Partial Differential Equation Description Research Articles

A new solution approach to proportion delayed and heat like fractional partial differential equations

The importance of fractional partial differential equations (FPDEs) may be observed in many fields of science and engineering. On the same hand their solutions and the approaches for the same are also very important to notice due to the effectiveness of the methods and accuracy of the results. This work discusses the diverse estimated analytic description of fractional partial differential equations (with proportion delay and heat like equation), applying the Iterative Laplace Transform Method. The specified method represents a significant advancement in the tool case of applied mathematicians and scientists. Its ability to efficiently and accurately solve complex differential equations, especially FPDEs. Here in this work, the solution of four test problems of FPDEs related to proportion delay and heat like equations is obtained for testing the validity and asset of the Iterative Laplace Transform Method. Further their numerical and graphical interpretations are also mentioned.

Використання клітинних автоматів при моделюванні процесів сушіння деревини у лісосушильній камері періодичної дії

In this work, research the essence of the wood drying process in a periodic wood drying chamber. This paper provides a mathematical model of a wood drying chamber, which describes the general essence of physical drying processes using the equipment available in the wood drying chamber. This approach allows to take into account the physical parameters of the necessary equipment, such as heaters, fans, humidifying nozzles or other. This approach also allows to ignore some design characteristics that may differ depending on the type of wood drying chamber. Considering this, the main task in this work is to determine the temperature and humidity of the drying agent and lumber in the stack, as well as the temperature of the main components of the wood drying chamber. Taking into account such a large number of input parameters and describing a complex non-stationary process of heat transfer, there is a need to create complicated mathematical models. The presence of such mathematical models greatly complicates their application and requires significant computer resources for their calculation. In this way, the mathematical description is reduced to the description of non-linear partial differential equations. To simplify and speed up the calculations of this mathematical model, the use of cellular automata is suggested. To do this, the 3D model of the wood drying chamber is represented as a cell-automatic field, which consists of cells of the same size but different types. As a result, neighboring cells contain local relationships that describe their general behavior. This behavior depends on the type of tangent cells and is described by transition rules based on a mathematical model. Through the use of the developed cell-automatic model and transition rules, it is possible to obtain the values of the temperature and moisture content of the wood in the stack, the drying agent in the chamber, as well as the temperature of the main components of the chamber. The work also shows the corresponding graphs of changes in temperature and moisture content. To check the adequacy and reliability, the obtained results were compared with the results of other authors' experiments. As a result of the verification, the values of the average absolute error aren't high, which confirms the adequacy of the mathematical model and the prospects of using the developed cell-automatic model.

Bayesian uncertainty quantification for data-driven equation learning.

Equation learning aims to infer differential equation models from data. While a number of studies have shown that differential equation models can be successfully identified when the data are sufficiently detailed and corrupted with relatively small amounts of noise, the relationship between observation noise and uncertainty in the learned differential equation models remains unexplored. We demonstrate that for noisy datasets there exists great variation in both the structure of the learned differential equation models and their parameter values. We explore how to exploit multiple datasets to quantify uncertainty in the learned models, and at the same time draw mechanistic conclusions about the target differential equations. We showcase our results using simulation data from a relatively straightforward agent-based model (ABM) which has a well-characterized partial differential equation description that provides highly accurate predictions of averaged ABM behaviours in relevant regions of parameter space. Our approach combines equation learning methods with Bayesian inference approaches so that a quantification of uncertainty can be given by the posterior parameter distribution of the learned model.

Modelling collective cell motion: are on- and off-lattice models equivalent?

