Value Problem For Parabolic Equation Research Articles

On the Second Initial-boundary Value Problem for Parabolic Equation, Containing Mixed Derivatives

In this work new analytical solution of the second initial-boundary value problem of the theory of thermal conductivity in a general anisotropic strip has been found considering that heat transfer characteristics are not scalar values but second symmetric tensors. The theorems about the existence and uniqueness of found solution have been proved.

Parabolic Regularity and Dirichlet boundary value problems

We study the relationship between the Regularity and Dirichlet boundary value problems for parabolic equations of the form Lu=div(A∇u)−ut=0 in Lip(1,1∕2) time-varying cylinders, where the coefficient matrix A=aij(X,t) is uniformly elliptic and bounded.We show that if the Regularity problem (R)p for the equation Lu=0 is solvable for some 1<p<∞ then the Dirichlet problem (D∗)p′ for the adjoint equation L∗v=0 is also solvable, where p′=p∕(p−1). This result is an analogue of the result established in the elliptic case by Kenig and Pipher (1993). In the parabolic settings in the special case of the heat equation in slightly smoother domains this has been established by Hofmann and Lewis (1996) and Nyström (2006) for scalar parabolic systems. In comparison, our result is abstract with no assumption on the coefficients beyond the ellipticity condition and is valid in more general class of domains.

Incomplete Iterative Implicit Schemes

Abstract In numerical solving boundary value problems for parabolic equations, two- or three-level implicit schemes are in common use. Their computational implementation is based on solving a discrete elliptic problem at a new time level. For this purpose, various iterative methods are applied to multidimensional problems evaluating an approximate solution with some error. It is necessary to ensure that these errors do not violate the stability of the approximate solution, i.e., the approximate solution must converge to the exact one. In the present paper, these questions are investigated in numerical solving a Cauchy problem for a linear evolutionary equation of first order, which is considered in a finite-dimensional Hilbert space. The study is based on the general theory of stability (well-posedness) of operator-difference schemes developed by Samarskii. The iterative methods used in the study are considered from the same general positions.

Fundamental mode exact schemes for unsteady problems

The problem of increasing the accuracy of an approximate solution is considered for boundary value problems for parabolic equations. For ordinary differential equations (ODEs), nonstandard finite difference schemes are in common use for this problem. They are based on a modification of standard discretizations of time derivatives and, in some cases, allow to obtain the exact solution of problems. For multidimensional problems, we can consider the problem of increasing the accuracy only for the most important components of the approximate solution. In the present work, new unconditionally stable schemes for parabolic problems are constructed, which are exact for the fundamental mode. Such two‐level schemes are designed via a modification of standard schemes with weights using Padé approximations. Numerical results obtained for a model problem demonstrate advantages of the proposed fundamental mode exact schemes.

Study of the Solvability of Some Volterra-Type Integral and Integro-Differential Equations of Third Kind

For Volterra integral equations of the third kind and for Volterra-type integrodifferential equations of the third kind, theorems on the existence of solutions in Sobolev spaces (i.e., regular solutions) are proved. The proofs are based on the theory of boundary value problems for degenerate ordinary differential equations and on the theory of boundary value problems for parabolic equations with a changing evolution direction.

Boundary value problems for parabolic equations of high order with a changing time direction

We analyze the solvability of boundary value problems for differential equations of high order with a changing direction of time. It is shown that a generalized solution exists in Sobolev spaces. Correctness of boundary value problems with a complete matrix of conjugation conditions is investigated. For such problems, it is shown that the Hölder classes of their solutions essentially depend on both the type of gluing conditions and on the non-integer index Hölder space.

Time step selection for the numerical solution of boundary value problems for parabolic equations

An algorithm is proposed for selecting a time step for the numerical solution of boundary value problems for parabolic equations. The solution is found by applying unconditionally stable implicit schemes, while the time step is selected using the solution produced by an explicit scheme. Explicit computational formulas are based on truncation error estimation at a new time level. Numerical results for a model parabolic boundary value problem are presented, which demonstrate the performance of the time step selection algorithm.

