Deterministic Differential Equations Research Articles

Convergence analysis of explicit stabilized integrators for parabolic semilinear stochastic PDEs

Abstract Explicit stabilized integrators are an efficient alternative to implicit or semiimplicit methods to avoid the severe time-step restriction faced by standard explicit integrators applied to stiff diffusion problems. In this paper we provide a fully discrete strong convergence analysis of a family of explicit stabilized methods coupled with finite element methods for a class of parabolic semilinear deterministic and stochastic partial differential equations. Numerical experiments including the semilinear stochastic heat equation with space-time white noise confirm the theoretical findings.

Structural Balance of Opinions.

The concept of Heider balance, usually applied to interpersonal relations, is generalized here to opinions gathered in surveys. At first, we compare four algorithms, which drive a matrix dataset to a balanced state. The criterion is that the final state obtained with an algorithm should be as close as possible to the initial state. The result is that deterministic differential equations work better than their Monte Carlo counterparts. Next, we apply the winning algorithms to the matrix of correlations between opinions gathered in American states between 1974 and 1998. The results are interpreted in terms of the classic comfort hypothesis (E. Babbie, 2007).

Dynamics of Tumor-Immune System with Random Noise

With deterministic differential equations, we can understand the dynamics of tumor-immune interactions. Cancer-immune interactions can, however, be greatly disrupted by random factors, such as physiological rhythms, environmental factors, and cell-to-cell communication. The present study introduces a stochastic differential model in infectious diseases and immunology of the dynamics of a tumor-immune system with random noise. Stationary ergodic distribution of positive solutions to the system is investigated in which the solution fluctuates around the equilibrium of the deterministic case and causes the disease to persist stochastically. In some conditions, it may be possible to attain infection-free status, where diseases die out exponentially with a probability of one. Some numerical simulations are conducted with the Euler–Maruyama scheme in order to verify the results. White noise intensity is a key factor in treating infectious diseases.

Stochastic dynamics, large deviation principle, and nonequilibrium thermodynamics.

By examining the deterministic limit of a general ε-dependent generator for Markovian dynamics, which includes the continuous Fokker-Planck equations and discrete chemical master equations as two special cases, the intrinsic connections among mesoscopic stochastic dynamics, deterministic ordinary differential equations or partial differential equations, large deviation rate functions, and macroscopic thermodynamic potentials are established. Our result not only solves the long-lasting question of the origin of the entropy function in classical irreversible thermodynamics, but also reveals an emergent feature that arises automatically during the deterministic limit, through its large deviation rate function, with both time-reversible dynamics equipped with a Hamiltonian function and time-irreversible dynamics equipped with an entropy function.

Mathematical Model of HBV/HCV Co-Infection

The co-infection of hepatitis B (HBV) and hepatitis C (HCV) virus is a complex clinical entity that has an estimated worldwide prevalence of 1--15%. In this paper HBV/HCV co-infection is modelled mathematically through the set of deterministic non-linear differential equations. This dynamical system has four equilibrium points i.e. disease-free, co-infection free, HCV free and endemic point. Reproduction number is computed for endemic equilibria. Local stability for all the equilibrium point is proved using Routh-Hurwitz criterion. Global stability is also studied for all the equilibria. The sensitivity analysis of relevant parameters in reproduction number is analyzed to see the effect of each parameter in disease spread.

A reduced modal subspace approach for damped stochastic dynamic systems

A novel method for characterising and propagating system uncertainty in structures subjected to dynamic actions is proposed, whereby modal shapes, frequencies and damping ratios constitute the random quantities. The latter, defined in the modal subspace rather than the full geometrical space, reduce the number of the random variables and the size of the dynamic problem. A numerical procedure is presented for their identification by calibrating their probabilistic definition in line with the geometrical space. A high-order perturbation technique is proposed for the multi-fidelity response quantification by means of an ad hoc extension of the conventional perturbation method. The approach involves a set of auxiliary deterministic differential equations to be adaptively solved with the piecewise exact method, and moment-cumulant relationships are employed to approximate high-order moments. Finally, a polynomial chaos expansion approach is adopted to complement the second-moment analysis for spectral quantification with the modal subspace reduction. Demonstrated on a multi-storey steel frame with semi-rigid connections and a simply supported bridge subjected to a moving load, the proposed variants exhibit improved performance with respect to the conventional second-order and improved perturbation, as well as increased flexibility, enabling the analyst to decide, on-demand, the level of fidelity, balancing accuracy and computational effort.

