Integro-differential Equation Of Parabolic Type Research Articles

Long-time numerical solution of a parabolic equation with memory

Long-time stability and convergence properties of two time-discretization methods for an integro-differential equation of parabolic type are studied. The methods are based on the standard backward Euler and second-order backward differencing methods. The memory term is approximated by a quadrature rule, with emphasis on such rules with reduced computational memory requirements. Discretization of the spatial partial differential operators by the finite element method is also considered.

Optimal Control for Integrodifferential Equations of Parabolic Type

Quadratic control problems for integrodifferential equations of parabolic type are considered. A state-space representation of the system is obtained by choosing an appropriate product space. By using the standard method based on Riccati equation, a unique optimal control over a finite horizon and under a stabilizability condition is obtained and the quadratic problem over an infinite horizon is solved. It is shown that the approach is also valid for some integrodifferential equations of different types. Two examples covered by the model are given.

Global Existence of Solutions of Volterra Integrodifferential Equations of Parabolic Type

Sufficient conditions for the global existence of a strong solution of the equation u t ( t, x) = ∫ t 0 k( t − s) σ( u x ( s, x)) x ds + f( t, x) are given. The kernel k satisfies R k̂( z) ≥ κ | I k̂( z) | and σ is increasing with sup {σ′ ( p) } < 1 + 2κ([formula] + κ).

On Fully Discrete Galerkin Approximations for Partial Integro-Differential Equations of Parabolic Type

The subject of this work is the application of fully discrete Galerkin finite element methods to initial-boundary value problems for linear partial integro-differential equations of parabolic type. We investigate numerical schemes based on the Padé discretization with respect to time and associated with certain quadrature formulas to approximate the integral term. A preliminary error estimate is established, which contains a term related to the quadrature rule to be specified. In particular, we consider quadrature rules with sparse quadrature points so as to limit the storage requirements, without sacrificing the order of overall convergence. For the backward Euler scheme, the Crank-Nicolson scheme, and a third-order (1,2) Padé-type scheme, the specific quadrature rules analyzed are based on the rectangular, the trapezoidal, and Simpson’s rule. For all the schemes studied, optimal-order error estimates are obtained in the case that the solution of the problem is smooth enough. Since this is important for our error analysis, we also discuss the regularity of the exact solutions of our equations. High-order regularity results with respect to both space and time are given for the solution of problems with smooth enough data.

On fully discrete Galerkin approximations for partial integro-differential equations of parabolic type

The subject of this work is the application of fully discrete Galerkin finite element methods to initial-boundary value problems for linear partial integro-differential equations of parabolic type. We investigate numerical schemes based on the Padé discretization with respect to time and associated with certain quadrature formulas to approximate the integral term. A preliminary error estimate is established, which contains a term related to the quadrature rule to be specified. In particular, we consider quadrature rules with sparse quadrature points so as to limit the storage requirements, without sacrificing the order of overall convergence. For the backward Euler scheme, the Crank-Nicolson scheme, and a third-order (1,2) Padé-type scheme, the specific quadrature rules analyzed are based on the rectangular, the trapezoidal, and Simpson’s rule. For all the schemes studied, optimal-order error estimates are obtained in the case that the solution of the problem is smooth enough. Since this is important for our error analysis, we also discuss the regularity of the exact solutions of our equations. High-order regularity results with respect to both space and time are given for the solution of problems with smooth enough data.

Finite element approximation of a parabolic integro-differential equation with a weakly singular kernel

We give error estimates for the numerical solution by means of the Galerkin finite element method of an integro-differential equation of parabolic type with a memory term containing a weakly singular kernel. Optimal-order estimates are shown for spatially semidiscrete and completely discrete methods. Special attention is paid to the regularity of the exact solution.

The stability of Ritz-Volterra projection and error estimates for finite element methods for a class of integro-differential equations of parabolic type

In this paper we first study the stability of Ritz-Volterra projection (see below) and its maximum norm estimates, and then we use these results to derive some $L^\infty$ error estimates for finite element methods for parabolic integro-differential equations.

Numerical Solution of Semilinear Integrodifferential Equations of Parabolic Type with Nonsmooth Data

Error estimates are derived for the discretization, in space by finite elements and in time by the backward Euler method, of a parabolic equation with a semilinear memory term. Special attention is paid to the case of nonsmooth data and the computation of the memory term by quadrature.

Smooth solutions for an integro-differential equation of parabolic type

Smooth solutions for an integro-differential equation of parabolic type

Error estimates for semidiscrete finite element methods for parabolic integro-differential equations

The purpose of this paper is to attempt to carry over known results for spatially discrete finite element methods for linear parabolic equations to integro-differential equations of parabolic type with an integral kernel consisting of a partial differential operator of order β ≤ 2 \beta \leq 2 . It is shown first that this is possible without restrictions when the exact solution is smooth. In the case of a homogeneous equation with nonsmooth initial data v , v ∈ L 2 v,v \in {L_2} , optimal O ( h r ) O({h^r}) convergence for positive time is possible in general only if r ≤ 4 − β r \leq 4 - \beta . This depends on the fact that the exact solution is then only in H 4 − β {H^{4 - \beta }} .

