Nonlinear Diffusion Model Research Articles

The Effect of Swabs on Modeling the First Wave of the COVID-19 Pandemic in Italy

The daily fluctuations in the released number of Covid-19 cases played a big role at the beginning of the pandemic, when local authorities in Italy had to decide whether imposing restrictive policies. When an increase/decrease was communicated, especially a large one, it was difficult to understand if it was due to a change in the epidemic evolution or if it was a fluctuation due to other reasons, such as an increase in the number of swabs or a delay in the swab processing. The aim of this paper is both to model the main trend of the outbreak evolution in the number of confirmed cases and to describe the daily fluctuations strongly dependent on the daily number of swabs. For our analysis, we propose a nonlinear asymmetric diffusion model, which includes information on the daily number of swabs, to describe daily fluctuations in the number of confirmed cases in addition to the main trend of the outbreak evolution. The proposed model is found to be the more efficient for prediction, as compared to 6 already existing models, including the SIRD and the logistic models. The new model combines the properties of innovation diffusion models with a parsimonious way to exploit information about swabs. Doi: 10.28991/esj-2021-SPER-04 Full Text: PDF

A mixture theory for the moisture transport in polyamide

Polyamide exhibits hygroscopic nature and can absorb up to 10% of moisture relative to its dry weight. The absorbed moisture increases the mobility of the molecular chains and causes a reduction in the glass transition temperature. Thus, depending on the moisture distribution, a polyamide component can show different stiffness and relaxation times. Moreover, the moisture distribution also depends on the mechanical loading of the material as the volumetric deformation results in a change of the available free volume for the moisture. Thus, a strongly coupled model is required to describe the material behaviour. In this work, a thermodynamically consistent coupled model within the framework of mixture theory is developed. The mechanical deformation of polyamide 6 (PA6) is based on a linear viscoelastic material model, and the moisture transport is based on a nonlinear diffusion model. The stiffness and the relaxation time of the viscoelastic model change with the moisture concentration. Furthermore, the moisture transport is affected by the pressure gradient generated by the mechanical loading of the material. This strongly coupled model has been implemented using the finite element method, and simulation results are presented for a three-point bending experiment.

Signal Smoothing with Time-Space Fractional Order Model

Abstract The time-space fractional-order model (TSFOM) is a generation of the classical diffusion model which is an excellent smoothing method. In this paper, the fractional-order derivative in the model is found to have good performance for peak-preserving. To check the validity and performance of the model, some noisy signals are smoothed by some commonly used smoothing methods and results are compared with those of the proposed model. The comparison result shows that the proposed method outperforms the classical nonlinear diffusion model and some commonly used smoothing methods.

Nonlinear parabolic predator–prey coupled system with convection

The aim of this work is to provide a formulation of two related nonlinear diffusive convective models in the form of coupled reaction-absorption equations. First, the postulated models are studied with an analytical approach. Later on, numerical evidences are considered to account for a precise characterization. The problem ([Formula: see text]) analyzed is of the form: [Formula: see text] Afterwards, a related problem [Formula: see text] is studied: [Formula: see text] The principal aspects for analysis are related to the existence and the derivation of particular solutions to reproduce the dynamic of the interacting species. For the problem [Formula: see text], we make use of the TW approach to study existence of solutions and precise evolution of profiles. Note that the term predator is used to refer to an invasive behavior, while the term prey is used for the invaded species.

Capillary transport in paper porous materials at low saturation levels: normal, fast or superfast?

The problem of capillary transport in fibrous porous materials at low levels of liquid saturation has been addressed. It has been demonstrated that the process of liquid spreading in this type of porous material at low saturation can be described macroscopically by a similar super-fast, nonlinear diffusion model to that which had been previously identified in experiments and simulations in particulate porous media. The macroscopic diffusion model has been underpinned by simulations using a microscopic network model. The theoretical results have been qualitatively compared with available experimental observations within the witness card technique using persistent liquids. The long-term evolution of the wetting spots was found to be truly universal and fully in line with the mathematical model developed. The result has important repercussions for the witness card technique used in field measurements of the dissemination of various low-volatility agents in imposing severe restrictions on collection and measurement times.

