Search Results (28)

The neutron survival probability (and related quantities including probabilities of extinction and initiation) is a central element of the broader stochastic theory of neutron populations and finds application in fields including reactor start-up, analysis of reactor power bursts and criticality accidents, and safeguards. In a full neutron transport formulation, the equation governing the single-neutron survival probability is a backward or adjoint-like integro-partial differential equation with the added complexity of being highly nonlinear. Analogous formulations of this equation exist in the context of many approximate theories of neutron transport, with the point kinetics formulation having received significant theoretical attention since the 1940s. This work continues this tradition by providing a novel analysis of the single-neutron survival probability equation using the tools of boundary layer theory. The analysis reveals that the “fully dynamic” solution of the single-neutron survival probability equation—and some key probability distributions derived from it—may be cast as a singular perturbation around the underlying quasi-static single-neutron probability of initiation. In this perturbation solution, the expansion parameter is the ratio of the neutron generation time to a macroscopic time scale characterizing the overall system evolution; this interpretation illuminates some of the fundamental structural aspects of neutron survival phenomena. Full article

This study delves into the spatial characteristics of solutions for a specific class of evolution equations that incorporate biharmonic operators. The process begins with the construction of an energy function. Subsequently, by employing an integro-differential inequality method, it is deduced that this energy function satisfies an integro-differential inequality. Resolving this inequality enables us to establish an estimate for the spatial decay of the solution. Ultimately, the finding affirms that the spatial exponential decay is reminiscent of Saint-Venant-type estimates. Full article

As a crucial modeling tool for stochastic financial markets, the Lévy risk model effectively characterizes the evolution of risks during enterprise operations. Through dynamic evaluation and quantitative analysis of risk indicators under specific dividend- distribution strategies, this model can provide theoretical foundations for optimizing corporate capital allocation. Addressing the inadequate adaptability of traditional single-period threshold strategies in time-varying market environments, this paper proposes a dividend strategy based on multiperiod dynamic threshold adjustments. By implementing periodic modifications of threshold parameters, this strategy enhances the risk model’s dynamic responsiveness to market fluctuations and temporal variations. Within the framework of the spectrally negative Lévy risk model, this paper constructs a stochastic control model for multiperiod threshold dividend strategies. We derive the integro-differential equations for the expected present value of aggregate dividend payments before ruin and the Gerber–Shiu function, respectively. Combining the methodologies of the discounted increment density, the operator introduced by Dickson and Hipp, and the inverse Laplace transforms, we derive the explicit solutions to these integro-differential equations. Finally, numerical simulations of the related results are conducted using given examples, thereby demonstrating the feasibility of the analytical method proposed in this paper. Full article

Evolution equations with fractional-time derivatives and singular memory kernels are used for modeling phenomena exhibiting hereditary properties, as they effectively incorporate memory effects into their formulation. Time-fractional partial integro-differential equations (FPIDEs) represent a significant class of such evolution equations and are widely used in diverse scientific and engineering fields. In this study, we use the $\mathrm{sinc}$-collocation and iterative Laplace transform methods to solve a specific FPIDE with a weakly singular kernel. Specifically, the $\mathrm{sinc}$-collocation method is applied to discretize the spatial domain, while a combination of numerical techniques is utilized for temporal discretization. Then, we prove the convergence analytically. To compare the two methods, we provide two examples. We notice that both the $\mathrm{sinc}$-collocation and iterative Laplace transform methods provide good approximations. Moreover, we find that the accuracy of the methods is influenced by fractional order $\alpha \in (0,1)$ and the memory-kernel parameter $\beta \in (0,1)$. We observe that the error decreases as $\beta$ increases, where the kernel becomes milder, which extends the single-value study of $\beta =1/2$ in the literature. Full article

In this paper, we study the existence of solutions for fractional integrodifferential equations with Hilfer derivatives. We establish some new existence theorems for mild solutions by using Schaefer’s fixed-point theorem, a measure of noncompactness, and the resolvent operators associated with almost sectorial operators. Our results improve and extend many known results in the relevant references by removing some strong assumptions. Furthermore, we propose new nonlocal initial conditions for Hilfer evolution equations and study the existence of mild solutions to nonlocal problems. Full article

