Linear Parabolic Differential Equations Research Articles

On the Cauchy-Dirichlet Problem in the Half Space for Parabolic SPDEs in Weighted Hoelder Spaces

We study the Cauchy-Dirichlet problem for a second order linear parabolic stochastic differential equation (SPDE) with constant coefficients in a half-space. Considering its solution as a function with values in a probability space and using the methods of deterministic partial differential equations, we establish the existence and uniqueness of a strong solution in Holder classes with weights.

Invariants of linear parabolic differential equations

The paper is dedicated to construction of invariants for the parabolic equation u t + a ( t , x ) u xx + b ( t , x ) u x + c ( t , x ) u = 0 . We consider the equivalence group given by point transformations and find all invariants up to seventh-order, i.e. the invariants involving the derivatives up to seventh-order of the coefficients a, b and c with respect to the independent variables t, x.

Fast Runge-Kutta approximation of inhomogeneous parabolic equations

The result after $N$ steps of an implicit Runge-Kutta time discretization of an inhomogeneous linear parabolic differential equation is computed, up to accuracy $\epsilon$, by solving only $$O\Big(\log N \log \frac1\epsilon \Big) $$ linear systems of equations. We derive, analyse, and numerically illustrate this fast algorithm.

Approximation of exact controllability problem involving parabolic differential equations

In this paper we examine computation of optimal control u* of the exact controllability problem (referred to as the constraint problem) governed by the following type of linear parabolic differential equations: (∂y/∂t) + Ay = u in Q y = 0 on ∑ y(0) = y0 on Ω where A is the second-order elliptic differential operator, Ω is a bounded domain in ℝk with smooth boundary ∂Ω, Q = (0, T) × Ω, ∑ = (0, T) × ∂Ω and T > 0. This is achieved by approximating u* through a sequence {un} of controls corresponding to unconstrained problems involving a penalty function arising from the controllability constraint.

A second-order Magnus-type integrator for nonautonomous parabolic problems

We analyse stability and convergence properties of a second-order Magnus-type integrator for linear parabolic differential equations with time-dependent coefficients, working in an analytic framework of sectorial operators in Banach spaces. Under reasonable smoothness assumptions on the data and the exact solution, we prove a second-order convergence result without unnatural restrictions on the time stepsize. However, if the error is measured in the domain of the differential operator, an order reduction occurs, in general. A numerical example illustrates and confirms our theoretical results.

Second order parabolic equations in Banach spaces with dynamic boundary conditions

In this paper, we exhibit a unified treatment of the mixed initial boundary value problem for second order (in time) parabolic linear differential equations in Banach spaces, whose boundary conditions are of a dynamical nature. Results regarding existence, uniqueness, continuous dependence (on initial data) and regularity of classical and strict solutions are established. Moreover, several examples are given as samples for possible applications.

On Cauchy–Dirichlet problem in half-space for parabolic SPDEs in weighted Hölder spaces

We study the Cauchy–Dirichlet problem for a second-order linear parabolic stochastic differential equation in the half-space with a zero-order noise term driven by a cylindrical Brownian motion. Considering its solution as a function with values in a probability space and using the methods of deterministic partial differential equations, we establish the existence and uniqueness of a strong solution in Hölder classes with weights.

On the Absolute Exponential Stability of Solutions of Systems of Linear Parabolic Differential Equations with One Delay

We investigate necessary conditions for the absolute exponential stability of a system of linear parabolic differential equations with one delay.

Absolute Exponential Stability of Solutions of Linear Parabolic Differential Equations with Delay

We find necessary and sufficient conditions for the absolute exponential stability of solutions of linear parabolic differential equations with delay in a pair of norms.

Numerical approximations of exact controllability problems by optimal control problems for parabolic differential equations

We consider the exact controllability problems for systems governed by linear parabolic differential equations. We prove that the solution of an associated optimal control problem converges to the solution of the exact controllability problem. The numerical solution of the exact controllability problem is then obtained by solving the optimal control problem numerically. Numerical experiments are presented.

