Partial Integro-differential Systems Research Articles

Results on Neutral Partial Integrodifferential Equations Using Monch-Krasnosel’Skii Fixed Point Theorem with Nonlocal Conditions

In this theory, the existence of a mild solution for a neutral partial integrodifferential nonlocal system with finite delay is presented and proved using the techniques of the Monch–Krasnosel’skii type of fixed point theorem, a measure of noncompactness and resolvent operator theory. For this work, we have introduced some sufficient conditions to confirm the existence of the neutral partial integrodifferential system. An illustration of the derived results is offered at the end with a filter system corresponding to our existence result.

Uniqueness and continuous dependence for a viscoelastic problem with memory in domains with time dependent cracks

We study some hyperbolic partial integro-differential systems in domains with time dependent cracks. In particular, we give conditions on the cracks which imply the uniqueness of the solution with prescribed initial-boundary conditions, and its continuous dependence on the cracks.

Hierarchical Structures and Leadership Design in Mean-Field-Type Games with Polynomial Cost

This article presents a class of hierarchical mean-field-type games with multiple layers and non-quadratic polynomial costs. The decision-makers act in sequential order with informational differences. We first examine the single-layer case where each decision-maker does not have the information about the other control strategies. We derive the Nash mean-field-type equilibrium and cost in a linear state-and-mean-field feedback form by using a partial integro-differential system. Then, we examine the Stackelberg two-layer problem with multiple leaders and multiple followers. Numerical illustrations show that, in the symmetric case, having only one leader is not necessarily optimal for the total sum cost. Having too many leaders may also be suboptimal for the total sum cost. The methodology is extended to multi-level hierarchical systems. It is shown that the order of the play plays a key role in the total performance of the system. We also identify a specific range of parameters for which the Nash equilibrium coincides with the hierarchical solution independently of the number of layers and the order of play. In the heterogeneous case, it is shown that the total cost is significantly affected by the design of the hierarchical structure of the problem.

Stackelberg Mean-Field-Type Games with Polynomial Cost

This article presents a class of Stackelberg mean-field-type games with multiple leaders and multiple followers. The decision-makers act in sequential order with informational differences. The state dynamics is driven by jump-diffusion processes and the cost function is non-quadratic and has a polynomial structure. The structures of Stackelberg strategies and costs of the leaders and followers are given in a semi-explicit way in state-and- mean-field-type feedback form. A sufficiency condition is provided using an infinite dimensional partial integro-differential system. The methodology is extended to multi-level hierarchical systems. It is shown that not only the set of decision-makers per level matters but also the number of hierarchical levels plays a key role in the global performance of the system. We also identify specific range of parameters for which the Nash equilibrium coincides with the hierarchical solution independently of the number of layers and the order of play.

Approximate controllability of a fractional stochastic partial integro-differential systems via noncompact operators

Objective: in this article, we discuss the approximate controllability problems of a new class of fractional impulsive stochastic partial integro-differential systems in separable Hilbert spaces. Methods: by applying the fractional calculus, the measure of noncompactness, properties of fractional resolvent operators and fixed point theorems. Results: we prove our main results without the hypotheses of compactness on the operator generated by the linear part of systems. Instead we suppose that the nonlinear term only satisfies a weakly compactness condition. Conclusion: the approximate controllability for the control systems with noncompact operators is established. Finally, an example is given for the illustration of the obtained theoretical results.

Passage to the limit in a singularly perturbed partial integro-differential system

We study an initial–boundary value problem for a singularly perturbed system of partial integro-differential equations. We prove a theorem on the passage to the limit. The result is used to decrease the dimension of a virus evolution model. We construct an asymptotic solution by the Tikhonov–Vasil’eva boundary function method. The analytic results obtained are compared with a numerical study of the system.

Boundary control for a class of reaction-diffusion systems

Boundary control for a class of partial integro-differential systems with space and time dependent coefficients is considered. A control law is derived via the partial differential equation (PDE) backstepping. The existence of kernel equations is proved. Exponential stability of the closed-loop system is achieved. Simulation results are presented through figures.

