Specific Initial Conditions Research Articles

A semi-analytical method for 1D, 2D and 3D time fractional second order dual-phase-lag model of the heat transfer

In this paper we present a new numerical technique for 1D, 2D, and 3D time-fractional second order dual-phase-lag model of heat transfer. The problem is formulated as a solution of a time fractional partial differential equation (TFPDE) subjected to specific initial and boundary conditions. In the framework of the presented technique we first find an analytical function which satisfies the boundary condition of the original problem at each time moment. Then, using this function we transform the original problem into the problem with zero boundary conditions. This problem admits of the approximate solution in the form of the Fourier series expansion. Applying the Fourier expansion, the original TFPDE is reduced into a sequence of independent multi-term fractional ordinary differential equations (FODEs). These FODEs are solved separately, using the efficient backward substitution method and the Mu¨ntz polynomials in the approximate solution of each FODE. The accuracy and convergence of the proposed method are examined experimentally with several examples. In particular, the equations with coefficients that correspond to the real thermophysical parameters of the biological tissue have been considered. In the last example we demonstrate the ability of the method presented to solve 3D FPDEs with time-dependent coefficients.

Hybrid reaction-diffusion and clock-and-wavefront model for the arrest of oscillations in the somitogenesis segmentation clock.

The clock and wavefront paradigm is arguably the most widely accepted model for explaining the embryonic process of somitogenesis. According to this model, somitogenesis is based upon the interaction between a genetic oscillator, known as segmentation clock, and a differentiation wavefront, which provides the positional information indicating where each pair of somites is formed. Shortly after the clock and wavefront paradigm was introduced, Meinhardt presented a conceptually different mathematical model for morphogenesis in general, and somitogenesis in particular. Recently, Cotterell et al. [A local, self-organizing reaction-diffusion model can explain somite patterning in embryos, Cell Syst. 1, 257-269 (2015)] rediscovered an equivalent model by systematically enumerating and studying small networks performing segmentation. Cotterell et al. called it a progressive oscillatory reaction-diffusion (PORD) model. In the Meinhardt-PORD model, somitogenesis is driven by short-range interactions and the posterior movement of the front is a local, emergent phenomenon, which is not controlled by global positional information. With this model, it is possible to explain some experimental observations that are incompatible with the clock and wavefront model. However, the Meinhardt-PORD model has some important disadvantages of its own. Namely, it is quite sensitive to fluctuations and depends on very specific initial conditions (which are not biologically realistic). In this work, we propose an equivalent Meinhardt-PORD model and then amend it to couple it with a wavefront consisting of a receding morphogen gradient. By doing so, we get a hybrid model between the Meinhardt-PORD and the clock-and-wavefront ones, which overcomes most of the deficiencies of the two originating models.

On the Non-Smooth Solutions of 3D Navier-Stokes Equations for the Incompressible Fluid Flows

In the paper non-stationary 3D incompressible viscous fluid flow over the point, the infinite line, the plane, the rectangular prism and the octahedron are studied. The corresponding Navier-Stokes equations (NSE) with the appropriate initial-boundary conditions are considered. NSE is a very important equation and has various applications in Plasma Physics, Astrophysics, magma physics, geophysical fluids, biophysics, nanofluids, etc. NSE describes significant characteristics of different fluids. The exact solutions are obtained in a very few cases and especially in 2D. In the paper the novel exact non-smooth solutions blow-up in time are obtained for the specific pressure and initial conditions by means of the methods of mathematical physics (the main result). Besides, the solutions for the turbulent flows are given. Those solutions are new and are applied to solving of the problem of some substance transportation in the space by means of the turbulent flow. The profiles of the velocity and substance distribution are constructed by means of “Maple” for the different parameters. The results have applications to the description of atmospheric and ocean currents, nanosciences.

VO2 Carbon Nanotube Composite Memristor-Based Cellular Neural Network Pattern Formation

A cellular neural network (CNN) based on a VO2 carbon nanotube memristor is proposed in this paper. The device is modeled by SPICE at first, and then the cell dynamic characteristics based on the device are analyzed. It is pointed out that only when the cell is at the sharp edge of chaos can the cell be successfully awakened after the CNN is formed. In this paper, we give the example of a 5 × 5 CNN, set specific initial conditions and observe the formed pattern. Because the generated patterns are affected by the initial conditions, the cell power supply can be pre-programmed to obtain specific patterns, which can be applied to the future information processing system based on complex space–time patterns, especially in the field of computer vision.

