On a McKean‐Vlasov Stochastic Integro‐differential Evolution Equation of Sobolev‐Type

Abstract

We establish the global existence and uniqueness of mild solutions for a class of first‐order abstract stochastic Sobolev‐type integro‐differential equations in a real separable Hilbert space in which we allow the nonlinearities at a given time t to depend not only on the state of the solution at time, t, but also on the corresponding probability distribution at time t. Results concerning the continuous dependence of solutions on the initial data and almost sure exponential stability, as well as an extension of the existence result to the case in which the classical initial condition is replaced by a so‐called nonlocal initial condition, are also discussed. Finally, an example is provided to illustrate the applicability of the general theory.     

On the solution of an initial‐boundary value problem that combines Neumann and integral condition for the wave equation

Numerical solution of hyperbolic partial differential equation with an integral condition continues to be a major research area with widespread applications in modern physics and technology. Many physical phenomena are modeled by nonclassical hyperbolic boundary value problems with nonlocal boundary conditions. In place of the classical specification of boundary data, we impose a nonlocal boundary condition. Partial differential equations with nonlocal boundary specifications have received much attention in last 20 years. However, most of the articles were directed to the second‐order parabolic equation, particularly to heat conduction equation. We will deal here with new type of nonlocal boundary value problem that is the solution of hyperbolic partial differential equations with nonlocal boundary specifications. These nonlocal conditions arise mainly when the data on the boundary can not be measured directly. Several finite difference methods have been proposed for the numerical solution of this one‐dimensional nonclassic boundary value problem. These computational techniques are compared using the largest error terms in the resulting modified equivalent partial differential equation. Numerical results supporting theoretical expectations are given. Restrictions on using higher order computational techniques for the studied problem are discussed. Suitable references on various physical applications and the theoretical aspects of solutions are introduced at the end of this article. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Boundary value problem for a global‐in‐time parabolic equation

The aim of this paper is to draw attention to an interesting semilinear parabolic equation that arose when describing the chaotic dynamics of a polymer molecule in a liquid. This equation is nonlocal in time and contains a term, called the interaction potential, that depends on the time‐integral of the solution over the entire interval of solving the problem. In fact, one needs to know the “future” in order to determine the coefficient in this term, that is, the causality principle is violated. The existence of a weak solution of the initial boundary value problem is proven. The interaction potential satisfies fairly general conditions and can have arbitrary growth at infinity. The uniqueness of this solution is established with restrictions on the length of the considered time interval.     

The Dirichlet problem for stable-like operators and related probabilistic representations

We study stochastic differential equations with jumps with no diffusion part, governed by a large class of stable-like operators, which may contain a drift term. For this class of operators, we establish the regularity of solutions to the Dirichlet problem up to the boundary as well as the usual stochastic characterization of these solutions. We also establish key connections between the recurrence properties of the jump process and the associated nonlocal partial differential operator. Provided that the process is positive (Harris) recurrent, we also show that the mean hitting time of a ball is a viscosity solution of an exterior Dirichlet problem.     

Computational Krylov‐based methods for large‐scale differential Sylvester matrix problems

In the present paper, we propose Krylov‐based methods for solving large‐scale differential Sylvester matrix equations having a low‐rank constant term. We present two new approaches for solving such differential matrix equations. The first approach is based on the integral expression of the exact solution and a Krylov method for the computation of the exponential of a matrix times a block of vectors. In the second approach, we first project the initial problem onto a block (or extended block) Krylov subspace and get a low‐dimensional differential Sylvester matrix equation. The latter problem is then solved by some integration numerical methods such as the backward differentiation formula or Rosenbrock method, and the obtained solution is used to build the low‐rank approximate solution of the original problem. We give some new theoretical results such as a simple expression of the residual norm and upper bounds for the norm of the error. Some numerical experiments are given in order to compare the two approaches.     

