Kolmogorov Equation Research Articles

Analytical Survival Analysis of the Non-autonomous Ornstein–Uhlenbeck Process

The survival probability for a periodic non-autonomous Ornstein–Uhlenbeck process is calculated analytically using two different methods. The first uses an asymptotic approach. We treat the associated Kolmogorov Backward Equation with an absorbing boundary by dividing the domain into an interior region, centered around the origin, and a “boundary layer” near the absorbing boundary. In each region we determine the leading-order analytical solutions, and construct a uniformly valid solution over the entire domain using asymptotic matching. In the second method we examine the integral relationship between the probability density function and the mean first passage time probability density function. These allow us to determine approximate analytical forms for the exit rate. The validity of the solutions derived from both methods is assessed numerically, and we find the asymptotic method to be superior.

Dynamics behaviours of N-kink solitons in conformable Fisher–Kolmogorov–Petrovskii–Piskunov equation

PurposeThis manuscript is related to compute $N$-kink soliton solutions for conformable Fisher–Kolmogorov equation (CFKE) by using the generalized extended direct algebraic method (EDAM). The considered problem has important applications in mathematical biology and reaction diffusion processes. Also, the mentioned problem has significant applications in population dynamics. The fractional order conformable derivative has many features as compared to the other fractional order differential operators. For instance, the chain, product and quotient procedures do not satisfy by other fractional differential operators, but conformable operators obey the mentioned rules. Hence, we compute the soliton solutions for the mentioned problem and present its various dynamical behaviours graphically.Design/methodology/approachThe generalized EDAM is used in this article to examine the calculation of N-kink soliton solutions for the CFKE. In mathematical biology and reaction-diffusion processes, the topic under consideration holds great significance, especially when considering population dynamics.FindingsThe results highlight the benefits of utilising conformable derivatives in mathematical modelling and further our understanding of fractional differential equations and their applications.Research limitations/implicationsThe work focuses primarily on N-kink soliton solutions, which may limit the examination of alternative types of solutions (e.g., multi-soliton or periodic solutions) that might give new insights into the dynamics of the CFKE.Practical implicationsThe generated N N-kink soliton solutions can enhance mathematical models in biological contexts, notably in modelling population dynamics, disease propagation and ecological interactions, leading to better forecasts and interventions.Social implicationsPublic health initiatives can benefit from the understanding of disease transmission and intervention efficacy that comes from modelling population dynamics and reaction-diffusion processes.Originality/valueThe use of the generalized EDAM to obtain solutions for N-kink soliton problems is an innovative method for solving the conformable Fisher–Kolmogorov equation, demonstrating the power of this mathematical tool.

Dynamic Programming-Based Approach to Model Antigen-Driven Immune Repertoire Synthesis

This paper presents a novel approach to modeling the repertoire of the immune system and its adaptation in response to the evolutionary dynamics of pathogens associated with their genetic variability. It is based on application of a dynamic programming-based framework to model the antigen-driven immune repertoire synthesis. The processes of formation of new receptor specificity of lymphocytes (the growth of their affinity during maturation) are described by an ordinary differential equation (ODE) with a piecewise-constant right-hand side. Optimal control synthesis is based on the solution of the Hamilton–Jacobi–Bellman equation implementing the dynamic programming approach for controlling Gaussian random processes generated by a stochastic differential equation (SDE) with the noise in the form of the Wiener process. The proposed description of the clonal repertoire of the immune system allows us to introduce an integral characteristic of the immune repertoire completeness or the integrative fitness of the whole immune system. The quantitative index for characterizing the immune system fitness is analytically derived using the Feynman–Kac–Kolmogorov equation.

The Reduced-Dimension Method for Crank–Nicolson Mixed Finite Element Solution Coefficient Vectors of the Extended Fisher–Kolmogorov Equation

A reduced-dimension (RD) method based on the proper orthogonal decomposition (POD) technology and the linearized Crank–Nicolson mixed finite element (CNMFE) scheme for solving the 2D nonlinear extended Fisher–Kolmogorov (EFK) equation is proposed. The method reduces CPU runtime and error accumulation by reducing the dimension of the unknown CNMFE solution coefficient vectors. For this purpose, the CNMFE scheme of the above EFK equation is established, and the uniqueness, stability and convergence of the CNMFE solutions are discussed. Subsequently, the matrix-based RDCNMFE scheme is derived by applying the POD method. Furthermore, the uniqueness, stability and error estimates of the linearized RDCNMFE solution are proved. Finally, numerical experiments are carried out to validate the theoretical findings. In addition, we contrast the RDCNMFE method with the CNMFE method, highlighting the advantages of the dimensionality reduction method.

