Class Of Parabolic Integro-differential Equations Research Articles

A Crank-Nicolson-type compact difference method with the uniform time step for a class of weakly singular parabolic integro-differential equations

This paper is concerned with an efficient numerical method for a class of parabolic integro-differential equations with weakly singular kernels. Due to the presence of the weakly singular kernel, the exact solution has singularity near the initial time t=0. A generalized Crank-Nicolson-type scheme for the time discretization is proposed by designing a product integration rule for the integral term, and a compact difference approximation is used for the space discretization. The proposed method is constructed on the uniform time mesh, but it can still achieve the second-order convergence in time for weakly singular solutions. The unconditional stability and convergence of the method is proved and the optimal error estimate in the discrete L2-norm is obtained. The error estimate shows that the method has the second-order convergence in time and the fourth-order convergence in space. The extension of the method to two-dimensional problems is also discussed. A simple comparison is made with several existing methods. Numerical results confirm the theoretical analysis result and show the effectiveness of the proposed method.

Global well-posedness for the Euler alignment system with mildly singular interactions

We consider the Euler alignment system with mildly singular interaction kernels. When the local repulsion term is of the fractional type, global in time existence of smooth solutions was proved in [, –]. Here, we consider a class of less singular interaction kernels and establish the global regularity of solutions as long as the interaction kernels are not integrable. The proof relies on modulus of continuity estimates for a class of parabolic integro-differential equations with a drift and mildly singular kernels.

Compact Alternating Direction Implicit Scheme for Integro-Differential Equations of Parabolic Type

In this paper, we present a fast and efficient numerical method to solve a class of parabolic integro-differential equations with weakly singular kernels, compact difference approach for spatial discretization and alternating direction implicit method in time, combined with second-order fractional quadrature rule suggested by Lubich approximating the integral term. The $$L^2$$ stability and convergence are derived. Two numerical examples with known exact solution are given to support the theoretical results.

Regularity for parabolic integro-differential equations with very irregular kernels

We prove H\"older regularity for a general class of parabolic integro-differential equations, which (strictly) includes many previous results. We present a proof which avoids the use of a convex envelop as well as give a new covering argument which is better suited to the fractional order setting. Our main result involves a class of kernels which may contain a singular measure, may vanish at some points, and are not required to be symmetric. This new generality of integro-differential operators opens the door to further applications of the theory, including some regularization estimates for the Boltzmann equation.

Finite element method for a class of parabolic integro-differential equations with interfaces

In this paper, convergence of finite element method for a class of parabolic integro-differential equations with discontinuous coefficients are analyzed. Optimal L2(L2) and L2 (H1) norms are shown to hold when the finite element space consists of piecewise linear functions on a mesh that do not require to fit exactly to the interface. Both continuous time and discrete time Galerkin methods are discussed for arbitrary shape but smooth interfaces.

An anisotropic nonconforming finite element scheme with moving grids for parabolic integro-differential equations

A Crank-Nicolson scheme based on nonconforming finite element with moving grids is discussed for a class of parabolic integro-differential equations under anisotropic meshes. The corresponding convergence analysis is presented and the error estimates are obtained by using the interpolation operator instead of the conventional elliptic projection which is an indispensable tool in the convergence analysis of traditional finite element methods in previous literature.

An $hp$-Version Discontinuous Galerkin Method for Integro-Differential Equations of Parabolic Type

We study the numerical solution of a class of parabolic integro-differential equations with weakly singular kernels. We use an $hp$-version discontinuous Galerkin (DG) method for the discretization in time. We derive optimal $hp$-version error estimates and show that exponential rates of convergence can be achieved for solutions with singular (temporal) behavior near $t=0$ caused by the weakly singular kernel. Moreover, we prove that by using nonuniformly refined time steps, optimal algebraic convergence rates can be achieved for the h-version DG method. We then combine the DG time-stepping method with a standard finite element discretization in space, and present an optimal error analysis of the resulting fully discrete scheme. Our theoretical results are numerically validated in a series of test problems.

Richardson extrapolation and defect correction of mixed finite element methods for integro-differential equations in porous media

Asymptotic error expansions in the sense of L ∞-norm for the Raviart-Thomas mixed finite element approximation by the lowest-order rectangular element associated with a class of parabolic integro-differential equations on a rectangular domain are derived, such that the Richardson extrapolation of two different schemes and an interpolation defect correction can be applied to increase the accuracy of the approximations for both the vector field and the scalar field by the aid of an interpolation postprocessing technique, and the key point in deriving them is the establishment of the error estimates for the mixed regularized Green’s functions with memory terms presented in R. Ewing at al., Int. J. Numer. Anal. Model 2 (2005), 301–328. As a result of all these higher order numerical approximations, they can be used to generate a posteriori error estimators for this mixed finite element approximation.

On a Class of Parabolic Integro-Differential Equations

On a Class of Parabolic Integro-Differential Equations

Global smooth solutions for a class of parabolic integrodifferential equations

The existence and uniqueness of smooth global large data solutions of a class of quasilinear partial integrodifferential equations in one space and one time dimension are proved, if the integral kernel behaves like t − α t^{-\alpha } near t = 0 t=0 with α > 2 / 3 \alpha > 2/3 . An existence and regularity theorem for linear equations with variable coefficients that are related to this type is also proved in arbitrary space dimensions and with no restrictions for α \alpha .

Stabilization of solutions for a class of parabolic integro-differential equations

Abstract : Integro-differential equations arise in the description of feed-back control systems, where the control variables are derived from filtered observations of the state or where the control mechanism possesses inertia. The author studies a model equation for a distributed control system (e.g., the state varies over some space-like domain) which contains also some diffusion effects and give conditions under which the state will tend to some limit, as time goes to infinity, regardless of the initial situation. The limit is shown to satisfy an elliptic differential equation. Convergence rates are also given; these show the slowing-down effect of a slow control mechanism on the convergence of the state variable. The problem under study can also be viewed as a natural extension of a type of reaction-diffusion equation that has received wide attention in the literature. (Author)