Counter rotating terms and dipole–dipole interaction effects on the entanglement and population inversion of two qubits interacting with a two-mode field

Recalling that counter rotating terms (CRTs) in atom–field interaction are not always ignorable, particularly in the strong coupling regimes of atom–field systems, in this paper we are going to study the interaction between two qubits (two-level atoms) with a two-mode field in a high quality cavity by considering CRTs while the dipole–dipole (D–D) interaction is also taken into account. Indeed, by performing an appropriate dynamical Hamiltonian associated with the Jaynes–Cummings and anti-Jaynes–Cummings terms, as well as the D–D interaction for the system, we solve the corresponding time-dependent Schrödinger equation with specific initial conditions for atoms and field. In the continuation, concurrence, as an appropriate measure of the degree of entanglement of the two qubits, and atomic population inversion of the qubits, as the rate of energy exchange between atoms and field, are evaluated. Meanwhile, we investigate the influence of CRTs and D–D interaction (as well as other involved parameters in the model interaction) on the mentioned quantities. Our numerical results indicate that collapse-revival phenomenon can be observed in the absence of D–D interaction with or without rotating wave approximation, however, CRTs cause an increase or decrease in the number of collapse-revivals in a finite interval of time, depending on the chosen parameters. In fact, by entering the D–D interaction term, this phenomenon is destroyed and the time behaviour of the quantities experiences irregular oscillations. Noticing that the initial field is considered in the vacuum state, the occurrence of collapse-revival in the population inversion of the two qubits is an interesting feature of this work. As is shown, the degree of entanglement and atomic population inversion of the qubits can generally be tuned, and the death of entanglement can be prevented, both by adjusting the evolved parameters in the model.

Jacobi Stability Analysis and the Onset of Chaos in a Two-Degree-of-Freedom Mechanical System

In this paper, the Jacobi stability of a two-degree-of-freedom mechanical system is studied by the innovative application of KCC-theory, namely differential geometric methods. We discuss the Jacobi stability of two equilibria and a periodic orbit by constructing geometric invariants. Both the regions of Jacobi stability and Lyapunov stability are presented to show the difference. We draw the phase portraits of the deviation vector near two equilibria under specific parameter values and initial conditions, and point out the sensitivity of deviation vector to initial conditions. In addition, the corresponding instability exponent and curvature are applicable for predicting the onset of chaos, which help us to detect chaotic behaviors quantitatively.

Optimal anticipatory control as a theory of motor preparation: A thalamo-cortical circuit model

SummaryAcross a range of motor and cognitive tasks, cortical activity can be accurately described by low-dimensional dynamics unfolding from specific initial conditions on every trial. These “preparatory states” largely determine the subsequent evolution of both neural activity and behavior, and their importance raises questions regarding how they are, or ought to be, set. Here, we formulate motor preparation as optimal anticipatory control of future movements and show that the solution requires a form of internal feedback control of cortical circuit dynamics. In contrast to a simple feedforward strategy, feedback control enables fast movement preparation by selectively controlling the cortical state in the small subspace that matters for the upcoming movement. Feedback but not feedforward control explains the orthogonality between preparatory and movement activity observed in reaching monkeys. We propose a circuit model in which optimal preparatory control is implemented as a thalamo-cortical loop gated by the basal ganglia.

Asymmetric synchronization in lattices of pinned spiral waves.

Networks of coupled oscillators show a wealth of fascinating dynamics and are capable of storing and processing information. In biological and social networks, the coupling is often asymmetric. We use the chirality of rotating spiral waves to introduce this asymmetry in an excitable reaction-diffusion model. The individual vortices are pinned to unexcitable disks and arranged at a constant spacing L along straight lines or simple geometric patterns. In the case of periodic boundaries or pinning disks arranged along the edge of a closed shape, small L values lead to synchronization via repeated wave collisions. The rate of synchronization as a function of L shows a single maximum and is determined by the dispersion behavior of a continuous wave train traveling along the system boundary. For finite systems, spirals are affected by their upstream neighbor, and a single dominant spiral exists along each chain. Specific initial conditions can decouple neighboring vortices even for small L values. We also present a time-delay differential equation that reproduces the phase dynamics in periodic systems.