On optimal control of coupled mean-field forward-backward stochastic equations

Abstract We consider a control problem for a mean-field coupled forward-backward stochastic differential equations, called also McKean–Vlasov equation (MF-FBSDE). For this type of equations, the coefficients depend not only on the state of the system, but also on its marginal distributions. They arise naturally in mean-field control problems and mean-field games. We consider the relaxed control problem where admissible controls are measure-valued processes. We prove the existence of a relaxed optimal control by using a suitable form of Skorokhod representation theorem and Jakubowski’s topology, on the space of càdlàg functions. We use martingale measure to define the relaxed state process. Our results extend to MF-FBSDEs those already known for forward and backward stochastic equations of Itô type.

Numerical Solution of the Forward Kolmogorov Equations in Population Genetics Using Eta Functions

This paper introduces a new numerical method for solving forward Kolmogorov equations in population genetics. Since there's no simple analytical expression for the distribution of allele frequencies (DAF), we use these equations to derive it. The accuracy of solving these equations depends on the choice of base functions, so we use Eta-based functions for better approximations. By employing the operational matrix of integral, we simplify the partial differential equation to an algebraic one. The method's error bounds, stability, and validity are demonstrated through numerical examples. Finally, we apply this method to analyze the behavior of forward Kolmogorov equations under various evolutionary forces.

Zero-sum games subject to uncertain stochastic noncausal systems considering polynomial control functions

An uncertain stochastic noncausal system (USNS) is an uncertain stochastic singular system that is expected to be regular but not impulse-free. This study investigates zero-sum games (ZSGs) under the constraint of nonlinear uncertain stochastic noncausal systems (USNSs) considering polynomial control functions. Firstly, a method is introduced to convert USNSs into subsystems, including forward uncertain stochastic difference equations as well as backward uncertain stochastic difference equations. Equilibrium equations are then derived to determine the saddle-point equilibrium solution of the zero-sum game (ZSG) for USNSs. Subsequently, a unified framework is developed to find the saddle-point equilibrium and equilibrium value of the ZSG. Additionally, a numerical example and an analysis of the treatment of industrial wastewater are provided to demonstrate the effectiveness of the proposed framework.

Beyond Time-Homogeneity for Continuous-Time Multistate Markov Models

–Multistate Markov models are a canonical parametric approach for data modeling of observed or latent stochastic processes supported on a finite state space. Continuous-time Markov processes describe data that are observed irregularly over time, as is often the case in longitudinal medical data, for example. Assuming that a continuous-time Markov process is time-homogeneous, a closed-form likelihood function can be derived from the Kolmogorov forward equations—a system of differential equations with a well-known matrix-exponential solution. Unfortunately, however, the forward equations do not admit an analytical solution for continuous-time, time- inhomogeneous Markov processes, and so researchers and practitioners often make the simplifying assumption that the process is piecewise time-homogeneous. In this article, we provide intuitions and illustrations of the potential biases for parameter estimation that may ensue in the more realistic scenario that the piecewise-homogeneous assumption is violated, and we advocate for a solution for likelihood computation in a truly time-inhomogeneous fashion. Particular focus is afforded to the context of multistate Markov models that allow for state label misclassifications, which applies more broadly to hidden Markov models (HMMs), and Bayesian computations bypass the necessity for computationally demanding numerical gradient approximations for obtaining maximum likelihood estimates (MLEs). Supplemental materials are available online.

Nonlinear evolution equations in dispersive media: Analyzing the semilinear dispersive-fisher model

The nonlinear Semilinear Dispersive-Fisher (SDF) model is an important equation describing wave dynamics in dispersive media influenced by nonlinear and diffusive effects. This study enhances both analytical and numerical methods to solve the SDF model and validate the solutions. We utilize the Modified Khater (MKhat) and Unified (UF) methods for analytical solutions, and He’s Variational Iteration (HVI) scheme for numerical approximations. Our investigation connects the SDF model to other nonlinear equations like the Korteweg–de Vries (KdV) and Fisher–Kolmogorov (FK) equations, demonstrating the consistency between analytical and numerical solutions. This research contributes to the understanding and modeling of wave phenomena in dispersive media with nonlinear and diffusive dynamics, offering refined analytical and numerical techniques specific to the SDF model. Key contributions include introducing the MKhat and UF methods for analytical solutions and employing the HVI scheme for numerical approximations, which are less represented in current literature. This work is positioned in mathematical physics, focusing on nonlinear evolution equations.