A sufficient condition for blowup of solutions to a class of pseudo‐parabolic equations with a nonlocal term

A sufficient condition for blowup of solutions to a class of pseudo‐parabolic equations with a nonlocal term is established in this paper. In virtue of the potential wells method, we first extend the results obtained by Xu and Su in [J. Funct. Anal., 264 (12): 2732‐2763, 2013] to the nonlocal case and describe successfully the behavior of solutions by using the energy functional, Nehari functional, and the ground state energy of the stationary equation. Sequently, we study the boundedness and convergency of any global solution. Finally, we achieve a criterion to guarantee the blowup of solutions without any limit of the initial energy.Copyright © 2015 John Wiley & Sons, Ltd.     

On the numerical solution of hyperbolic PDEs with variable space operator

The first and second order of accuracy in time and second order of accuracy in the space variables difference schemes for the numerical solution of the initial‐boundary value problem for the multidimensional hyperbolic equation with dependent coefficients are considered. Stability estimates for the solution of these difference schemes and for the first and second order difference derivatives are obtained. Numerical methods are proposed for solving the one‐dimensional hyperbolic partial differential equation. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2009     

A fractional splitting algorithm for nonoverlapping domain decomposition for parabolic problem

In this article we study the convergence of the nonoverlapping domain decomposition for solving large linear system arising from semi‐discretization of two‐dimensional initial value problem with homogeneous boundary conditions and solved by implicit time stepping using first and two alternatives of second‐order FS‐methods. The interface values along the artificial boundary condition line are found using explicit forward Euler's method for the first‐order FS‐method, and for the second‐order FS‐method to use extrapolation procedure for each spatial variable individually. The solution by the nonoverlapping domain decomposition with FS‐method is applicable to problems that requires the solution on nonuniform meshes for each spatial variable, which will enable us to use different time‐stepping over different subdomains and with the possibility of extension to three‐dimensional problem. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 609–624, 2002     

Implicit–explicit schemes for nonlinear nonlocal equations with a gradient flow structure in one space dimension

Nonlinear convection–diffusion equations with nonlocal flux and possibly degenerate diffusion arise in various contexts including interacting gases, porous media flows, and collective behavior in biology. Their numerical solution by an explicit finite difference method is costly due to the necessity of discretizing a local spatial convolution for each evaluation of the convective numerical flux, and due to the disadvantageous Courant–Friedrichs–Lewy (CFL) condition incurred by the diffusion term. Based on explicit schemes for such models devised in the study of Carrillo et al. a second‐order implicit–explicit Runge–Kutta (IMEX‐RK) method can be formulated. This method avoids the restrictive time step limitation of explicit schemes since the diffusion term is handled implicitly, but entails the necessity to solve nonlinear algebraic systems in every time step. It is proven that this method is well defined. Numerical experiments illustrate that for fine discretizations it is more efficient in terms of reduction of error versus central processing unit time than the original explicit method. One of the test cases is given by a strongly degenerate parabolic, nonlocal equation modeling aggregation in study of Betancourt et al. This model can be transformed to a local partial differential equation that can be solved numerically easily to generate a reference solution for the IMEX‐RK method, but is limited to one space dimension.     

Optimal control problem of the uncertain second‐order circuit based on first hitting criteria

First hitting criteria of a system are to initially achieve some performance indeces of the target state set. This paper primarily investigates the optimal control problem of the uncertain second‐order circuit based on first hitting criteria. First, considering time efficiency and different from the ordinary expected utility criterion over an infinite time horizon, two first hitting criteria which are reliability index and reliable time criteria are innovatively proposed. Second, assuming the circuit output voltage as an uncertain variable when the historical data is lacking, we better model the real circuit system with the uncertain second‐order differential equation which is essentially the uncertain fractional‐order differential equation. Then, based on the first hitting time theorem of the uncertain fractional‐order differential equation, the distribution function of the first hitting time under the second‐order circuit system is proposed and the uncertain second‐order circuit optimal control model (reliability index and reliable time‐based model) is transformed into corresponding crisp optimal problem. Lastly, analytic expressions of the optimal control for the reliability index model are obtained. Meanwhile, sufficient condition and guidance for parameters for the optimal solution of the reliable time‐based model are derived, and corresponding numerical examples are also given to demonstrate the fluctuation of our optimal solution for different parameters.     