Biological processes, such as embryonic development, wound repair and cancer invasion, or bacterial swarming and fruiting body formation, involve collective motion of cells as a coordinated group. Collective cell motion of eukaryotic cells often includes interactions that result in polar alignment of cell velocities, while bacterial patterns typically show features of apolar velocity alignment. For analysing the population-scale effects of these different alignment mechanisms, various on- and off-lattice agent-based models have been introduced. However, discriminating model-specific artefacts from general features of collective cell motion is challenging. In this work, we focus on equivalence criteria at the population level to compare on- and off-lattice models. In particular, we define prototypic off- and on-lattice models of polar and apolar alignment, and show how to obtain an on-lattice from an off-lattice model of velocity alignment. By characterizing the behaviour and dynamical description of collective migration models at the macroscopic level, we suggest the type of phase transitions and possible patterns in the approximative macroscopic partial differential equation descriptions as informative equivalence criteria between on- and off-lattice models. This article is part of the theme issue 'Multi-scale analysis and modelling of collective migration in biological systems'.

Time–space coupled learning method for model reduction of distributed parameter systems with encoder‐decoder and RNN

AbstractModel reduction of a high‐dimensional distributed parameter system (DPS) reduces the complexity of the system for various applications, from monitoring to model predictive control, while retaining its intrinsic properties. Unfortunately, the assumption of time–space separability usually fails to hold for popular time–space separation model reduction methods because the space and time of the DPS are inherently coupled. In this study, a time–space coupled learning method for a data‐driven model reduction of the DPS is presented. The proposed method has the advantage of preserving the time–space coupling characteristics and increasing the number of degrees of freedom during the model reduction learning process. A novel deep‐learning architecture is presented by combining encoder‐decoder networks with recurrent neural networks. Given a high‐dimensional system without an exact partial differential equation description, the dimension‐reduced model and its temporal dynamics are jointly learned using the collected input and output data. The learned model is then applied to predict the low‐dimensional representations and reconstruct the high‐dimensional outputs. The proposed method was demonstrated on the catalytic rod in a tubular reactor with recycle, the results of which indicate a better modeling accuracy and lower intrinsic dimensionality compared with classical time–space separation model reduction methods.

THE HIERARCHY OF ASSOCIATIVITY EQUATIONS FOR n=3 CASE WITH AN METRIC ƞ11≠0

This paper describes the hierarchy for N = 2 and n=3 case with an metric ƞ11≠0 when V0 = 0 of associativity equations. The equation of associativity arose from the 2D topological field theory. 2D topological field theory represent the matter sector of topological string theory. These theories covariant before coupling to gravity due to the presence of a nilpotent symmetry and are therefore often referred to as cohomological field theories. We give a description of nonlinear partial differential equations of associativity in 2D topological field theories as integrable nondiagonalizable weakly nonlinear homogeneous system of hydrodynamic type. The article discusses nonlinear equations of the third order for a function f = f(x,t)) of two independent variables x, t. In this work we consider the associativity equation for n=3 case with an a metric 0 11   . The solution of some cases of hierarchy when N = 2 and V0 = 0 equations of associativity illustrated

Advanced Hepatitis C Virus Replication PDE Models within a Realistic Intracellular Geometric Environment.

The hepatitis C virus (HCV) RNA replication cycle is a dynamic intracellular process occurring in three-dimensional space (3D), which is difficult both to capture experimentally and to visualize conceptually. HCV-generated replication factories are housed within virus-induced intracellular structures termed membranous webs (MW), which are derived from the Endoplasmatic Reticulum (ER). Recently, we published 3D spatiotemporal resolved diffusion–reaction models of the HCV RNA replication cycle by means of surface partial differential equation (sPDE) descriptions. We distinguished between the basic components of the HCV RNA replication cycle, namely HCV RNA, non-structural viral proteins (NSPs), and a host factor. In particular, we evaluated the sPDE models upon realistic reconstructed intracellular compartments (ER/MW). In this paper, we propose a significant extension of the model based upon two additional parameters: different aggregate states of HCV RNA and NSPs, and population dynamics inspired diffusion and reaction coefficients instead of multilinear ones. The combination of both aspects enables realistic modeling of viral replication at all scales. Specifically, we describe a replication complex state consisting of HCV RNA together with a defined amount of NSPs. As a result of the combination of spatial resolution and different aggregate states, the new model mimics a cis requirement for HCV RNA replication. We used heuristic parameters for our simulations, which were run only on a subsection of the ER. Nevertheless, this was sufficient to allow the fitting of core aspects of virus reproduction, at least qualitatively. Our findings should help stimulate new model approaches and experimental directions for virology.