Linear growth of a liquid droplet divided from its vapour by a “soap bubble”-like fluid interface

The theory proposed in [1,3] is particularized to describe the spherically symmetric growth, in the neighbourhood of an equilibrium state far from the critical temperature, of a liquid incompressible droplet surrounded by its vapour when the interface between the phases behaves as a fluid membrane which resembles a soap bubble. A free moving boundary problem for an integro-differential equation of parabolic type is deduced in which the second-order time derivative of the radius R(t) of the droplet appears, together with a volume source term which depends on the history of R.

Solvability on the real line of a class of linear volterra integrodifferential equations of parabolic type

We consider an abstract parabolic integrodifferential equation with infinite delay in general Banach space X: $$u'(t) = Au(t) + \int\limits_{ - \infty }^t {K(t - s)u(s)ds + f(t),t \in R}$$ (*) where A: D(A) ⊂ X → X generates an analytic semigroup, and K(t) e L(D(A), X) for every t ⩾ 0. Under suitable assumptions on the kernel K, we extend to equation (*) the well known results about bounded solutions, periodic solutions, and solutions with exponential growth, of the abstract parabolic equation: $$v'(t) = Av(t) + f(t),t \in R.$$ (**)

Non-classicalH 1 projection and Galerkin methods for non-linear parabolic integro-differential equations

In this paper the Galerkin method is analyzed for the following nonlinear integro-differential equation of parabolic type: $$c(u)u_t = \nabla \cdot \{ a(u)\nabla u + \int_0^t {b(x, t, r, u(x, r))} \nabla u(x, r) dr\} + f (u)$$ Optimal L 2 error estimates for Crank-Nicolson and extrapolated Crank-Nicolson approximations are derived by using a non-classicalH 1 projection associated with the above equation. Both schemes result in procedures which are second order correct in time, but the latter requires the solution of a linear algebraic system only once per time step.

The classical solutions for nonlinear parabolic integro- differential equations

In this paper, we consider the solvability in the classical sense of a class of nonlinear one-dimensional integrodifferential equations of parabolic type. The motivation for studying this problem comes from the many physical models in such fields as heat transfer, nuclear reactor dynamics and thermoelasticity. One of the characteristics of this kind of equation is that the maximum principle is no longer valid in general. We combine the integral estimate method and Schauder estimate theory for a linear parabolic equation to derive an a priori bound for the solution of our nonlinear problem in the norm of the Banach space C 2 (QT). The method of continuity then allows us to establish the global existence of the solution. For completeness, we also demonstrate the uniqueness and continuous dependence of the solution.

Time Discretization of an Integro-Differential Equation of Parabolic Type

The stability and convergence properties of two time discretizations of an integro-differential equation of parabolic type are studied. The methods reduce to the backward-Euler and Crank–Nicolson methods if the integral term is absent. The integral term is approximated in each case by a quadrature rule with relatively high-order truncation error, so that a relatively large time step can be used for the quadrature, in order to reduce the memory and computational requirements of the method.

C α-regularity for non-autonomous linear integrodifferential equations of parabolic type

Linear integrodifferential equations in general Banach space are studied and applications are given to linear integrodifferential partial differential equations.

Asymptotic stability of an integro-differential equation of parabolic type

Asymptotic stability of an integro-differential equation of parabolic type

Existence and regularity for a class of integrodifferential equations of parabolic type

The method we use rests on the inversion of the formal Laplace transform of R and allows one to analyze the regularity of the solution U. Problem (1.1) has been studied by several authors using various methods (see, for instance, [l-18, 211). The Laplace transform approach presented here has been used in [S, 7-9, 14, 153. In our previous papers [S, 7-93 we have in fact analyzed a more general situation while here we take full 36 0022-247X/85 $3.00

Estimates of the solutions of two-point boundary-value problems for systems of first-order integrodifferential equations and the method of lines

One proves theorems on the estimates of the solutions of the systems of first-order integrodifferential equations with the boundary conditions On the basis of these theorems, one suggests a method for estimating the norms of integrodifferential equations by the method of the lines for the solutions of the periodic boundary-value problems for second-order integrodifferential equations of parabolic type. On the basis of the established theorem, on the solvability and on the estimate of the solution of the nonlinear equation $$Tx + F\left( x \right) = 0$$ in a Banach space X, where T is a linear unbounded operator, one investigates the convergence of the method of lines for solving the periodic boundary-value problem for a second-order nonlinear integrodifferential equation of parabolic type.

Uniqueness of the solution of a tangential derivative problem for an infinite system of nonlinear integro-differential equations of parabolic type

Uniqueness of the solution of a tangential derivative problem for an infinite system of nonlinear integro-differential equations of parabolic type