A finite difference scheme for the nonlinear time-fractional partial integro-differential equation

In this paper, a finite difference scheme is proposed for solving the nonlinear time-fractional integro-differential equation. This model involves two nonlocal terms in time, ie, a Caputo time-fractional derivative and an integral term with memory. The existence of numerical solutions is shown by the Leray-Schauder theorem. And we obtain the discrete L₂ stability and convergence with second order in time and space by the discrete energy method. Then the uniqueness of numerical solutions is derived. Moreover, an iterative algorithm is designed for solving the derived nonlinear system. Numerical examples are presented to validate the theoretical findings and the efficiency of the proposed algorithm.     

Global Existence and blowup phenomenon for a 1D radiation hydrodynamics model problem

We consider the global existence of classical solutions and blowup phenomena for a spatially one‐dimensional radiation hydrodynamics model problem, which consists of a scalar Burgers‐type equation coupled with a nonlocal advection‐reaction equation for radiation intensity. The model can be seen as an extension of the well‐known Hamer model that includes additionally the effects of scattering. It is well‐known that the initial value problem for Burgers' equation cannot be solved classically as soon as the derivative of the initial datum is negative somewhere. For our model problem, there is a critical negative number such that if the spatial derivative of the initial function is larger than this number, the associated initial‐value problem admits a global classical solution. However, when the spatial derivative of the initial data is below another negative threshold number, the initial value problem can also not be solved classically. Moreover, when there does not exist a global classical solution, it is shown that the first spatial derivative of solution must blow up in finite time. The results of the paper generalize the findings of Kawashima and Nishibata for the Hamer model. Copyright © 2012 John Wiley & Sons, Ltd.     

Envelope viscosity solutions of first- and second-order PDEs with u-dependence

General envelope methods are introduced which may be used to embed equations with u-dependence into equations without solution dependence. Furthermore, these methods present a rigorous way to consider so-called nodal solutions. That is, if w(t,x,z) is the viscosity solution of some pde, the nodal solution of an associated pde is a function u(t,x) so that w(t,x,u(t,x)) = 0. Examples are given to first- and second-order pdes arising in optimal control, differential games, minimal time problems, scalar conservation laws, geometric-type equations, and forward backward stochastic control.     

Identification of a time-dependent source term for a time fractional diffusion problem

In this paper, we study an inverse problem of identifying a time-dependent term of an unknown source for a time fractional diffusion equation using nonlocal measurement data. Firstly, we establish the conditional stability for this inverse problem. Then two regularization methods are proposed to for reconstructing the time-dependent source term from noisy measurements. The first method is an integral equation method which formulates the inverse source problem into an integral equation of the second kind; and a prior convergence rate of regularized solutions is derived with a suitable choice strategy of regularization parameters. The second method is a standard Tikhonov regularization method and formulates the inverse source problem as a minimizing problem of the Tikhonov functional. Based on the superposition principle and the technique of finite-element interpolation, a numerical scheme is proposed to implement the second regularization method. One- and two-dimensional examples are carried out to verify efficiency and stability of the second regularization method.     

A multigrid solver to the Helmholtz equation with a point source based on travel time and amplitude

The Helmholtz equation arises when modeling wave propagation in the frequency domain. The equation is discretized as an indefinite linear system, which is difficult to solve at high wave numbers. In many applications, the solution of the Helmholtz equation is required for a point source. In this case, it is possible to reformulate the equation as two separate equations: one for the travel time of the wave and one for its amplitude. The travel time is obtained by a solution of the factored eikonal equation, and the amplitude is obtained by solving a complex‐valued advection–diffusion–reaction equation. The reformulated equation is equivalent to the original Helmholtz equation, and the differences between the numerical solutions of these equations arise only from discretization errors. We develop an efficient multigrid solver for obtaining the amplitude given the travel time, which can be efficiently computed. This approach is advantageous because the amplitude is typically smooth in this case and, hence, more suitable for multigrid solvers than the standard Helmholtz discretization. We demonstrate that our second‐order advection–diffusion–reaction discretization is more accurate than the standard second‐order discretization at high wave numbers, as long as there are no reflections or caustics. Moreover, we show that using our approach, the problem can be solved more efficiently than using the common shifted Laplacian multigrid approach.     