Modeling of an aircraft fire extinguishing process with a porous medium equation

The aim of this work is to provide a formulation of a non-linear diffusion model in the form of a Porous Medium Equation (PME) with application to a fire extinguishing process in an aircraft engine nacelle. The work starts by describing some relevant publications currently related to fire suppression modelling methods with emphasis in the aerospace sector. The PME is then introduced highlighting some key relevant features (particularly the finite speed of propagation) compared to the classical Heat Equation (HE). We will refer as u to the extinguisher or suppressor concentration in the media, which is postulated to be governed by a PME equation of the form: 1 $$\begin{aligned} u_{t}=\Delta u^m + \vert x \vert ^\sigma u^p, \end{aligned}$$ 2 $$\begin{aligned} u\left( x,0\right) = u_0(x) \in \mathbb {L}_{loc}^\infty (\mathbb {R^N}), \end{aligned}$$ where 3 $$\begin{aligned} m>1, \sigma >0, p<1, \end{aligned}$$ 4 $$\begin{aligned} (x,t) \in Q_T = \mathbb {R^N} \times (0,T) \end{aligned}$$ Without losing generality and in virtue of the mass transfer application, we will consider that any solution is $$u \ge 0$$ . The set of equation and conditions expressed from (1) to (4) will be referred as problem P. From a mathematical perspective, the main areas of analysis are related to the existence of solutions, the obtaining of particular solutions as asymptotic approach and the application or particularization to a representative aircraft engine nacelle domain where a fire may happen.

An efficient local meshless approach for solving nonlinear time-fractional fourth-order diffusion model

This paper adopts an efficient meshless approach for approximating the nonlinear fractional fourth-order diffusion model described in the Riemann–Liouville sense. A second-order difference technique is applied to discretize temporal derivatives, while the radial basis function meshless generated the finite difference scheme approximates the spatial derivatives. One key advantage of the local collocation method is the approximation of the derivatives via the finite difference formulation, for each local-support domain, by deriving the basis functions expansion. Another advantage of this method is that it can be applied in problems with non-regular geometrical domains. For the proposed time discretization, the unconditional stability is examined and an error bound is obtained. Numerical results illustrate the applicability and validity of the scheme and confirm the theoretical formulation.

Evaluating young fluvial terrace riser degradation using a nonlinear transport model: Application to the Kongur Normal Fault in the Pamir, northwest China

AbstractSelecting an appropriate hillslope transport model and calibrating model parameters are essential for morphological dating of fault and fluvial scarps. In this paper, we refine the method of profile‐based morphologic dating by updating the representation of nonlinearity in sediment flux dependence on the hillslope gradient. We apply this revised method to fluvial scarps bounding fluvial terraces offset along the Kongur Normal Fault in the semi‐arid high‐altitude Pamir mountains, northwestern China. One of these terraces, the T3 surface, is dated to 7.0+1.9/−1.6 kyr using 10Be cosmogenic depth profile analysis. Well‐preserved, dated terraces make this an ideal site to test the utility of morphological modelling in constraining ages of the young terrace risers. To do this, 35 topographic swath profiles across the terrace risers are extracted from a 0.2 m‐resolution digital elevation model produced using structure from motion from photos collected by an unmanned aerial vehicle. The best estimates of morphological age are 13.9 ± 1.3 m2 for the riser T3/T4 and 11.9 ± 1.3 m2 for T2/T3 using a linear diffusion approach. These two morphological ages overlap within uncertainty and fail to distinguish between two young terrace risers. Alternatively, we employed a nonlinear diffusion model, calibrated with transport constant k = 1 m2 kyr−1, nonlinearity n = 2, and critical gradient Sc = tan(33°). This nonlinear model produces ages of 7.3 ± 0.5 kyr for T3/T4 and 4.0 ± 0.2 kyr for T2/T3; these ages are consistent with terrace surface ages deduced by using vertical offset divided by independently determining average throw rate. This comparison shows the advantage of a nonlinear model in defining ages of young scarps. Furthermore, we explored the minor effect of heterogeneous degradation along steep sections of the scarp profiles. The nonlinear scarp modelling scheme we develop in this paper is suitable for studying scarp degradation in other regions. © 2020 John Wiley &amp; Sons, Ltd.

Nuclear crowding and nonlinear diffusion during interkinetic nuclear migration in the zebrafish retina.

