Algeria, we developed a deterministic model for the transmission dynamics of the disease. The model includes an incidental host for human which acts only as a sink of infection, a primary reservoir host for rodent which acts as a source and a sink of infection, and a secondary reservoir host for

P. papatasi

which have a role in transmission by acting as the liaison between incidental host and primary reservoir. The global stability of the equilibria of the proposed model shows that the threshold conditions for disease persistence are completely determined by the reproduction number and do not explicitly include parameters relating to the dynamic transmission in the incidental hosts, which means that the disease becomes endemic if it persists endemically in the primary reservoir hosts, and therefore the control measures should be directed towards reservoir hosts. This is illustrated via numerical simulations of the model using parameters generated from data from M’Sila province in Algeria.

Characteristics of scientific phenomenon are commonly investigated using mathematical tools in science and engineering to develop our conceptual understanding. However, computational thinking (CT) and modeling with simulations can result in a more advanced understanding of scientific concepts and offer an effective learning experience for students with various backgrounds. In this chapter, we show how a simulation tool, Scratch, can be used to unfold the abstract side of science through project-based visualizations in fun and engaging ways. It can be an effective approach in attracting young talented students to science and technology by motivating their natural imagination to probe scientific abstraction.

This work is devoted to computational modeling of stochastic processes of the electron multiplication in electron amplifiers in order to reduce the noise factor which is a measure of the loss of available information. The effects of the processes, arising when a layer with increased secondary emission yield is formed at the entrance of the channel, are investigated.

A computational method for simulation of stochastic processes of an electron multiplication in microchannel electron amplifiers is developed. It is based on 3D Monte Carlo (MC) simulations and theorems about serial and parallel amplification stages proposed by the author. Splitting a stochastic process into a number of different stages, enables a contribution of each stage to the entire process to be easily investigated. The method provides a high calculation accuracy with minimal cost of computations. The computational model easily implements new experimental data without any changes in the algorithm.

The problem of estimating parameters from time series data is considered. A parameter range reduction scheme is employed to quickly reduce a priori ranges of parameters. The effectiveness of the scheme is tested in the presence of partial data sets using an SIR model test case. The algorithm is shown to make substantial reductions of parameter ranges when limited time series data is available. Such reductions are shown to be of benefit to traditional parameter estimation techniques.

In this chapter, the stabilization of nonlinear impulsive systems under time-dependent switching control is investigated. In the open-loop approach, the switching rule is programmed in advance and the switched system is composed entirely of unstable subsystems. Sufficient conditions are found that establish the existence of stabilizing time-dependent switching rules using the Campbell–Baker–Hausdorff formula and Lyapunov stability theory.

Mathematics in Industry Study Groups (MISG) have been an annual event in Australia and New Zealand since 1984. Projects from the last decade are considered. Among the industries involved are those of steel, electricity and agriculture.

The gravitational N-body problem has long been a source of theoretical investigation with application to astronomical systems. There is a rich and varied dynamics. With systems of four bodies arranged symmetrically, the symmetry tends to reduce the complexity of the system so that it is perhaps more similar to one with three bodies, although such systems also provide a starting point for our understanding of more general four-body systems.

We present a simple method for quasilinearity (QL) analysis of differential-algebraic equations (DAEs). It uses the signature matrix and offsets computed by Pryce’s structural analysis and determines if a DAE is QL in its leading derivatives. Our method is suitable for an implementation through operator overloading or source code translation.

We consider that nondeterministic programs behave as badly as they can and loop forever whenever they have the possibility to do so. We deal with a relational algebra model to define a nondeterministic refinement fuzzy ordering (

nondeterministic fuzzy inclusion

) and also the associated fuzzy operations which are fuzzy nondeterministic join (

$\sqcup_{fuz}$

), fuzzy nondeterministic meet (

$\sqcap_{fuz}$

), and fuzzy nondeterministic composition (

${\square \mbox{\tiny}}_{fuz}$

). We also give some properties of these operations and illustrate them with simple examples.

Fretwell and Lucas [3] introduced the Ideal Free Distribution (IFD) to predict how birds establish themselves among habitats. It has been shown that the IFD is an evolutionarily stable strategy (ESS) of the habitat selection game when fitness is a decreasing function of patch density. We develop a formula for the IFD when there are an arbitrary number of habitats, and fitness functions are linearly decreasing in the population size (i.e., density) in each habitat. We also explore the IFD when fitness functions increase with population size until some maximum threshold is reached (Allee Effect) and examine whether an IFD still is an ESS in this case.

The disposal of waste of electrical and electronic equipments (WEEE) represents the loss of large amounts of valuable resources, in particular metals and plastics. If these were to be recycled, it would not only divert the waste from disposal by limiting waste flows damage but would also reduce the need to use virgin raw materials. In this study, we focus on waste management and we concentrate on the recycling of mobile phones. Mobile phone components and their requirements in production phases such as energy, labor, and know-how are depicted in a matrix form inspired from the seminal work of Leontief and input–output (I–O) methodology which appears to be appropriate for analyzing waste management problem is presented. Thus, we propose numerically static I–O solutions in order to demonstrate the contribution of recycling of mobile phones into the economy. Particularly, we concentrate on the monetary IO table (MIOT) and environmental IO table (EIOT) with recycling and balance equations. By defining waste outputs as a new vector class, the classical I–O model has been improved.

This chapter deals with a mathematical problem related to the equilibrium analysis of a membrane with rigid and cable boundary. The membrane and its boundary are respectively identified with a regular surface and a set of regular curves. The equilibrium is directly expressed by means of an elliptic problem, in terms of the shape of the membrane and its stress tensor; therefore, a free boundary numerical resolution procedure is presented and applied in a particular case.