Effect mechanism of Nb addition on grain refinement and inhibition of discontinuous precipitation of Cu–15Ni–8Sn alloy

Cu–15Ni–8Sn-xNb alloys (x = 0, 0.2, 0.6 wt%) were prepared using a medium-frequency induction melting furnace. The effect of Nb addition on grain refinement and inhibition of discontinuous precipitation (DP) in Cu–15Ni–8Sn alloys was systematically studied to elucidate the mechanism by which microstructural characteristics contribute to strength improvement. The results indicate that the increase the Nb content from 0 to 0.6 wt% reduces the average grain size of the as-cast alloy from approximately 524.8 μm to approximately 81.3 μm, and significantly decreases the lamellar transition structure region (α+γ). During solution treatment, dispersed needle-like NbNi3 phases were observed in the Cu–15Ni–8Sn-0.2Nb alloy. After aging, the DP growth rate in the Cu–15Ni–8Sn-0.2Nb alloy was notably slower than those of the other alloys. This was attributed to NbNi3 phases at the grain boundaries hindering DP nucleation, with intragranular NbNi3 phases inhibiting DP growth. The phase transformation order of the solid solution Cu–15Ni–8Sn-0.2Nb at 673 K aging is: spinodal structure → D022 ordered phase → L12 ordered phase → DP. The hardness and tensile strength of the Cu–15Ni–8Sn-0.2Nb alloy peaked at 338.3 HV and 725.42 MPa, respectively, after aging for 120 min. Using the Johnson–Mehl–Avrami–Kolmogorov (JMAK) and Arrhenius equations, the activation energies of DP in Cu–15Ni–8Sn-xNb (x = 0, 0.2, 0.6 wt%) alloys were calculated to be 93.19, 148.64, and 98.33 kJ/mol, respectively. These values suggest that the diffusion of DP atoms in the Nb-containing alloys is hindered, which effectively inhibits DP formation.

Intelligent seismic AVO inversion method for brittleness index of shale oil reservoirs

The brittleness index (BI) is crucial for predicting engineering sweet spots and designing fracturing operations in shale oil reservoir exploration and development. Seismic amplitude variation with offset (AVO) inversion is commonly used to obtain the BI. Traditionally, velocity, density, and other parameters are firstly inverted, and the BI is then calculated, which often leads to accumulated errors. Moreover, due to the limited of well-log data in field work areas, AVO inversion typically faces the challenge of limited information, resulting in not high accuracy of BI derived by existing AVO inversion methods. To address these issues, we first derive an AVO forward approximation equation that directly characterizes the BI in P-wave reflection coefficients. Based on this, an intelligent AVO inversion method, which combines the advantages of traditional and intelligent approaches, for directly obtaining the BI is proposed. A TransU-net model is constructed to establish the strong nonlinear mapping relationship between seismic data and the BI. By incorporating a combined objective function that is constrained by both low-frequency parameters and training samples, the challenge of limited samples is effectively addressed, and the direct inversion of the BI is stably achieved. Tests on model data and applications on field data demonstrate the feasibility, advancement, and practicality of the proposed method.

Linear quadratic optimal control of stochastic 2-D Roesser models

This paper investigates the linear quadratic (LQ) optimal control problem for the stochastic two-dimensional (2-D) systems governed by Roesser models with multiplicative noise. The main contribution is to give the necessary and sufficient optimality condition by proposing a set of novel forward and backward stochastic partial difference equations (FBSPDE), and to further present the explicitly optimal feedback control laws on the finite horizon and on the infinite horizon based on the Riccati-like difference equations and the algebraic equation, respectively. Several numerical simulations are provided to illustrate the performance of the designed controllers.

Simplified Direct Strength Method for the design of cold-formed steel channels with circular web openings under two-flange loadings

Previous studies have put forward reduction factor equations to determine the web crippling strength of cold-formed steel (CFS) channels with circular web openings. However, these equations overlook vital parameters such as material yield strength, flange width, and fillet radius of CFS channels. Consequently, this paper introduces simplified Direct Strength Method (DSM) equations, which account for plastic loads (Py) and elastic buckling loads (Pcr), integrating the aforementioned crucial parameters. Eight unique yield line models were devised, considering two-flange loadings: interior-two-flange (ITF) and end-two-flange (ETF), as well as the positioning of circular web openings, to compute the yield line length for Py. Similarly, new Pcr equations were also proposed. The reliability of the proposed equations was evaluated using a dataset of 236 test results sourced from the literature. This assessment aims to demonstrate the suitability of the equations for incorporation into future revisions of international design codes.

Probability Bracket Notation for Probability Modeling

Following Dirac’s notation in Quantum Mechanics (QM), we propose the Probability Bracket Notation (PBN), by defining a probability-bra (P-bra), P-ket, P-bracket, P-identity, etc. Using the PBN, many formulae, such as normalizations and expectations in systems of one or more random variables, can now be written in abstract basis-independent expressions, which are easy to expand by inserting a proper P-identity. The time evolution of homogeneous Markov processes can also be formatted in such a way. Our system P-kets are identified with probability vectors and our P-bra system is comparable with Doi’s state function or Peliti’s standard bra. In the Heisenberg picture of the PBN, a random variable becomes a stochastic process, and the Chapman–Kolmogorov equations are obtained by inserting a time-dependent P-identity. Also, some QM expressions in Dirac notation are naturally transformed to probability expressions in PBN by a special Wick rotation. Potential applications show the usefulness of the PBN beyond the constrained domain and range of Hermitian operators on Hilbert Spaces in QM all the way to IT.