An important question in early neural development is the origin of stochastic nuclear movement between apical and basal surfaces of neuroepithelia during interkinetic nuclear migration. Tracking of nuclear subpopulations has shown evidence of diffusion - mean squared displacements growing linearly in time - and suggested crowding from cell division at the apical surface drives basalward motion. Yet, this hypothesis has not yet been tested, and the forces involved not quantified. We employ long-term, rapid light-sheet and two-photon imaging of early zebrafish retinogenesis to track entire populations of nuclei within the tissue. The time-varying concentration profiles show clear evidence of crowding as nuclei reach close-packing and are quantitatively described by a nonlinear diffusion model. Considerations of nuclear motion constrained inside the enveloping cell membrane show that concentration-dependent stochastic forces inside cells, compatible in magnitude to those found in cytoskeletal transport, can explain the observed magnitude of the diffusion constant.

Decision-time statistics of nonlinear diffusion models: Characterizing long sequences of subsequent trials

Cognitive models of decision making are an important tool in studies of cognitive psychology and have been successfully used to fit experimental data and to relate them to neurophysiological mechanisms in the brain. One of the most important models for binary decision making is the diffusion-decision model (DDM) in which a diffusion process that models the accumulation of perceived evidence yields the decisions upon reaching one of two thresholds. Due to its simplicity, the model is analytically tractable and has been used to bridge the gap between implementations of decision making in neurobiologically plausible neural networks and experiments. However, biologically realistic network models exhibit nonlinear dynamics that yield via mean-field-reduction techniques a nonlinear DDM for which analytical solutions and proper numerical tools in general are not known. Furthermore, although often agents have to make a number of subsequent decisions, the statistics of such sequences of decisions (containing information on whether the decisions are correct or incorrect and on their timing) are so far poorly understood. Here we introduce the decision trains, sequences of negative or positive spikes at the decision times with the sign corresponding to the correctness of the decision. The decision trains enable a proper characterization of experiments in which many trials are performed consecutively. For the principal reference case of independent decisions (renewal statistics), we derive relations between the second-order statistics of the decision trains (i.e. their power spectra) and the response-time densities. Most importantly, we extend an efficient numerical procedure for spiking neuron models, the threshold-integration method, to determine the temporal statistics of nonlinear DDMs. The threshold-integration method provides the temporal statistics, i.e. decision rates, decision-time densities and the decision-train power spectra. Moreover, the procedure is used for the calculation of the linear response to a sinusoidal modulation. We compare all results with direct simulations of the stochastic model.

An investigation of nonlinear time-fractional anomalous diffusion models for simulating transport processes in heterogeneous binary media

In this work, we consider two of the most frequently used two-dimensional nonlinear time-fractional anomalous sub-diffusion models for simulating transport phenomena in heterogeneous binary media—a variable-order model and a generalised transport equation based on the Riemann-Liouville fractional operator. Our computational modelling framework uses a second order modified weighted shifted Grünwald-Letnikov scheme with correction terms to approximate the time-fractional derivative and nonlinear source term, together with an unstructured mesh control volume method for the spatial discretisation that accommodates the heterogeneous model properties. The resulting nonlinear system of algebraic equations is then stepped in time using an efficient Jacobian-free, Newton-Krylov solver to determine the set of discrete solution unknowns. We derive a semi-analytical solution and mass balance equation for a class of two-layered problems to evaluate the accuracy of the different computational models. A key contribution of our work is to identify the correct form of the interfacial boundary conditions to impose for the flux terms within the time-fractional framework and to illustrate the significant impact that the time-fractional indices have on the mass transfer. We show that the generalised transport model not only exhibits the correct physical solution behaviour, it produces a more accurate overall mass balance in comparison to the variable-order time-fractional model which is unable to resolve the abrupt changes in the solution behaviour at the interface between two different media having contrasting diffusivity, fractional index and source term. This finding indicates that this generalised model is an effective tool for characterising anomalous transport processes in heterogeneous systems. Finally, a series of numerical examples are presented to verify the theoretical analysis and to highlight the capability of the generalised transport model.

Global Existence and Nonexistence of Solutions to a Cross Diffusion System with Nonlocal Boundary Conditions

Mathematical models of nonlinear cross diffusion are described by a system of nonlinear partial parabolic equations associated with nonlinear boundary conditions. Explicit analytical solutions of such nonlinearly coupled systems of partial differential equations are rarely existed and thus, several numerical methods have been applied to obtain approximate solutions. In this paper, based on a self-similar analysis and the method of standard equations, the qualitative properties of a nonlinear cross-diffusion system with nonlocal boundary conditions are studied. We are constructed various self-similar solutions to the cross diffusion problem for the case of slow diffusion. It is proved that for certain values of the numerical parameters of the nonlinear cross-diffusion system of parabolic equations coupled via nonlinear boundary conditions, they may not have global solutions in time. Based on a self-similar analysis and the comparison principle, the critical exponent of the Fujita type and the critical exponent of global solvability are established. Using the comparison theorem, upper bounds for global solutions and lower bounds for blow-up solutions are obtained.