Parameter estimation (inverse) problems are ubiquitous in many fields, including spatial econometrics. Global optimization can provide good parameter estimates for many such problems for which traditional, analytic estimation methods fail, or that are otherwise intractable. Stochastic global methods inspired by natural processes have recently gained popularity for difficult optimization problems characterized by imprecise measurements or local optima. In this chapter, one such approach, particle swarm optimization (PSO), is used to estimate parameters of the time series cross-sectional spatiotemporal autoregressive model, a particulary difficult and computationally intensive problem arising in spatial econometrics. Preliminary results are promising, and suggest that stochastic global approaches, and global optimization in general, can successfully address some of these intractable problems.

In this work, the Block Davidson and the residual minimization-direct inversion in the iterative subspace (RMM-DIIS) algorithms are used to diagonalize the Hamilton matrices arising from antiferromagnetic spin-

$\frac{1}{2}$

Heisenberg models. The results show that both algorithms find reliably the lowest eigenvalues but the computational costs are smaller for the RMM-DIIS method. In addition to this, the authors show that the new Intel Xeon Phi coprocessor can be used efficiently for this type of problems.

A compartmental model of the human circulatory system that illustrates Cheyne–Stokes respiration (CSR) is presented. Clinical evidence suggests that patient body position can influence the likelihood of experiencing CSR, and this model is analyzed to see if blood fluid shifts associated with body position could be the means of this influence. It is shown that lying down causes a shift in the location of the Hopf bifurcation curve associated with the onset of CSR, making it more likely.

We present a novel methodology to compute the transient thermal condition of a set of objects in an open space environment. The governing energy equation and the convective energy transfer are solved by the sparse iterative solvers. The average radiating energy on a set of surfaces is represented by a linear system of the radiosity equations, which is factorized by an out-of-core parallel Cholesky decomposition solver. The coupling and interplay of the direct radiosity solver using graphics processing units (GPUs) and the CPU-based sparse solver are handled by a light weight software integrator called Interoperable Executive Library (IEL). IEL manages the distribution of data and memory, coordinates communication among parallel processes, and also directs execution of the set of loosely coupled physics tasks as warranted by the thermal condition of the simulated object and its surrounding environment.

The human newborn is a reflection of the entirety of nutrients transferred from the maternal to the fetal circulation across the placenta during gestation. By extension, birth weight and newborn health depend on placental function. The goal of this chapter is to introduce the use of optimal transport modeling to study the expected effects of (i) placental size, (ii) placental shape (separate from size), and (iii) the position of insertion of the umbilical cord, on birth weight and placental functional efficiency. For each placenta (N = 1110), a total transport cost based on all measurements (i), (ii), and (iii) is given by the model. This computed cost is highly correlated with measured birth weight, placenta weight, the fetal–placental weight ratio (FPR), and the metabolic scaling factor beta. Next, a shape factor is calculated in a model of the total transport cost if each placenta were rescaled to have a unit area chorionic plate (thus separating shape from size). This shape factor is also highly correlated with birth weight, and after adjustment for placental weight, is highly correlated with the metabolic scaling factor beta.

In this chapter, we describe an image representation framework based on which a robust, nonparametric interpolation method for filling in “missing” information of an image can be performed. As in an earlier work, this approach utilizes a class of localized band-limited functions that are compact in both image and frequency domains. However, the current algorithm may be carried without a statistical classifier such as a K-means algorithm, which was employed in our previous work. After a brief description of our approach, results are given to show its efficacy in a few use cases.

The

Wall Street Journal

’s “Best on the Street,” StarMine and many other systems measure analyst stock-rating performance using variations on a method we term the “portfolio method,” whereby a synthetic portfolio is formed to track the analyst’s ratings. At the end of the evaluation period, analysts are compared by their respective portfolio returns. Of the pitfalls to this method, one most troubling is that the analysts are generally covering different sets of stocks over different time periods. Thus, each analyst has access to different opportunities and just comparing portfolio values is unfair. In response, we present a Monte Carlo (MC) method where, for each analyst, we generate numerous “pseudo-analysts” with the same coverage over the same time periods as the real analyst. Using this method, we are better able to compare analysts, adjusted for their individual opportunities. We draw comparisons between our results and the results from existing systems, showing that those systems are less precise in reflecting analyst performance.

The steady flow of an incompressible viscous fluid past an elliptic cylinder with minor-to-major axis ratio of 0.2 and at incidence to the free stream is considered. Numerical results for Reynolds number up to 450 and inclination angle varying from 0° to 20° are presented which permit completion of a bifurcation diagram describing the wake topology behind the cylinder in terms of three regions: Region I with no separation; Region II with a single recirculatory region attached to the cylinder; and Region III with two recirculatory regions, one attached and one unattached.

Chaos entanglement is a new approach to systematically generate chaotic dynamics by entangling two or multiple stable linear systems with periodic nonlinear coupling functions such that each of them evolves in a chaotic manner. In this study, chaos entanglement is extended to unstable linear systems by introducing a well-defined bound function to guarantee the boundedness of each unstable linear system. A novel 6-scroll attractor is obtained by entangling three identical unstable linear systems with sine function. It is verified that this attractor possesses a positive Lyapunov exponent and its trajectories are bounded. The Lyapunov spectra and bifurcation diagram reveal the chaotic behaviors of this new attractor.

This chapter investigates the impulsive control and synchronization problem of spatiotemporal chaos in the Gray–Scott model. Based on the Lyapunov function method, a class of pinning impulsive controller is designed to stabilize and synchronize the spatiotemporal chaos in the Gray–Scott model. The approach allows us to analyze the stability and synchronization problem of other spatiotemporal chaotic systems with the same structure. Numerical simulations are provided to illustrate the theoretical results.