A free boundary model of epithelial dynamics

In this work we analyse a one-dimensional, cell-based model of an epithelial sheet. In the model, cells interact with their nearest neighbouring cells and move deterministically. Cells also proliferate stochastically, with the rate of proliferation specified as a function of the cell length. This mechanical model of cell dynamics gives rise to a free boundary problem. We construct a corresponding continuum-limit description where the variables in the continuum limit description are expanded in powers of the small parameter 1/N, where N is the number of cells in the population. By carefully constructing the continuum limit description we obtain a free boundary partial differential equation description governing the density of the cells within the evolving domain, as well as a free boundary condition that governs the evolution of the domain. We show that care must be taken to arrive at a free boundary condition that conserves mass. By comparing averaged realisations of the cell-based model with the numerical solution of the free boundary partial differential equation, we show that the new mass-conserving boundary condition enables the coarse-grained partial differential equation model to provide very accurate predictions of the behaviour of the cell-based model, including both evolution of the cell density, and the position of the free boundary, across a range of interaction potentials and proliferation functions in the cell based model.

Stochastic models of cell invasion with fluorescent cell cycle indicators

Fluorescent cell cycle labelling in cell biology experiments provides real time information about the location of individual cells, as well as the phase of the cell cycle of individual cells. We develop a stochastic, lattice-based random walk model of a two-dimensional scratch assay where the total population is composed of three distinct subpopulations which we visualise as red, yellow and green subpopulations. Our model mimics FUCCI technology in which cells in the G1 phase of the cell cycle fluoresce red, cells in the early S phase fluoresce yellow, and cells in the S/G2/M phase fluoresce green. The model is an exclusion process so that any potential motility or proliferation event that would place an agent on an occupied lattice site is aborted. Using experimental images and previous experimental measurements we explain how to apply the stochastic model to simulate a scratch assay initialised with a low to moderate density monolayer of human melanoma cell line. We obtain additional mathematical insight by deriving an approximate partial differential equation (PDE) description of the stochastic model, leading to a novel system of three coupled nonlinear reaction diffusion equations. Comparing averaged simulation data with the solution of the continuum limit model confirms that the PDE description is accurate for biologically-relevant parameter combinations.

Physiological factors leading to a successful vaccination: A computational approach

The immune system mounts a response to an infection by activating T cells. T cell activation occurs when dendritic cells, which have already interacted with the pathogen, scan a T cell that is cognate for (responsive to) the pathogen. This often occurs inside lymph nodes. The time it takes for this scanning event to occur, indeed the probability that it will occur at all, depends on many factors, including the rate that T cells and dendritic cells enter and leave the lymph node as well as the geometry of the lymph node and of course other cellular and molecular parameters. In this paper, we develop a hybrid stochastic-deterministic mathematical model at the tissue scale of the lymph node and simulate dendritic cells and cognate T cells to investigate the most important physiological factors leading to a successful and timely immune response after a vaccination. We use an agent-based model to describe the small population of cognate naive T cells and a partial differential equation description for the concentration of mature dendritic cells. We estimate the model parameters based on the known literature and measurements previously taken in our lab. We perform a parameter sensitivity analysis to quantify the sensitivity of the model results to the parameters. The results show that increasing T cell inflow through high endothelial venules, restricting cellular egress via the efferent lymph and increasing the total dendritic cell count by improving vaccinations are the among the most important physiological factors leading to an improved immune response. We also find that increasing the physical size of lymph nodes improves the overall likelihood that an immune response will take place but has a fairly weak effect on the response rate. The nature of dendritic cell trafficking through the LN (either passive or active transport) seems to have little effect on the overall immune response except if a change in overall egress time is observed.