Dynamics of Multiple-Choice Decision Making

Neuroscientists have carried out comprehensive experiments to reveal the neural mechanisms underlying the perceptual decision making that pervades daily life. These experiments have illuminated salient features of decision making, including probabilistic choice behavior, the ramping activity of decision-related neurons, and the dependence of decision time and accuracy on the difficulty of the task. Spiking network models have reproduced these features, and a two-dimensional mean field model has demonstrated that the saddle node structure underlies two-alternative decision making. Here, we reduced a spiking network model to an analytically tractable, partial integro-differential system and characterized not only multiple-choice decision behaviors but also the time course of neural activities underlying decisions, providing a mechanistic explanation for the observations noted in the experiments. First, we observed that a two-bump unstable steady state of the system is responsible for two-choice decision making, similar to the saddle node structure in the two-dimensional mean field model. However, for four-choice decision making, three types of unstable steady states collectively predominate the time course of the evolution from the initial state to the stable states. Second, the time constant of the unstable steady state can explain the fact that four-choice decision making requires a longer time than two-choice decision making. However, the quicker decision, given a stronger motion strength, cannot be explained by the time constant of the unstable steady state. Rather, the decision time can be attributed to the projection coefficient of the difference between the initial state and the unstable steady state on the eigenvector corresponding to the largest positive eigenvalue.

Stable periodic traveling waves for a predator–prey model with non-constant death rate and delay

In this paper we will consider a predator–prey model with a non-constant death rate and distributed delay, described by a partial integro-differential system. The main goal of this work is to prove that the partial integro-differential system has periodic orbitally asymptotically stable solutions in the form of periodic traveling waves; i.e. N( x, t) = N( σt − μ · x), P( x, t) = P( σt − μ · x), where σ > 0 is the angular frequency and μ is the vector number of the plane wave, which propagates in the direction of the vector μ with speed c = σ/∥ μ∥; and N( x, t) and P( x, t) are the spatial population densities of the prey and the predator species, respectively. In order to achieve our goal we will use singular perturbation’s techniques.

The existence of weak solutions to single porosity and simple dual-porosity models of two-phase incompressible flow

Abstract. It is shown that there exists a weak solution to a degenerate and singular elliptic-parabolic partial integro-differential system of equations. These equations model two-phase incompressible flow of immiscible fluids in either an ordinary porous medium or in a naturally fractured porous medium. The full model is of dual-porosity type, though the single porosity case is covered by setting the matrixto-fracture flow terms to zero. This matrix-to-fracture flow is modeled simply in terms of fracture quantities; that is, no distinct matrix equations arise. The equations are considered in a global pressure formulation that is justified by appealing to a physical relation between the degeneracy of the wetting fluid’s mobility and the singularity of the capillary pressure function. In this formulation, the elliptic and parabolic parts of the system are separated; hence, it is natural to consider various boundary conditions, including mixed nonhomogeneous, saturation dependent ones of the first three types. A weak solution is obtained as a limit of solutions to discrete time problems. The proof makes no use of the corresponding regularized system. The hypotheses required for various earlier results on single-porosity systems are weakened so that only physically relevant assumptions are made. In particular, the results cover the cases of a singular capillary pressure function, a pure Neumann boundary condition, and an arbitrary initial condition.

Simulation of batch crystallization

The closed mathematical model of a well-mixed batch crystallizer has been presented. This model takes into consideration crystals growth rate fluctuations producing an increased spread of crystal sizes. The numerical methods for solving partial integro-differential system of this model equations are proposed for the cases of when growth rate is expressible as product of supersaturation and size functions. Simultaneous population and mass balance equations have been calculated together with power law kinetics of nucleation and growth to determine crystal-size distribution functions (CSD), variation coefficients (CV), produced crystal sizes and numbers. The case of size-dependent crystal growth determined by sequential processes of diffusion and surface reaction is also investigated. The computer results are presented in the dimensionless form as functions on both diffusion and fluctuation Peclet numbers and dimensionless cooling rate. In particular, it has been established that the CV of weight CSD increases approximately twice due to fluctuations and becomes close to 40%; the CV also increases owing to the decrease of cooling rate and approaches 28%. Some formulae are suggested for predicting the kinetics of crystallization (or precipitation) and determining the kinetic parameters of crystals growth and nucleation.

Crystallization of salts from supersaturated solutions; Diffusion kinetics

The kinetics of a spontaneous isothermic crystallization is calculated for a growth rate inversely proportional to the crystal size. Formulas are derived for predicting the kinetics and determining the crystal growth and nucleation rate constants for both the power and Gibbs-Volmer-Weber nucleation laws. A numerical method for solving partial integro-differential system describing the diffusion crystallization is presented. The method is tested against a problem whose solution is known so that numerical accuracy can be ascertained.