An Approach to Optimal Filtering of Time-Variant Systems via Finite Measurements

Fast optimal estimates are often required in control and signal processing. In this paper, we discuss an approach to optimal finite impulse response (OFIR) filtering for discrete time-variant systems using finite measurements. The mean square error is minimized to obtain the batch OFIR algorithm which requires measurements on an a finite horizon of N points. Fast iterative algorithm is found using recursions. It is shown that each recursion has a predictor/corrector Kalman filter (KF)-like format with special initial conditions. In this sense, the KF is considered as a special case of the proposed iterative OFIR filtering algorithm when N approaches infinity for known initial conditions. It has been confirmed by simulation that the iterative form of the OFIR filter operates much faster than the batch form.

Physical Determinism, Zygote-Manipulation and Responsible Agency

Agents have no control over the formation of their own zygote. Others may do. According to a well-known argument, the so-called Zygote Argument for incompatibilism, these facts, together with a prima facie plausible further assumption, are sufficient to prove that human agents cannot be responsible for their actions if they live in a deterministic universe. This paper argues that the lack of agents’ control over the constitution of their own zygote can undermine their responsibility only in exceptional conditions and that the occurrence or non-occurrence of those conditions has nothing to do with the truth or falsity of determinism. What undermines agents’ responsibility in the situations described by the Zygote Argument is the occurrence of some specific initial conditions which may render the manipulation of agents’ behaviour possible, and not the truth of determinism.

Numerical analysis of fractional human liver model in fuzzy environment

Many papers have shown that fractional derivatives are preferable to other operators when the data or information is exact, but this is not the case in practice because we live in an uncertain environment. Fuzzy operators are the best option for modelling in this situation. In this paper, we use the fuzzy fractional Caputo's derivative to generalize the fractional-order human liver model. We consider both types of H-differentiability (type 1 and type 2). We establish a general procedure of solution under the concept of H-differentiability through fuzzy Laplace transform. We implement the proposed scheme to derive the numerical results of the model. We present the archived theoretical solution via two- and three-dimensional graphs at different values of fractional orders and specific fuzzy triangular initial conditions. We present the evolution of the proposed model for some values of to see the effect of uncertainty on the secretion of Bromsulphthalein in the blood and liver.

Eigenrays in 3D heterogeneous anisotropic media, Part II: Dynamics

ABSTRACTThis paper is the second in a sequel of two papers and dedicated to the computation of paraxial rays and dynamic characteristics along the stationary rays obtained in the first paper. We start by formulating the linear, second‐order, Jacobi dynamic ray tracing equation. We then apply a similar finite‐element solver, as used for the kinematic ray tracing, to compute the dynamic characteristics between the source and any point along the ray. The dynamic characteristics in our study include the relative geometric spreading and the phase correction due to caustics (i.e. the amplitude and the phase of the asymptotic form of the Green's function for waves propagating in 3D heterogeneous general anisotropic elastic media). The basic solution of the Jacobi equation is a shift vector of a paraxial ray in the plane normal to the ray direction at each point along the central ray. A general paraxial ray is defined by a linear combination of up to four basic vector solutions, each corresponds to specific initial conditions related to the ray coordinates at the source. We define the four basic solutions with two pairs of initial condition sets: point–source and plane‐wave. For the proposed point–source ray coordinates and initial conditions, we derive the ray Jacobian and relate it to the relative geometric spreading for general anisotropy. Finally, we introduce a new dynamic parameter, similar to the endpoint complexity factor, presented in the first paper, used to define the measure of complexity of the propagated wave/ray phenomena. The new weighted propagation complexity accounts for the normalized relative geometric spreading not only at the receiver point, but along the whole stationary ray path. We propose a criterion based on this parameter as a qualifying factor associated with the given ray solution. To demonstrate the implementation of the proposed method, we use several isotropic and anisotropic benchmark models. For all the examples, we first compute the stationary ray paths, and then compute the geometric spreading and analyse these trajectories for possible caustics. Our primary aim is to emphasize the advantages, transparency and simplicity of the proposed approach.