Comments on the description and initialization of fractional partial differential equations using Riemann–Liouville’s and Caputo’s definitions

In this paper, it is shown that neither Riemann–Liouville’s nor Caputo’s definitions can be used to properly take into account initial conditions in the description of a fractional partial differential equation. This demonstration is done on a counterexample. Some suggestions are made to address this problem.

Misconceptions in using Riemann-Liouville’s and Caputo’s definitions for the description and initialization of fractional partial differential equations

In this paper, it is shown that neither Riemann-Liouville’s nor Caputo’s definitions can be used to properly take into account initial conditions in the description of a fractional partial differential equation. This demonstration is done on a counterexample. Some suggestions are made to address this problem.

Concurrent Efficient Evaluation of Small-Change Parameters and Green’s Functions for TCAD Device Noise and Variability Analysis

We present here an efficient numerical approach for the concurrent evaluation of the small-change deterministic device parameters and of the relevant Green’s functions exploited in the simulation of device small-signal (SS), small-signal large-signal (SSLS conversion matrix), stationary and cyclostationary noise, and variability properties of semiconductor devices through the solution of physics-based models based on a partial-differential equation description of charged carrier transport. The proposed technique guarantees a significant advantage in computation time with respect to the currently implemented solutions. The accuracy is the same as for the standard technique.

Dynamical Modelling & Estimation in Wastewater Treatment Processes

Environmental quality is becoming an increasing concern in our society. In that context, waste and wastewater treatment, and more specifically biological wastewater treatment processes play an important role. In this book, we concentrate on the mathematical modelling of these processes. The main purpose is to provide the increasing number of professionals who are using models to design, optimise and control wastewater treatment processes with the necessary background for their activities of model building, selection and calibration.The book deals specifically with dynamic models because they allow us to describe the behaviour of treatment plants under the highly dynamic conditions that we want them to operate (e.g. Sequencing Batch Reactors) or we have to operate them (e.g. storm conditions, spills). Further extension is provided to new reactor systems for which partial differential equation descriptions are necessary to account for their distributed parameter nature (e.g. settlers, fixed bed reactors).The model building exercise is introduced as a step-wise activity that, in this book, starts from mass balancing principles. In many cases, different hypotheses and their corresponding models can be proposed for a particular process. It is therefore essential to be able to select from these candidate models in an objective manner. To this end, structure characterisation methods are introduced. Important sections of the book deal with the collection of high quality data using optimal experimental design, parameter estimation techniques for calibration and the on-line use of models in state and parameter estimators.ISBN: 9781900222501 (Print)ISBN: 9781780403045 (eBook)

Facility location design under continuous traffic equilibrium

This paper presents two modeling approaches for median-type facility location design under elastic customer demand and traffic equilibrium in a continuous space. The first approach, following the continuum approximation scheme, builds upon the special case of an infinite homogeneous plane where traffic equilibrium can be described by an ordinary differential equation. The solution to this homogeneous case, sometimes in a closed form, is then used to develop approximate solutions to more general cases (e.g., those in a heterogeneous space). This model provides a computationally efficient way to obtain managerial insights and near-optimal solutions, especially for large problem instances. We also develop a more traditional discrete location model in the form of a mixed-integer program, which builds directly upon a nonlinear partial differential equation description of customer traffic equilibrium. We develop a Lagrangian relaxation based solution approach with an embedded finite-element method subroutine, to separate and solve the location decisions as well as the traffic equilibrium. Numerical experiments are conducted to illustrate applicability of the proposed models and to compare performance of the two complementing modeling approaches.

Modelling the movement of interacting cell populations: a moment dynamics approach.