Optimization of experiment methodology based on identification of parameters in concrete drying

The water transport especially drying process in concrete has a significant influence on its structural durability and stability. However, the inconsistent values of parameters in drying model studies increases the difficulty in predicting the drying process of cementitious materials. In this work, an experimental-numerical-identification coupling approach is developed, based on the non-linear diffusion model, in order to optimize the parameters in drying model, and find out the optimal experiment methodology of concrete drying. Two different geometries of concrete samples are involved, which represent for two drying patterns (prismatic lateral drying and cylindrical radius drying). In the experiment test, both global (mass loss) and local (relative humidity) information are monitored, and the corresponding simulation results are obtained by CAST3M with finite element method. At last, an identification process is performed based on Levenberg-Marquart algorithm, and the optimization combination of parameters are obtained. The analysis of the relative variation of parameters allows to find out the optimization of experiment approach, including test duration and numbers and positions of humidity sensors.

Modeling the Shape and Evolution of Normal‐Fault Facets

Facets formed along the footwalls of active normal‐fault blocks display a variety of longitudinal profile forms, with variations in gradient, shape, degree of soil cover, and presence or absence of a slope break at the fault trace. We show that a two‐dimensional, process‐oriented cellular automaton model of facet profile evolution can account for the observed morphologic diversity. The model uses two dimensionless parameters to represent fault slip, progressive rock weathering, and downslope colluvial‐soil transport driven by gravity and stochastic disturbance events. The parameters represent rock weathering and soil disturbance rates, respectively, scaled by fault slip rate; both can be derived from field‐estimated rate coefficients. In the model's transport‐limited regime, slope gradient depends on the ratio of disturbance to slip rate, with a maximum that represents the angle of repose for colluvium. In this regime, facet evolution is consistent with nonlinear diffusion models of soil‐mantled hillslope evolution. Under the weathering‐limited regime, bedrock becomes partly exposed but microtopography helps trap some colluvium even when facet gradient exceeds the threshold angle. Whereas the model predicts a continuous gradient from footwall to colluvial wedge under transport‐limited behavior, fully weathering‐limited facets tend to develop a slope break between footwall and basal colluvium as a result of reduced transport efficiency on the rocky footwall slope. To the extent that the model provides a reasonable analogy for natural facets, its behavior suggests that facet profile morphology can provide useful constraints on relative potential rates of rock weathering, soil disturbance, and fault slip.

Numerical modeling of water uptake in white rice (Oryza sativa L.) using variable diffusivity approach

Soaking and drying of rice kernels are two very important processes in parboiling. Since diffusion of water through the rice kernels plays a crucial role in the design and operation of parboiling plants, it becomes essential to develop models to predict accurately the process of water uptake or removal under various operating conditions. In this study, moisture absorption by white rice (Oryza sativa L.) has been studied using a nonlinear model of Fickian diffusion. The moisture diffusion coefficient has been varied as a function of both moisture content and temperature. The proposed model has been tried with two different boundary conditions: (i) immediate surface saturation and (ii) two layers with different diffusion properties, using finite difference approach, and fitted into previously published experimental data for a temperature range of 25–80 °C. The model, using variable diffusivity and a fixed grid step method, explains the absorption process and the phenomenological effect of swelling (expansion) due to absorption comparatively better than a model which uses effective (lumped) diffusivity. An optimal set of process parameter values are determined for this temperature range. The proposed model is validated with hydration and swelling characteristics of chickpeas (Cicer arietinum L.) using experimental results obtained from literature.