A Small Time Approximation for the Solution to the Zakai Equation

We propose a novel small time approximation for the solution to the Zakai equation from nonlinear filtering theory. We prove that the unnormalized filtering density is well described over short time intervals by the solution of a deterministic partial differential equation of Kolmogorov type; the observation process appears in a pathwise manner through the degenerate component of the Kolmogorov’s type operator. The rate of convergence of the approximation is of order one in the lenght of the interval. Our approach combines ideas from Wong-Zakai-type results and Wiener chaos approximations for the solution to the Zakai equation. The proof of our main theorem relies on the well-known Feynman-Kac representation for the unnormalized filtering density and careful estimates which lead to completely explicit bounds.

Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials

Abstract Fourier approximation plays a key role in qualitative theory of deterministic and random differential equations. In this paper, we will develop a new approximation tool. For an m m -order differentiable function f f on [ 0 , 1 0,1 ], we will construct an m m -degree algebraic polynomial P m {P}_{m} depending on values of f f and its derivatives at ends of [ 0 , 1 0,1 ] such that the Fourier coefficients of R m = f − P m {R}_{m}=f-{P}_{m} decay fast. Since the partial sum of Fourier series R m {R}_{m} is a trigonometric polynomial, we can reconstruct the function f f well by the combination of a polynomial and a trigonometric polynomial. Moreover, we will extend these results to the case of random processes.

Langevin original approach and Ornstein–Uhlenbeck-type processes

This paper applies Langevin idea to describe the Brownian motion of a particle characterized by an Ornstein–Uhlenbeck-type process. The original and clever method proposed by Langevin is based on Newton’s second law plus a fluctuating force whose solution for the mean square displacement consists in separating the fluctuating force from his equation to obtain a deterministic equation for the relevant physical variable. In this work the Langevin original idea is applied to calculate the mean square velocity for a field free particle case; then it is extended for a charged particle in a constant magnetic field. In a similar way, the strategy is also applied to calculate the mean square displacement for a Brownian harmonic oscillator in the overdamped regime, and also when a magnetic field is present. In particular, it is shown in the field free case that Langevin’s original strategy leads to the same results as those obtained using the statistical properties of a Gaussian white noise. All the theoretical results are compared with both the numerical simulation of the Langevin equation, and numerical solution of the corresponding deterministic differential equations.

BLUES function method applied to partial differential equations and analytic approximants for interface growth under shear

An iteration sequence based on the BLUES (beyond linear use of equation superposition) function method is presented for calculating analytic approximants to solutions of nonlinear partial differential equations. This extends previous work using this method for nonlinear ordinary differential equations with an external source term. Now, the initial condition plays the role of the source. The method is tested on three examples: a reaction-diffusion-convection equation, the porous medium equation with growth or decay, and the nonlinear Black-Scholes equation. A comparison is made with three other methods: the Adomian decomposition method (ADM), the variational iteration method (VIM), and the variational iteration method with Green function (GVIM). As a physical application, a deterministic differential equation is proposed for interface growth under shear, combining Burgers and Kardar-Parisi-Zhang nonlinearities. Thermal noise is neglected. This model is studied with Gaussian and space-periodic initial conditions. A detailed Fourier analysis is performed and the analytic coefficients are compared with those of ADM, VIM, GVIM, and standard perturbation theory. The BLUES method turns out to be a worthwhile alternative to the other methods. The advantages that it offers ensue from the freedom of choosing judiciously the linear part, with associated Green function, and the residual containing the nonlinear part of the differential operator at hand.

Modeling and analysis of COVID-19 in India with treatment function through different phases of lockdown and unlock

India, unlike several other countries, has witnessed a greater challenge in overcoming the pandemic during the second wave crisis. The strategies on lockdown imposition during the first wave of the pandemic were well implemented due to which the year-long wave did not witness uncontrollable surges in the infections. In this study, we present a detailed study on the disease spread during the first and second wave of COVID-19, by performing numerical simulation in a phase-wise manner as per the lockdown and unlock phases implemented in India. The inclusion of a piecewise treatment function in the framed epidemiological model is a noteworthy aspect of the study since this function takes into consideration the availability of medical equipment, based upon the threshold number of infections. This function framed in this model first grows linearly reaching a peak, then it declines due to shortage of medical resources and finally gets saturated. Analysis of the impact of lockdown is presented for each phase of the pandemic in India. Though the data considered for the study is for a period that marked the beginning of the pandemic, major analysis and predictions are presented based on the second wave data in terms of sensitivity analysis and time series behavior. A comparison of deterministic and stochastic differential equations is presented with simulation results on certain parameter set to examine variation in treatment and recovery. The simulations are performed using MATLAB and R softwares. The work is validated with the real data and model fitting is done applying the Maximum Likelihood method. The study implies that, under accurate lockdown strategies and sufficient medical care, the peak in cases would be attained by 16 May 2021, after which a decline in the cases could be observed.