Population balance equations may be employed to handle a wide variety of particle processes has certainly received unprecedented attention, but the absence of explicit exact solutions necessitates the use of numerical approaches. In this paper, a (2 + 1) dimensional population balance equation with aggregation, nucleation, growth and breakage processes is solved analytically by use of the methods of scaling transformation group, observation and trial function. Symmetries, reduced equations, invariant solutions, exact solutions, existence of solutions, evolution analysis of dynamic behavior for solutions are presented. The exact solutions obtained can be compared with the numerical scheme. The obtained results also show that the method of scaling transformation group can be applied to study integro-partial differential equations. Full article

This article is mainly concerned with the approximate controllability for some semi-linear fractional integro-differential impulsive evolution equations of order $1<\alpha <2$ with delay in Banach spaces. Firstly, we study the existence of the $PC$-mild solution for our objective system via some characteristic solution operators related to the Mainardi’s Wright function. Secondly, by using the spatial decomposition techniques and the range condition of control operator B, some new results of approximate controllability for the fractional delay system with impulsive effects are obtained. The results cover and extend some relevant outcomes in many related papers. The main tools utilized in this paper are the theory of cosine families, fixed-point strategy, and the Grönwall-Bellman inequality. At last, an example is given to demonstrate the effectiveness of our research results. Full article

In this paper, we analytically study the two-dimensional unsteady irrotational flow of an ideal incompressible fluid in a half-plane whose boundary is assumed to be a linear sink. It is shown that the nonlinear evolution of perturbations of the initial uniform flow is described by a one-dimensional integro-differential equation, which can be considered as a nonlocal generalization of the Hopf equation. This equation can be reduced to a system of ordinary differential equations (ODEs) in the cases of spatially localized or spatially periodic perturbations of the velocity field. In the first case, ODEs describe the motion of a system of interacting virtual point vortex-sinks/sources outside the flow domain. In the second case, ODEs describe the evolution of a finite number of harmonics of the velocity field distribution; this is possible due to the revealed property of the new equation that the interaction of initial harmonics does not lead to generation of new ones. The revealed reductions made it possible to effectively study the nonlinear evolution of the system, in particular, to describe the effect of nonlinearity on the relaxation of velocity field perturbations. It is shown that nonlinearity can significantly reduce the relaxation rate by more than 1.5 times. Full article

We consider a simple model for QCD dynamics in which DGLAP integro-differential equation may be solved analytically. This is a gauge model which possesses dominant evolution of gauge boson (gluon) distribution and in which the gauge coupling does not run. This may be $\mathcal{N}=4$ supersymmetric gauge theory with softly broken supersymmetry, other finite supersymmetric gauge theory with a lower level of supersymmetry, or topological Chern–Simons field theories. We maintain only one term in the splitting function of unintegrated gluon distribution and solve DGLAP analytically for this simplified splitting function. The solution is found using the Cauchy integral formula. The solution restricts the form of the unintegrated gluon distribution as a function of momentum transfer and of Bjorken x. Then, we consider an almost realistic splitting function of unintegrated gluon distribution as an input to DGLAP equation and solve it by the same method which we have developed to solve DGLAP equation for the toy-model. We study a result obtained for the realistic gluon distribution and find a singular Bessel-like behavior in the vicinity of the point $x=0$ and a smooth behavior in the vicinity of the point $x=1$. Full article

At present, a significant part of oil is extracted from difficult-to-develop reservoirs with low permeability. Hydraulic fracturing is one of the most important methods of production stimulation. Scientific articles do not describe the connections between the flow-rate in the well and the change in pressure in the hydraulic fracture or between the changing pressure in the well and the pressure in the hydraulic fracture, except in some cases of constant fluid-flow in the well and constant production. We obtained both the exact analytical solutions and the simple approximate solutions which describe the connection between the well-fluid flow-rate and the pressure evolution in a fracture. The work has also solved the inverse problem: how to determine the parameters of a hydraulic fracture, knowing the change in well flow-rate and the change in fluid-flow. The results show good comparison with practical data. Full article

