Linear Parabolic Differential Equations Research Articles

Approximate controllability of linear parabolic equation with memory

In this paper, we consider an optimal control problem governed by linear parabolic differential equations with memory. Under the assumption that the corresponding linear parabolic differential equation without memory term is approximately controllable, it is shown that the set of approximate controls is nonempty. The problem is first viewed as a constrained optimal control problem, and then it is approximated by an unconstrained problem with a suitable penalty function. The optimal pair of the constrained problem is obtained as the limit of the optimal pair sequence of the unconstrained problem. The result is proved by using the theory of strongly continuous semigroups and the Banach fixed point theorem. The approximation theorems, which guarantee the convergence of the numerical scheme to the optimal pair sequence, are also proved. Finally, we present an algorithm to compute an optimal control with a numerical example.

Delay-dependent elliptic reconstruction and optimal 𝐿^{∞}(𝐿²) a posteriori error estimates for fully discrete delay parabolic problems

We derive optimal order a posteriori error estimates for fully discrete approximations of linear parabolic delay differential equations (PDDEs), in the L ∞ ( L 2 ) L^\infty (L^2) -norm. For the discretization in time we use Backward Euler and Crank-Nicolson methods, while for the space discretization we use standard conforming finite element methods. A novel space-time reconstruction operator is introduced, which is a generalization of the elliptic reconstruction operator, and we call it as delay-dependent elliptic reconstruction operator. The related a posteriori error estimates for the delay-dependent elliptic reconstruction play key roles in deriving optimal order a posteriori error estimates in the L ∞ ( L 2 ) L^\infty (L^2) -norm. Numerical experiments verify and complement our theoretical results.

Numerical method for a boundary value problem for a linear singularly perturbed parabolic delay differential equation of reaction-diffusion type with discontinuous source term

A singularly perturbed boundary value problem for a linear parabolic second order delay differential equation of reaction-diffusion type with discontinuous source term is considered in the rectangle domain Ω = {(x, t) : 0 < x < 2,0 < t ⩽ T}. Discontinuity occurs in the source term at (d, t) < ∈ Ω. As the highest order space derivative is multiplied by a singular perturbation parameter, the components of the solution exhibit boundary layers at (x,t) = (0,t) and (x,t) = (2,t) and interior layers at (x, t) = (1, t) and/or at (x,t) = (d,t) and (x,t) = (1 + d, t) with respect to the position of (d, t) in Ω. A numerical method which uses classical finite difference scheme on a Shishkin piecewise uniform mesh is suggested to approximate the solution. The method is proved to be first order convergent uniformly for all the values of singular perturbation parameters. Numerical illustrations are presented so that the theoretical results are supported.

Linear parabolic equations with strong boundary degeneration

As an application of the theory of linear parabolic differential equations on noncompact Riemannian manifolds, developed in earlier papers, we prove a maximal regularity theorem for nonuniformly parabolic boundary value problems in Euclidean spaces. The new feature of our result is the fact that—besides of obtaining an optimal solution theory—we consider the ‘natural’ case where the degeneration occurs only in the normal direction.

Numerical Solution for One Dimensional Linear Types of Parabolic Partial Differential Equation and Application to Heat Equation

In this paper, present solution of one-dimensional linear parabolic differential equation by using Forward difference, backward difference, and Crank Nicholson method. First, the solution domain is discretized using the uniform mesh for step length and time step. Then applying the proposed method, we discretize the linear parabolic equation at each grid point and then rearranging the obtained discretization scheme we obtain the system of equation generated with tri-diagonal coefficient matrix. Now applying inverse matrixes method and writing MATLAB code for this inverse matrixes method we obtain the solution of one-dimensional linear parabolic differential equation. The stability of each scheme analyses by using Von-Neumann stability analysis technique. To validate the applicability of the proposed method, two model example are considered and solved for different values of mesh sizes in both directions. The convergence has been shown in the sense of maximum absolute error (E∞) and Root mean error (E2). Also, condition number (K(A)) and Order of convergence are calculated. The stability of this Three class of numerical method is also guaranteed and also, the comparability of the stability of these three methods is presented by using the graphical and tabular form. The proposed method is validated via the same numerical test example. The present method approximate exact solution very well.