Simulation of Hydrogen Fuel Combustion in Neon-oxygen Circulated Compression Ignition Engine

The elimination of NOx emissions for hydrogen combustion in the compression ignition (CI) engine is the primary concern, therefore replacing nitrogen with noble gas is one of the solutions to the eliminated the NOx emission. Current research trend focuses on oxygen-argon as the most suitable working. When using argon, external heating for intake is required for low compression ratio (CR) engines. Hence, neon is another potential solution because it has a high specific heat ratio that is as high as other noble gasses with the specific heat capacity, Cp, which is almost identical to nitrogen. This advantage pointed to neon as the best nitrogen replacement option. This paper aims to study neon-oxygen as the working gas for stabilization of the hydrogen ignition with the standard ambient intake condition. In addition, the optimum intake temperature and suitable CR was also determined from the analysis of the combustion properties. A computational analysis was performed using Converge CFD software with specific initial temperature and CR conditions base on Yanmar NF19SK engine. The study showed that the mean initial hydrogen temperature in the neon-oxygen atmosphere was lower than that of oxygen-argon. However, the initial minimum temperature needed for compression ratio 10:1 is 310 K, with a slightly unstable ignition and potential of knock. Hence, the most suitable initial temperature is at 340K. For higher CR, external heating on intake gas is no longer required; hence the engine output was increased. Neon-oxygen is available in CI engine hydrogen combustion at a higher compression ratio with no modification. Analysis of the injection parameters and control of heat loss in the neon-oxygen atmosphere is needed for hydrogen combustion strategies for this type of engine.

MASS TRANSFER PROCESSES DURING CORROSION OF CEMENT CONCRETE OF ROAD PAVEMENT AND AIRFIELD COVERINGS MODELING

Varieties of corrosion processes under the action of various corrosive environments on the cement concretes for pavement and airfield coatings within the classification by V.M. Moskvin are considered and their distinctive features are described. It is noted in the work that the rate of destruction of cement concretes is determined by the processes of mass transfer and chemical reactions. On this basis, we provide the substantiation of a generalized methodological approach to modeling the processes of mass transfer during liquid corrosion of building materials. For corrosion types I and II, analytical solutions are obtained for specific initial and boundary conditions of the problem.
 Development of mathematical models of concrete corrosion processes is based on the physical models of diffusion of components that are imbedded in the porous structure of concrete and the mathematical apparatus of boundary problems of mass transfer using differential equations in partial derivatives of parabolic type. The works, to which the authors refer, provide the results of the development of mathematical models of the processes of corrosion of concrete of the first type, as well as the results of the practical application of these mathematical models. However, the results obtained cannot withstand scrutiny in terms of methods and approaches of classical mathematical physics.
 In this study, these errors, inaccuracies, inconsistencies and incorrectness of the abovementioned works are eliminated. At the same time, models of corrosion of concretes of the first and second types are taken into account. An undoubtedly positive aspect of the dependences obtained in the study is the possibility of solving the inverse problem, when the actual experimental data using this model make it possible to predict the numerical values of free calcium hydroxide, which, as a result, with a minimum error allows to predict the long-term durability of structures and buildings.

Analytical Model for Volatile Organic Compound Transport in the Coupled Vadose Zone-Groundwater System.

A 3D mathematical model that describes transport of volatile organic compounds in a coupled vadose-saturated zone system is proposed. The subsurface processes incorporated in the model include advection, dispersion, interphase mass transfer, and diffusive mass exchange between two horizontal porous media formations, as well as the time-dependent mass loading from a source zone. The analytical solutions are derived subject to the specific initial and boundary conditions. The solutions are evaluated by numerical Laplace inverse transform. The model solutions can be used to study the fate and transport in subsurface formations composed of a vadose zone and a water table aquifer, where the volatile organic compound is released from entrapped nonaqueous phase liquid in the vadose zone, or the dissolved volatile organic compound transports with groundwater accompanied by diffusive mass transfer into the overlying soil formations. Mass transfer between two layers is demonstrated to have back-diffusion characteristics, which results in secondary contamination and retains low levels of contaminant concentrations over a prolonged period of time. The model solutions are specifically useful in assessing the vapor intrusion process in a contaminated site where a vadose zone is underlain by a water table aquifer contaminated with volatile organic compounds.