Regularity preservation in Kolmogorov equations for non-Lipschitz coefficients under Lyapunov conditions

Regularity preservation in Kolmogorov equations for non-Lipschitz coefficients under Lyapunov conditions

Insights of infected Schwann cells extinction and inherited randomness in a stochastic model of leprosy

Investigating disease progression, transmission of infection and impacts of Multidrug Therapy (MDT) to inhibit demyelination in leprosy involves a certain amount of difficulty in terms of the in-built uncertain complicated and complex intracellular cell dynamical interactions. To tackle this scenario and to elucidate a more realistic, rationalistic approach of examining the infection mechanism and associated drug therapeutic interventions, we propose a four-dimensional ordinary differential equation-based model. Stochastic processes has been employed on this deterministic system by formulating the Kolmogorov forward equation introducing a transition state and the quasi-stationary distribution, exact distribution analysis have been investigated which allow us to estimate an expected time to extinction of the infected Schwann cells into the human body more prominently. Additionally, to explore the impact of uncertainty in the key intracellular factors, the stochastic system is investigated incorporating random perturbations and environmental noises in the disease dissemination, proliferation and reinfection rates. Rigorous numerical simulations validating the analytical outcomes provide us significant novel insights on the progression of leprosy and unravelling the existing major treatment complexities. Analytical experiments along with the simulations utilizing Monte-Carlo method and Euler–Maruyama scheme involving stochasticity predicts that the bacterial density is underestimated due to the recurrence of infection and suggests that maintaining a drug-efficacy rate in the range 0.6−0.8 would be substantially efficacious in eradicating leprosy.

Weak convergence of the split-step backward Euler method for stochastic delay integro-differential equations

In this paper, our primary objective is to discuss the weak convergence of the split-step backward Euler (SSBE) method, renowned for its exceptional stability when used to solve a class of stochastic delay integro-differential equations (SDIDEs) characterized by global Lipschitz coefficients. Traditional weak convergence analysis techniques are not directly applicable to SDIDEs due to the absence of a Kolmogorov equation. To bridge this gap, we employ modified equations to establish an equivalence between the SSBE method used for solving the original SDIDEs and the Euler–Maruyama method applied to modified equations. By demonstrating first-order strong convergence between the solutions of SDIDEs and the modified equations, we establish the first-order weak convergence of the SSBE method for SDIDEs. Finally, we present numerical simulations to validate our theoretical findings.

Error analysis of the fully Galerkin approximations for the nonlinear extended-Fisher–Kolmogorov equation

Error analysis of the fully Galerkin approximations for the nonlinear extended-Fisher–Kolmogorov equation

Computational Methods for Stationary and Nonstationary Fokker–Planck–Kolmogorov Equations with Random Coefficients

In this paper, methodological approaches for distribution functions of nonlinear stochastic differential equations (SDEs) with uncertain parameters are elaborated. The stationary and nonstationary Fokker–Planck–Kolmogorov (FPK) equations associated with the considered random equations are investigated. A semi-analytical approach based on the extended exponential closure handling the parameters uncertainty effects is presented. A new numerical method coupling the polynomial chaos with the differential quadrature method (DQM) is elaborated for the stationary case. A compact matrix formulation is established allowing considering a large number of generalized polynomial chaos. For nonstationary random FPK equations, the polynomial chaos is coupled with the quadrature discretization. A symmetric discretization of the main Fokker–Planck operator using a tensor product of the Hilbert state space and the Hilbert probabilistic space has resulted. The obtained eigenvalues and random eigenfunctions are used for a spectral decomposition. The time solution is obtained by the spectral expansion as well as by the random Euler stochastic method. The accuracy, effectiveness and advantages of the developed procedure in analyzing the stationary and nonstationary probabilistic solutions of nonlinear stochastic dynamical systems with uncertain parameters excited by a Gaussian white noise are demonstrated.

Observability inequality of backward stochastic heat equations with Lévy process for measurable sets and its applications

This paper endeavours to directly establish the observability inequality of backward stochastic heat equations involving a Lévy process for measurable sets. As an immediate application, we attain the approximate controllability of forward stochastic heat equations featuring a Lévy process. Subsequently, we embark upon the formulation of a nuanced optimization problem, encompassing both the determination of an optimal actuator location and its associated minimum norm control. This formulation is elegantly cast as a two-person zero-sum game problem. We then proceed to articulate a set of conditions–both sufficient and necessary–required for optimizing the solution through the prism of a Nash equilibrium. At last, we prove that the relaxed optimal solution is an optimal actuator location for the classical problem.