Sufficient Conditions for the Solutions of General Parabolic Initial-Boundary-Value Problems to be Classical

We establish new sufficient conditions under which the generalized solutions of initial-boundary-value problems for linear parabolic differential equations of any order with complex-valued coefficients are classical. These conditions are formulated in the terms of belonging of the right-hand sides of this problem to certain anisotropic Hӧrmander spaces. In the definition of classical solution, its continuity on the line connecting the lateral surface of the cylinder (in which the problem is considered) with its base is not required.

Classical Solutions of Parabolic Initial-Boundary-Value Problems and HӧRmander Spaces

For the second-order linear parabolic differential equations with complex-valued coefficients, we establish new sufficient conditions under which the generalized solutions of these problems are continuous. The conditions are formulated in the terms of belonging of the right-hand sides of these problems to certain anisotropic Hӧrmander spaces.

Cauchy problems for parabolic equations in Sobolev–Slobodeckii and Hölder spaces on uniformly regular Riemannian manifolds

In this paper we establish optimal solvability results, that is, maximal regularity theorems, for the Cauchy problem for linear parabolic differential equations of arbitrary order acting on sections of tensor bundles over boundaryless complete Riemannian manifolds with bounded geometry. We employ an anisotropic extension of the Fourier multiplier theorem for arbitrary Besov spaces introduced in earlier by the author. This allows for a unified treatment of Sobolev-Slobodeckii and little H\"older spaces. In the flat case we recover classical results for Petrowskii-parabolic Cauchy problems.

Unconditionally Explicit Stable Difference Schemes for Solving Some Linear and Non-Linear Parabolic Differential Equation

We present the numerical method for solution of some linear and non-linear parabolic equation. Using idea [1], we will present the explicit unconditional stable scheme which has no restriction on the step size ratio k/h2 where k and h are step sizes for space and time respectively. We will also present numerical results to justify the present scheme.

Universal boundary value problem for equations of mathematical physics

A new statement of a boundary value problem for partial differential equations is discussed. An arbitrary solution to a linear elliptic, hyperbolic, or parabolic second-order differential equation is considered in a given domain of Euclidean space without any constraints imposed on the boundary values of the solution or its derivatives. The following question is studied: What conditions should hold for the boundary values of a function and its normal derivative if this function is a solution to the linear differential equation under consideration? A linear integral equation is defined for the boundary values of a solution and its normal derivative; this equation is called a universal boundary value equation. A universal boundary value problem is a linear differential equation together with a universal boundary value equation. In this paper, the universal boundary value problem is studied for equations of mathematical physics such as the Laplace equation, wave equation, and heat equation. Applications of the analysis of the universal boundary value problem to problems of cosmology and quantum mechanics are pointed out.

A STUDY ON UNCONDITIONALLY STABLE EXPLICIT DIFFERENCE SCHEMES FOR THE VARIABLE COEFFICIENTS PARABOLIC DIFFERENTIAL EQUATION

In (2), we have presented some new algorithms for solving the linear variable coefficients parabolic differential equations. The proposed scheme is stable without any restriction to step-sizes of space and time. The scheme is required the condition of step size ratio k h2,→ 0 as k,h → 0 in the convergence, where k and h are step sizes for space and time respectively. In this paper, we will present the explicit unconditional stable scheme which has no restriction on the step size ratio k h2 in the convergence. We will also present analysis for the present scheme.

Backward difference time discretization of parabolic differential equations on evolving surfaces

A linear parabolic differential equation on a moving surface is discretized in space by evolving surface finite elements and in time by backward difference formulas (BDF). Using results from Dahlquist's G-stability theory and Nevanlinna & Odeh's multiplier technique together with properties of the spatial semi-discretization, stability of the full discretization is proven for the BDF methods up to order 5 and optimal-order convergence is shown. Numerical experiments illustrate the behaviour of the fully discrete method.

The oscillations of solutions of initial value problems for parabolic equations by HPM

In this paper, we have used the homotopy perturbation method in order to find the analytical solutions of some linear and nonlinear parabolic differential equations. The method does not need linearization or weak nonlinearity assumptions. In this scheme, solution is calculated in the form of a convergent power series with easily computable components. Mathematics Subject Classification: 47J30, 35G25, 35Q53

On a posteriori estimates of inverse operators for linear parabolic initial-boundary value problems

We present numerically verified a posteriori estimates of the norms of inverse operators for linear parabolic differential equations. In case that the corresponding elliptic operator is not coercive, existing methods for a priori estimates of the inverse operators are not accurate and, usually, exponentially increase in time variable. We propose a new technique for obtaining the estimates of the inverse operator by using the finite dimensional approximation and error estimates. It enables us to obtain very sharp bounds compared with a priori estimates. We will give some numerical examples which confirm the actual effectiveness of our method.