A linear implicit Euler method for the finite element discretization of a controlled stochastic heat equation

AbstractWe consider a numerical approximation of a linear quadratic control problem constrained by the stochastic heat equation with nonhomogeneous Neumann boundary conditions. This involves a combination of distributed and boundary control, as well as both distributed and boundary noise. We apply the finite element method for the spatial discretization and the linear implicit Euler method for the temporal discretization. Due to the low regularity induced by the boundary noise, convergence orders above $1/2$ in space and $1/4$ in time cannot be expected. We prove such optimal convergence orders for our full discretization when the distributed noise and the initial condition are sufficiently smooth. Under less smooth conditions the convergence order is further decreased. Our results only assume that the related (deterministic) differential Riccati equation can be approximated with a certain convergence order, which is easy to achieve in practice. We confirm these theoretical results through a numerical experiment in a two-dimensional domain.

Periodic solution of stochastic process in the distributional sense

In this paper, we aim to study a stochastic process from a macro point of view, and thus, periodic solution of a stochastic process in distributional sense is introduced. We first give the definition and then establish the existence of periodic solution on bounded domain. Lastly, for the case that probability density function exists, we obtain the existence periodic solutions of the probability density function corresponding to the stochastic process by using the technique of deterministic partial differential equations.

Connection of CTMC process, deterministic and stochastic differential equations in modeling of epidemics

Mathematical modeling has been used in many fields of study including in epidemiology. The main objective of this study is to show the connection of three mathematical models often used to study the dynamics of disease spread in the natural world; i.e., a stochastic process (CTMC), deterministic model (ODEs) and stochastic differential equation (SDEs). We show that, by proper scaling technique, it is possible to derive the deterministic analogue of a CTMC. Its stochastic differential equation (SDE) version can be obtained by adding a white noise or Weinner process in the deterministic model with proper means and covariance. We demonstrate all three models with the dynamics of SIR epidemics followed by several numerical experiments to show how accurate the trajectories of ODEs follow the sample paths of both CTMC and SDEs.

Poly-Sinc Solution of Stochastic Elliptic Differential Equations

In this paper, we introduce a numerical solution of a stochastic partial differential equation (SPDE) of elliptic type using polynomial chaos along side with polynomial approximation at Sinc points. These Sinc points are defined by a conformal map and when mixed with the polynomial interpolation, it yields an accurate approximation. The first step to solve SPDE is to use stochastic Galerkin method in conjunction with polynomial chaos, which implies a system of deterministic partial differential equations to be solved. The main difficulty is the higher dimensionality of the resulting system of partial differential equations. The idea here is to solve this system using a small number of collocation points in space. This collocation technique is called Poly-Sinc and is used for the first time to solve high-dimensional systems of partial differential equations. Two examples are presented, mainly using Legendre polynomials for stochastic variables. These examples illustrate that we require to sample at few points to get a representation of a model that is sufficiently accurate.

A full probabilistic analysis of a randomized kinetic model for reaction–deactivation of hydrogen peroxide decomposition with applications to real data

The classical kinetic equation has been broadly used to describe reaction and deactivation processes in chemistry. The mathematical formulation of this deterministic nonlinear differential equation depends on reaction and deactivation rate constants. In practice, these rates must be calculated via laboratory experiments, hence involving measurement errors. Therefore, it is more realistic to treat these rates as random variables rather than deterministic constants. This leads to the randomization of the kinetic equation, and hence its solution becomes a stochastic process. In this paper we address the probabilistic analysis of a randomized kinetic model to describe reaction and deactivation by catalase of hydrogen peroxide decomposition at a given initial concentration. In the first part of the paper, we determine closed-form expressions for the probability density functions of important quantities of the aforementioned chemical process (the fractional conversion of hydrogen peroxide, the time until a fixed quantity of this fractional conversion is reached and the activity of the catalase). These expressions are obtained by taking extensive advantage of the so called Random Variable Transformation technique. In the second part, we apply the theoretical results obtained in the first part together with the principle of maximum entropy to model the hydrogen peroxide decomposition and aspergillus niger catalase deactivation using real data excerpted from the recent literature. Our results show full agreement with previous reported analysis but having as additional benefit that they provide a more complete description of both model inputs and outputs since we take into account the intrinsic uncertainties involved in modelling process.

A Stochastic Differential Equations Model for the Spread of Coronavirus COVID-19): The Case of Iraq

In this paper, we model the spread of coronavirus (COVID -19) by introducing stochasticity into the deterministic differential equation susceptible -infected-recovered (SIR model). The stochastic SIR dynamics are expressed using Itô's formula. We then prove that this stochastic SIR has a unique global positive solution I(t).The main aim of this article is to study the spread of coronavirus COVID-19 in Iraq from 13/8/2020 to 13/9/2020. Our results provide a new insight into this issue, showing that the introduction of stochastic noise into the deterministic model for the spread of COVID-19 can cause the disease to die out, in scenarios where deterministic models predict disease persistence. These results were also clearly illustrated by Computer simulation.