Mathematical models describing the movement of multiple interacting subpopulations are relevant to many biological and ecological processes. Standard mean-field partial differential equation descriptions of these processes suffer from the limitation that they implicitly neglect to incorporate the impact of spatial correlations and clustering. To overcome this, we derive a moment dynamics description of a discrete stochastic process which describes the spreading of distinct interacting subpopulations. In particular, we motivate our model by mimicking the geometry of two typical cell biology experiments. Comparing the performance of the moment dynamics model with a traditional mean-field model confirms that the moment dynamics approach always outperforms the traditional mean-field approach. To provide more general insight we summarise the performance of the moment dynamics model and the traditional mean-field model over a wide range of parameter regimes. These results help distinguish between those situations where spatial correlation effects are sufficiently strong, such that a moment dynamics model is required, from other situations where spatial correlation effects are sufficiently weak, such that a traditional mean-field model is adequate.

Impact Mechanics of Elastic Structures With Point Contact

This paper introduces a modeling framework that is suitable to resolve singularities of impact phenomena encountered in applications. The method involves an exact transformation that turns the continuum, often partial differential equation description of the contact problem into a delay differential equation. The new form of the physical model highlights the source of singularities and suggests a simple criterion for regularity. To contrast singular and regular behavior the impacting Euler–Bernoulli and Timoshenko beam models are compared.

Biased random walks, partial differential equations and update schemes

There is much interest within the mathematical biology and statistical physics community in converting stochastic agent-based models for random walkers into a partial differential equation description for the average agent density. Here a Fokker-Planck equation is used to describe the evolution of a collection of noninteracting asymmetric random walkers. The resulting continuum description is highly dependent on the simulation update schemes. The theoretical results are confirmed with examples. These findings provide insight into the importance of updating schemes to the macroscopic description of stochastic local movement rules in agent-based models. doi:10.1017/S1446181113000369

Lattice-free descriptions of collective motion with crowding and adhesion

Cell-to-cell adhesion is an important aspect of malignant spreading that is often observed in images from the experimental cell biology literature. Since cell-to-cell adhesion plays an important role in controlling the movement of individual malignant cells, it is likely that cell-to-cell adhesion also influences the spatial spreading of populations of such cells. Therefore, it is important for us to develop biologically realistic simulation tools that can mimic the key features of such collective spreading processes to improve our understanding of how cell-to-cell adhesion influences the spreading of cell populations. Previous models of collective cell spreading with adhesion have used lattice-based random walk frameworks which may lead to unrealistic results, since the agents in the random walk simulations always move across an artificial underlying lattice structure. This is particularly problematic in high-density regions where it is clear that agents in the random walk align along the underlying lattice, whereas no such regular alignment is ever observed experimentally. To address these limitations, we present a lattice-free model of collective cell migration that explicitly incorporates crowding and adhesion. We derive a partial differential equation description of the discrete process and show that averaged simulation results compare very well with numerical solutions of the partial differential equation.

Incorporating spatial correlations into multispecies mean-field models

In biology, we frequently observe different species existing within the same environment. For example, there are many cell types in a tumour, or different animal species may occupy a given habitat. In modeling interactions between such species, we often make use of the mean-field approximation, whereby spatial correlations between the locations of individuals are neglected. Whilst this approximation holds in certain situations, this is not always the case, and care must be taken to ensure the mean-field approximation is only used in appropriate settings. In circumstances where the mean-field approximation is unsuitable, we need to include information on the spatial distributions of individuals, which is not a simple task. In this paper, we provide a method that overcomes many of the failures of the mean-field approximation for an on-lattice volume-excluding birth-death-movement process with multiple species. We explicitly take into account spatial information on the distribution of individuals by including partial differential equation descriptions of lattice site occupancy correlations. We demonstrate how to derive these equations for the multispecies case and show results specific to a two-species problem. We compare averaged discrete results to both the mean-field approximation and our improved method, which incorporates spatial correlations. We note that the mean-field approximation fails dramatically in some cases, predicting very different behavior from that seen upon averaging multiple realizations of the discrete system. In contrast, our improved method provides excellent agreement with the averaged discrete behavior in all cases, thus providing a more reliable modeling framework. Furthermore, our method is tractable as the resulting partial differential equations can be solved efficiently using standard numerical techniques.