The Cauchy-Dirichlet Problem for a Class of Linear Parabolic Differential Equations with Unbounded Coefficients in an Unbounded Domain

We consider the Cauchy-Dirichlet problem in [0,∞)×D for a class of linear parabolic partial differential equations. We assume that D⊂ℝd is an unbounded, open, connected set with regular boundary. Our hypotheses are unbounded and locally Lipschitz coefficients, not necessarily differentiable, with continuous data and local uniform ellipticity. We construct a classical solution to the nonhomogeneous Cauchy-Dirichlet problem using stochastic differential equations and parabolic differential equations in bounded domains.

Asymptotic stability analysis of the linear θ-method for linear parabolic differential equations with delay

This paper is concerned with asymptotic stability property of linear θ-method for partial functional differential equations with delay. A sufficient condition for the underlying partial functional differential equations to be asymptotically stable is presented. We investigate numerical stability of the linear θ-method by using the spectral radius condition. When θ ∈ [0, 1/2), a sufficient and necessary condition for the linear θ-method to be asymptotically stable is established. When θ ∈ [1/2, 1], the linear θ-method is unconditionally asymptotically stable. The behaviour of the norm of the iteration matrix when the linear θ-method is asymptotically stable is studied by using Kreiss resolvent condition. Numerical experiments have been implemented to confirm the derived stability properties of the numerical method.

Estimation of base settlement from the surface subsidence profile: Plane‐field of displacements

AbstractAccording to Litwiniszyn's theory, subsidence over a yielding underground geo‐structure is seen as a stochastic (Markov) process. This theory leads to a single, linear parabolic differential equation of diffusion–convection type (D–C equation) in the plane‐field of displacements. If the boundary conditions for the governing D–C equation are prescribed along the shear bands, i.e. at ‘moving’ boundaries—it has been observed from small‐scale model experiments that the subsiding process is always confined between a set of inclined shear bands—then the resulting equation is nonlinear. The inverse problem for this nonlinear equation, i.e. the problem of determining the base displacement using the surface subsidence as ‘initial’ condition, is ill‐posed and estimation of the base displacement from a given surface subsidence profile is not possible. In the present paper the domain of integration of the governing D–C equation is fixed (and bounded)—the boundaries are not evolving. Hence, the governing equation remains linear parabolic. The advantage is that this linear differential equation admits an analytical solution, under the trap‐door mechanism assumption, that enables a direct solution to the inverse problem. Copyright © 2008 John Wiley &amp; Sons, Ltd.

Asymptotic stability of numerical methods for linear delay parabolic differential equations

This paper is concerned with the asymptotic stability property of some numerical processes by discretization of parabolic differential equations with a constant delay. These numerical processes include forward and backward Euler difference schemes and Crank–Nicolson difference scheme which are obtained by applying step-by-step methods to the resulting systems of delay differential equations. Sufficient and necessary conditions for these difference schemes to be delay-independently asymptotically stable are established. It reveals that an additional restriction on time and spatial stepsizes of the forward Euler difference scheme is required to preserve the delay-independent asymptotic stability due to the existence of the delay term. Numerical experiments have been implemented to confirm the asymptotic stability of these numerical methods.

Absolute asymptotic stability of solutions of linear parabolic differential equations with delay

We establish necessary and sufficient conditions for the absolute asymptotic stability of solutions of linear parabolic differential equations with delay.

Internal layers of parabolic singularly perturbed problems

AbstractThe aim of this work is to show, through analyzing some incorrect results in the literature, (i) the internal layer structure is not so predictable for the linear parabolic differential equations with constant coefficients; and (ii) the location of the viscous shock of Burgers' equation in the quarter plane depends on the viscosity, but not in the half plane. Our methods are based on asymptotics of integrals together with special functions to unveil these distinguished features. Contrary to the exponential function as the sole building block of boundary layer functions for differential equations and partial differential equations near the outflow boundary, the class of the complementary error function and its iterated integrals becomes fundamental in this investigation.