Stochastic functional differential equations modelling materials with selective recall

We study a problem in stochastic functional differential equations which, in addition to a standard one-one-parameter noise term involves a random perturbation of the memory. This problem can also be regarded as a first order hyperbolic system of stochastic partial differential equations with given initial data and nonlocal boundary data. Existence and uniqueness of a solution is established and the generator of the associated Markov process is analyzed. Thereafter, for two model problems arising from first- and second-order integro-differential equations suggested by physical applications we establish asymptotic stability in probability of the associated stochastic processes.     

Solvability of nonlinear boundary-value problems arising in modeling plasma diffusion across a magnetic field and its equilibrium configurations

We study the simplest one-dimensional model of plasma density balance in a tokamak type system, which can be reduced to an initial boundary-value problem for a second-order parabolic equation with implicit degeneration containing nonlocal (integral) operators. The problem of stabilizing nonstationary solutions to stationary ones is reduced to studying the solvability of a nonlinear integro-differential boundary-value problem. We obtain sufficient conditions for the parameters of this boundary-value problem to provide the existence and the uniqueness of a classical stationary solution, and for this solution we obtain the attraction domain by a constructive method.Translated from Matematicheskie Zametki, vol. 77, no. 2, 2005, pp. 219–234.Original Russian Text Copyright © 2005 by G. A. Rudykh, A. V. Sinitsyn.This revised version was published online in April 2005 with a corrected issue number.     

Composite spectral method for solution of the diffusion equation with specification of energy

Many physical subjects are modeled by nonclassical parabolic boundary value problems with nonlocal boundary conditions replacing the classic boundary conditions. In this article, we introduce a new numerical method for solving the one‐dimensional parabolic equation with nonlocal boundary conditions. The approximate proposed method is based upon the composite spectral functions. The properties of composite spectral functions consisting of terms of orthogonal functions are presented and are utilized to reduce the problem to some algebraic equations. The method is easy to implement and yields very accurate result. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Positive and unbounded solution of stochastic delayed evolution equations

This article is concerned with the blowup phenomenon of stochastic delayed evolution equations. We first establish the sufficient condition to ensure the existence of a unique nonnegative solution of stochastic parabolic equations. Then the problem of blow-up solutions in mean L^q-norm, q ? 1, in a finite time is considered. The main aim in this article is to investigate the effect of time delay and stochastic term. A new result shows that the stochastic delayed term can induce singularities.     

A robust finite difference scheme for strongly coupled systems of singularly perturbed convection‐diffusion equations

This paper is devoted to developing an Il'in‐Allen‐Southwell (IAS) parameter‐uniform difference scheme on uniform meshes for solving strongly coupled systems of singularly perturbed convection‐diffusion equations whose solutions may display boundary and/or interior layers, where strong coupling means that the solution components in the system are coupled together mainly through their first derivatives. By decomposing the coefficient matrix of convection term into the Jordan canonical form, we first construct an IAS scheme for 1D systems and then extend the scheme to 2D systems by employing an alternating direction technique. The robustness of the developed IAS scheme is illustrated through a series of numerical examples, including the magnetohydrodynamic duct flow problem with a high Hartmann number. Numerical evidence indicates that the IAS scheme appears to be formally second‐order accurate in the sense that it is second‐order convergent when the perturbation parameter ϵ is not too small and when ϵ is sufficiently small, the scheme is first‐order convergent in the discrete maximum norm uniformly in ϵ.