Comparison among simultaneous confidence regions for nonlinear diffusion models

Accuracy measures for parameter estimates represent a tricky issue in nonlinear models. Practitioners often use separate marginal confidence intervals for each parameter in place of a simultaneous confidence region (sCR). However, this can be extremely misleading due to the curvature of the parameter space of the nonlinear model. For low parameter dimensions, routines for evaluating approximate sCRs are available in the most common software programs; however, the degree of accuracy depends on the curvature of the parameter space and the sample size. Exact sCRs are computationally intensive, and for this reason, in the past, they did not receive much attention. In this paper, we perform a comparison among exact, asymptotic exact, approximate sCRs, and marginal confidence intervals. More modern regions based on bootstrap are also examined as an alternative approach (both parametric and nonparametric). Their degree of accuracy is compared with both real data and simulation results. Among the nonlinear models, in this paper, the focus is on two of the most widespread diffusion models of products and technologies, that is, the Bass and Generalized Bass models. Three different empirical studies are analyzed here. Simulation studies are also performed for lifecycles with the same diffusion characteristics as those of the empirical studies. Our results show that, as the parameter dimension increases, overlapping among the alternative sCRs reduces. The approximate sCR shows inadequate values of overlapping with the exact sCR, even for moderate parameter dimension. Bootstrap regions also exhibit good performance in describing the shape of the exact region when curvature is present, but they fail to spread up to its boundary. The coverage probability of each region is assessed with simulations. We observe that the coverage probability of the approximate sCR decreases rapidly, even for moderate parameter dimension, and it is smaller than the nominal level for bootstrap regions.

Multi-attribute decision-making method based on a novel distance measure of linguistic intuitionistic fuzzy sets

Linguistic intuitionistic fuzzy sets can qualitatively rather than quantitatively express data in the form of membership degree. But quantitative tools are required to handle qualitative information. Therefore, an improved linguistic scale function, which can more accurately manifest the subjective feelings of decision-makers, is employed to deal with linguistic intuitionistic information. Subsequently, due to some commonly used distance measures do not comprehensively evaluate the information of linguistic intuitionistic fuzzy sets, an improved distance measure of linguistic intuitionistic fuzzy sets is designed. It considers the cross-evaluation information to get more realistic reasoning results. In addition, a new similarity measure defined by nonlinear Gaussian diffusion model is proposed, which can provide different response scales for different information between various schemes. The properties of these measures are also studied in detail. On this basis, a method in linguistic intuitionistic fuzzy environment is developed to handle multi-attribute decision-making problems. Finally, an illustrative example is given to demonstrate the effectiveness of the proposed method and the influence of the parameters is analyzed.

Bifurcation behaviour of a nonlinear innovation diffusion model with external influences

A nonlinear form of Bass model for innovation diffusion consisting of a system of two variables viz. for adopters and nonadopters population is proposed to lay stress on the evaluation period. The local stability of a positive equilibrium and the existence of Hopf bifurcation are demonstrated by analysing the associated characteristic equation. The critical value of evaluation period is determined beyond which small amplitude oscillations of the adopter and nonadopters population occur, and this critical value goes on decreasing with the increase in carrying capacity of the non-adopters population. The direction and the stability of bifurcating periodic solutions is determined by using the normal form theory and centre manifold theorem. It is observed that the cumulative density of external influences has a significant role in developing the maturity stage (final adoption stage) in the system. Numerical computations are executed to confirm the correctness of theoretical investigations.

Bifurcation behaviour of a nonlinear innovation diffusion model with external influences

A nonlinear form of Bass model for innovation diffusion consisting of a system of two variables viz. for adopters and nonadopters population is proposed to lay stress on the evaluation period. The local stability of a positive equilibrium and the existence of Hopf bifurcation are demonstrated by analysing the associated characteristic equation. The critical value of evaluation period is determined beyond which small amplitude oscillations of the adopter and nonadopters population occur, and this critical value goes on decreasing with the increase in carrying capacity of the non-adopters population. The direction and the stability of bifurcating periodic solutions is determined by using the normal form theory and centre manifold theorem. It is observed that the cumulative density of external influences has a significant role in developing the maturity stage (final adoption stage) in the system. Numerical computations are executed to confirm the correctness of theoretical investigations.

Analytical Solutions of a Modified Predator-Prey Model through a New Ecological Interaction.

The predator-prey model is a common tool that researchers develop continuously to predict the dynamics of the animal population within a certain phenomenon. Due to the sexual interaction of the predator in the mating period, the males and females feed together on one or more preys. This scenario describes the ecological interaction between two predators and one prey. In this study, the nonlinear diffusive predator-prey model is presented where this type of interaction is accounted for. The influence of this interaction on the population of predators and preys is predicted through analytical solutions of the dynamical system. The solutions are obtained by using two reliable and simple methods and are presented in terms of hyperbolic functions. In addition, the biological relevance of the solutions is discussed.