NONWEAK/STRONG SOLUTIONS OF LINEAR AND NONLINEAR HYPERBOLIC AND PARABOLIC EQUATIONS RESULTING FROM A SINGLE CONSERVATION LAW

This paper presents an investigation of the numerical computations of nonweak/strong solutions of linear and nonlinear hyperbolic and parabolic differential and partial differential equations resulting from a single conservation law using C1p-version least squares finite element formulation (LSFEF) and C11p-version space time least squares finite element formulation (STLSFEF) for stationary and time dependent processes. It is demonstrated that with this approach it is possible to compute nonweak/strong solutions in the sense that the computed solutions possess the same orders of continuity in space and time as the strong solutions and satisfy nonweak (or residual) form of the governing differential equations (GDE's). Other benefits of this approach over weak solutions are also discussed and demonstrated. Stationary and time dependent convection-diffusion and Burgers equations are used as model problems.

On the Cauchy problem for parabolic SPDEs in Hölder classes

We study Cauchy’s problem for certain second-order linear parabolic stochastic differential equation (SPDE)driven by a cylindrical Brownian motion.Considering its solution as a function with values in a probability space and using the methods of deterministic partial differential equations, we establish the existence and uniqueness of a strong solution in Hölder classes.

Identifying unknown terms in hyperbolic and parabolic equations

We recover unknown source terms in nonlinear hyperbolic differential equations and in nonlinear parabolic integro-differential equations in one space variable under the assumption of knowing a first integral (in the hyperbolic case) or the value of the solution at a point inside the domain (in the parabolic case). For this class of problems we prove existence results in classes of smooth solutions. Moreover, for linear hyperbolic and parabolic differential equations in one space variable we recover some characteristic parameters.

Time-periodic solutions of linear parabolic differential equations

Time-periodic solutions of linear parabolic differential equations

Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in $ℝ^{n}$

We study existence, uniqueness, and smoothing properties of the solutions to a class of linear second order elliptic and parabolic differential equations with unbounded coefficients in $ℝ^n$. The main results are global Schauder estimates, which hold in s

The time-dependent Schrödinger equation at turning points

The Lagrange Manifold (WKB) formalism enables the determination of the asymptotic series solution of linear hyperbolic and parabolic differential equations at turning points. Here this formalism is applied to determine the asymptotic solution of the time-dependent Schrödinger equation at turning points. We also use this approach to study some physical phenomenology associated with the time-dependent Schrödinger equation at caustics, the locus of turning points.

Homogeneous reonomic contact transformations and path reparametrization in functional integrals with respect to quasimeasures

Homogeneous reonomic contact transformations and path reparametrizations in functional integrals with respect to quasimeasures are considered for linear fourth-order parabolic differential equations with coefficients explicitly depending on time. An integral relation between the Green’s functions is found for fourth-order differential equations that correspond to each other. In a special case, the differential operator of the transformed equation coincides with the Bol conformally covariant operator, which is well known in exactly integrable and conformal models.

Path Integral Transformations and Stochastic Processes

The path integral transformation that changes the evolution parameters of the paths is considered in the path integrals for the systems of two differential equations in one space dimension and in the path integrals related with the fourth-order linear parabolic differential equations. In both cases, the coefficients of differential equations have an explicit time dependence. By using the procedure of the stochastic change of time and phase-space transformation of the stochastic processes the integral relations between the Green functions of the corresponding differential equations have been obtained. As a consequence the generalization of the Shepp formula for the Green function is obtained. It is shown, that paths reparametrization in the higher-order path integrals leads to the Green function for the differential equation with the Bol operator.

State Elimination and Reachability of Infinite-Dimensional Time Varying Linear Systems

In this paper the approximate controllability (reachability) of a class of time varying infinite-dimensional linear systems is characterized by a generalized Triggiani's rank condition. It is supposed that the structure operators of the systems generate a finite-dimensional Lie algebra of unbounded operators and a vector space of bounded operators respectively. We also suppose the differentially-analytical independence of the time varying coefficients with respect to the basis of the Lie algebra and the vector space, respectively. Linear parabolic differential equations with time varying coefficients, associated to the Lie algebra of the symplectic group are also included in this class of problems.

Difference scheme for an initial-boundary value problem for linear coefficient-varied parabolic differential equation with a nonsmooth boundary layer function

In this paper, using nonuniform mesh and exponentially fitted difference method, a uniformly convergent difference scheme for an initial-boundary value problem of linear parabolic differential equation with the nonsmooth boundary layer function with respect to small parameter e is given, and error estimate and numerical result are also given.