In this paper, we derive the boundary integral equation (BIE), a single integrodifferential equation governing the evolutionary behavior of the interface function, paying special attention to the nonlinear liquidus equation and atomic kinetics. As a result, the BIE is found for a thermodiffusion problem of binary melt crystallization with convection. Analyzing this equation coupled with the selection criterion for a stationary dendritic growth in the form of a parabolic cylinder, we show that nonlinear effects stemming from the liquidus equation and atomic kinetics play a decisive role. Namely, the dendrite tip velocity and diameter, respectively, become greater and lower with the increasing deviation of the liquidus equation from a linear form. In addition, the dendrite tip velocity can substantially change with variations in the power exponent of the atomic kinetics. In general, the theory under consideration describes the evolution of a curvilinear crystallization front, as well as the growth of solid phase perturbations and patterns in undercooled binary melts at local equilibrium conditions (for low and moderate Péclet numbers). In addition, our theory, combined with the unsteady selection criterion, determines the non-stationary growth rate of dendritic crystals and the diameter of their vertices. Full article

This study addresses a magneto-viscoelasticity problem, considering the one-dimensional case. The system under investigation is given by the coupling a non-linear partial differential equation with a linear integro-differential equation. The system models a viscoelastic body whose mechanical behavior, described by the linear integro-differential equation, is also influenced by an external magnetic field. The model here investigated aims to consider the concomitance of three different effects: viscoelasticity, aging and magnetization. In particular, the viscoelastic behavior is represented via an integro-differential equation whose kernel characterizes the properties of the material. In a viscoelastic material subject to the effects of aging, all changes in the response to deformation are due not only to the intrinsic memory of the material but also to deterioration with the age of the material itself. Thus, the relaxation function is not assumed to depend on the two times, present and past, via their difference, but to depend on both the present and past times as two independent variables. The sensibility to an external magnetic field is modeled by a non-linear partial differential equation taking its origin in the Landau–Lifschitz magnetic model. This investigation is part of a long-term research project aiming to provide new insight in the study of materials with memory and, in particular, viscoelastic materials. Specifically, the classical model of viscoelastic body introduced by Boltzmann represents the fundamental base from which a variety of generalizations have been considered in the literature. In particular, the effects on the viscoelastic body due to interaction with an external magnetic field are studied. The new aspect under investigation is the combined presence of the external magnetic field with the effect of aging. Indeed, the coupling of viscoelasticity, which takes into account the deterioration of the material with time, with the presence of an external magnetic field, was never considered in previous research. An existence and uniqueness result is proved under suitable regularity assumptions. Full article

A method is presented for generating free energies relating to nonlinear constitutive equations with memory from known free energies associated with hereditary linear theories. Some applications to viscoelastic solids and hereditary electrical conductors are presented. These new free energies are then used to obtain estimates for nonlinear integro-differential evolution problems describing the behavior of nonlinear plasmas with memory. Full article

In this paper, by using the cosine family theory, measure of non-compactness, the Mönch fixed point theorem and the method of estimate step by step, we establish the existence theorems of mild solutions for fractional impulsive integro-differential evolution equations of order $1<\beta \le 2$ with nonlocal conditions in Banach spaces under some weaker conditions. The results obtained herein generalizes and improves some known results. Finally, an example is presented for the demonstration of obtained results. Full article

Motivated by a wide range of applications in various fields of physics and materials science, we consider a generalized approach to the evolution of a polydisperse ensemble of spherical particles in metastable media. An integrodifferential system of governing equations, consisting of a kinetic equation for the particle-size distribution function (Fokker–Planck type equation) and a balance equation for the temperature (concentration) of a metastable medium, is formulated. The kinetic equation takes into account fluctuations in the growth/reduction rates of individual particles, the velocity of particles in a spatial direction, the withdrawal of particles of a given size from the metastable medium, and their source/sink term. The heat (mass) balance equation takes into account the growth/reduction of particles in a metastable system as well as heat (mass) exchange with the environment. A generalized system of equations describes various physical and chemical processes of phase transformations, such as the growth and dissolution of crystals, the evaporation of droplets, the boiling of liquids and the combustion of a polydisperse fuel. The ways of analytical solution of the formulated integrodifferential system of equations based on the saddle-point technique and the separation of variables method are considered. The theory can be applied when describing the evolution of an ensemble of particles at the initial and intermediate stages of phase transformation when the distances between the particles are large enough, and interactions between them can be neglected. Full article