Kinetic modelling of colonies of myxobacteria

A new kinetic model for the dynamics of myxobacteria colonies on flat surfaces is derived formally, and first analytical and numerical results are presented. The model is based on the assumption of hard binary collisions of two different types: alignment and reversal. We investigate two different versions: a) realistic rod-shaped bacteria and b) artificial circular shaped bacteria called Maxwellian myxos in reference to the similar simplification of the gas dynamics Boltzmann equation for Maxwellian molecules. The sum of the corresponding collision operators produces relaxation towards nematically aligned equilibria, i.e. two groups of bacteria polarized in opposite directions. For the spatially homogeneous model a global existence and uniqueness result is proved as well as exponential decay to equilibrium for special initial conditions and for Maxwellian myxos. Only partial results are available for the rod-shaped case. These results are illustrated by numerical simulations, and a formal discussion of the macroscopic limit is presented. Erratum: The month information has been corrected from January to February. We apologize for any inconvenience this may cause.

Dynamical chaos in a simple model of a knuckleball

The knuckleball is perhaps the most enigmatic pitch in baseball. Relying on the presence of raised seams on the surface of the ball to create asymmetric flow, a knuckleball’s trajectory has proven very challenging to predict compared to other baseball pitches, such as fastballs or curveballs. Previous experimental tracking of large numbers of knuckleballs has shown that they can move in essentially any direction relative to what would be expected from a drag-only trajectory. This has led to speculation that knuckleballs exhibit chaotic motion. Here we develop a relatively simple model of a knuckleball that includes quadratic drag and lift from asymmetric flow which is taken from experimental measurements of slowly rotating baseballs. Our models can indeed exhibit dynamical chaos as long as torque on the ball from the asymmetric flow is included. In contrast, models that omit torques on the ball in flight do not show chaotic behavior. Uncertainties in the phase space position of the knuckleball are shown to grow by factors as large as 106 over the flight of the ball from the pitcher to home plate. We quantify the impact of our model parameters on the chaos realized in our models, specifically showing that maximum Lyapunov exponent is roughly proportional to the square root of the effective lever arm of the torque, and also roughly proportional to the initial velocity of the pitch. We demonstrate the existence of bifurcations that can produce changes in the location of the ball when it reaches the plate of as much as 1.2 m for specific initial conditions similar to those used by professional knuckleball pitchers. As we introduce additional complexity in the form of more faithful representations of the empirical asymmetry force measurements, we find that a larger fraction of the possible initial conditions result in dynamical chaos. The chaotic behavior seen in our simplified model combined with the additional complex and nonlinear effects likely present in real knuckleballs provide strong evidence that knuckleballs are in fact chaotic, and that the chaotic uncertainty is likely a significant factor in the unpredictability of the pitch.

Application of Ateb and generalized trigonometric functions for nonlinear oscillators

In this paper, the new application of the generalized trigonometric function (GTF) and of the Ateb function in strong nonlinear dynamic systems is considered. It is found that the GTF and the Ateb function represent the closed-form solution of the purely nonlinear one-degree of freedom oscillator with specific initial conditions. Definition of the GTF and Ateb functions is introduced. In spite of the fact that both functions use the incomplete Beta function and its inverse form, the difference exists according to the definition of both of these functions. The correlation between these two types of functions is exposed. Main properties of the Ateb function and of the special GTF function with parameters $$a = \frac{1}{2}$$ and $$b = \frac{1}{\alpha +1}$$ , which are the solution of the pure nonlinear oscillator, are compared and the value of the functions are calculated. Special attention is directed toward the sine GTF and the cosine Ateb function. Advantages and disadvantages of the both type of solutions are discussed.

Pre-inflation dynamical behavior of warm inflation in loop quantum cosmology

Considering a constant dissipative coefficient [Formula: see text], the pre-inflation dynamical behaviors of warm inflation in the loop quantum cosmology scenario are discussed. We consider three sets of initial conditions. The evolution of the background can always be divided into three phases, namely super-inflation, damping, and slow-roll inflation phases, with the duration of each phase depending on the initial conditions. As an example, we compare the background evolution between [Formula: see text] and [Formula: see text] under special initial conditions and find that there is no slow-roll inflation phase for [Formula: see text] while the number of e-folds is about 60.209 for [Formula: see text].

On the Characteristic Polynomial of the Generalized k-Distance Tribonacci Sequences

In 2008, I. Włoch introduced a new generalization of Pell numbers. She used special initial conditions so that this sequence describes the total number of special families of subsets of the set of n integers. In this paper, we prove some results about the roots of the characteristic polynomial of this sequence, but we will consider general initial conditions. Since there are currently several types of generalizations of the Pell sequence, it is very difficult for anyone to realize what type of sequence an author really means. Thus, we will call this sequence the generalized k-distance Tribonacci sequence (Tn(k))n≥0.