Combinatorial mathematical modelling approaches to interrogate rear retraction dynamics in 3D cell migration.

Cell migration in 3D microenvironments is a complex process which depends on the coordinated activity of leading edge protrusive force and rear retraction in a push-pull mechanism. While the potentiation of protrusions has been widely studied, the precise signalling and mechanical events that lead to retraction of the cell rear are much less well understood, particularly in physiological 3D extra-cellular matrix (ECM). We previously discovered that rear retraction in fast moving cells is a highly dynamic process involving the precise spatiotemporal interplay of mechanosensing by caveolae and signalling through RhoA. To further interrogate the dynamics of rear retraction, we have adopted three distinct mathematical modelling approaches here based on (i) Boolean logic, (ii) deterministic kinetic ordinary differential equations (ODEs) and (iii) stochastic simulations. The aims of this multi-faceted approach are twofold: firstly to derive new biological insight into cell rear dynamics via generation of testable hypotheses and predictions; and secondly to compare and contrast the distinct modelling approaches when used to describe the same, relatively under-studied system. Overall, our modelling approaches complement each other, suggesting that such a multi-faceted approach is more informative than methods based on a single modelling technique to interrogate biological systems. Whilst Boolean logic was not able to fully recapitulate the complexity of rear retraction signalling, an ODE model could make plausible population level predictions. Stochastic simulations added a further level of complexity by accurately mimicking previous experimental findings and acting as a single cell simulator. Our approach highlighted the unanticipated role for CDK1 in rear retraction, a prediction we confirmed experimentally. Moreover, our models led to a novel prediction regarding the potential existence of a ‘set point’ in local stiffness gradients that promotes polarisation and rapid rear retraction.

Potential and Challenges of Investigating Intrinsic Uncertainty of Hydrological Models With Stochastic, Time‐Dependent Parameters

AbstractStochastic hydrological process models have two conceptual advantages over deterministic models. First, even though water flow in a well‐defined environment is governed by deterministic differential equations, a hydrological system, at the level we can observe it, does not behave deterministically. Reasons for this behavior are unobserved spatial heterogeneity and fluctuations of input, unobserved influence factors, heterogeneity and variability in soil and aquifer properties, and an imprecisely known initial state. A stochastic model provides thus a more realistic description of the system than a deterministic model. Second, hydrological models simplify real processes. The resulting structural deficits can better be accounted for by stochastic than by deterministic models because they, even for given parameters and input, allow for a probability distribution of different system evolutions rather than a single trajectory. On the other hand, stochastic process models are more susceptible to identifiability problems and Bayesian inference is computationally much more demanding. In this paper, we review the use of stochastic, time‐dependent parameters to make deterministic models stochastic, discuss options for the numerical implementation of Bayesian inference, and investigate the potential and challenges of this approach with a case study. We demonstrate how model deficits can be identified and reduced, and how the suggested approach leads to a more realistic description of the uncertainty of internal and output variables of the model compared to a deterministic model. In addition, multiple stochastic parameters with different correlation times could explain the variability in the time scale of output error fluctuations identified in an earlier study.

Lagrangian models of particle-laden flows with stochastic forcing: Monte Carlo, moment equations, and method of distributions analyses

Deterministic Eulerian–Lagrangian models represent the interaction between particles and carrier flow through the drag force. Its analytical descriptions are only feasible in special physical situations, such as the Stokes drag for low Reynolds number. For high particle Reynolds and Mach numbers, where the Stokes solution is not valid, the drag must be corrected by empirical, computational, or hybrid (data-driven) methods. This procedure introduces uncertainty in the resulting model predictions, which can be quantified by treating the drag as a random variable and by using data to verify the validity of the correction. For a given probability density function of the drag coefficient, we carry out systematic uncertainty quantification for an isothermal one-way coupled Eulerian–Lagrangian system with stochastic forcing. The first three moment equations are analyzed with a priori closure using Monte Carlo computations, showing that the stochastic solution is highly non-Gaussian. For a more complete description, the method of distributions is used to derive a deterministic partial differential equation for the evolution of the joint PDF of the particle phase and drag coefficient. This equation is solved via Chebyshev spectral collocation method, and the resulting numerical solution is compared with Monte Carlo computations. Our analysis highlights the importance of a proper approximation of the Dirac delta function, which represents deterministic (known with certainty) initial conditions. The robustness and accuracy of our PDF equation were tested on one-dimensional problems in which the Eulerian phase represents either a uniform flow or a stagnation flow.