Monte Carlo method for solution of initial–boundary value problem for nonlinear parabolic equations

In this paper we will consider the initial–boundary value problem for a parabolic equation with a polynomial non-linearity relative to the unknown function. First we will derive a probabilistic representation of our problem. The representation of the solution of this problem is given in the form of a mathematical expectation, which is determined based on trajectories of branching processes. Under the assumption of the existence of the solution, an unbiased estimator is built using trajectories of a branching process. We will use a mean value theorem to write out a special integral equation, that equates the value of the unknown function u(x,t) with its integral over a spheroid and balloid with center at the point (x,t). A probabilistic representation of the solution to the problem in the form of mathematical expectation of some random variables is obtained. This probabilistic representation uses a branching process whose trajectories are used in the contraction of an unbiased estimator for the solution. The derived unbiased estimator has a finite variance, and is built up from trajectories of branching processes with a finite average number of branches. Finally, the results of numerical experiments and application to the practical problems are discussed.

Problems for parabolic equations with variable exponents of nonlinearity and time delay

The initial-boundary value problems for parabolic equations with variable exponents of nonlinearity and time depended delay are considered. Existence and uniqueness of solutions of these problems are proved.

Justification of Application of the Fourier Method for Parabolic Equations with Random Boundary Conditions from Orlicz Spaces

A new method of constructing solutions of boundary value problems for parabolic equations with random boundary value conditions is proposed. We assume that the boundary conditions are stochastic processes from the Orlicz space of random variables (including processes with zero mean values).

REGULARIZATION OF PONTRYAGIN MAXIMUM PRINCIPLE IN OPTIMAL CONTROL OF DISTRIBUTED SYSTEMS

This article is devoted to studying dual regularization method applied to parametric convex optimal control problem of controlled third boundary–value problem for parabolic equation with boundary control and with equality and inequality pointwise state constraints. This dual regularization method yields the corresponding necessary and sufficient conditions for minimizing sequences, namely, the stable, with respect to perturbation of input data, sequential or, in other words, regularized Lagrange principle in nondifferential form and Pontryagin maximum principle for the original problem. Regardless of the fact that the stability or instability of the original optimal control problem, they stably generate a minimizing approximate solutions in the sense of J. Warga for it. For this reason, we can interpret these regularized Lagrange principle and Pontryagin maximum principle as tools for direct solving unstable optimal control problems and reducing to them unstable inverse problems.

A PRIORI ESTIMATION OF A TIME STEP FOR NUMERICALLY SOLVING PARABOLIC PROBLEMS

This work deals with the problem of choosing a time step for the numerical solution of boundary value problems for parabolic equations. The problem solution is derived using the fully implicit scheme, whereas a time step is selected via explicit calculations. The selection strategy consists of the following two stages. At the first stage, we employ explicit calculations for selecting the appropriate time step. At the second stage, using the implicit scheme, we calculate the solution at a new time level. This solution should be close to the solution of our problem at this time level with a prescribed accuracy. Such an algorithm leads to explicit formulas for the calculation of the time step and takes into account both the dynamics of the problem solution and changes in coefficients of the equation and in its right-hand side. The same formulas for the evaluation of the time step are obtained by using a comparison of two approximate solutions, which are obtained using the explicit scheme with the primary time step and the step that is reduced by half. Numerical results are presented for a model parabolic boundary value problem, which demonstrate the robustness of the developed algorithm for the time step selection.

Three-level schemes of the alternating triangular method

In this paper, the schemes of the alternating triangular method are set out in the class of splitting methods used for the approximate solution of Cauchy problems for evolutionary problems. These schemes are based on splitting the problem operator into two operators that are conjugate transposes of each other. Economical schemes for the numerical solution of boundary value problems for parabolic equations are designed on the basis of an explicit-implicit splitting of the problem operator. The alternating triangular method is also of interest for the construction of numerical algorithms that solve boundary value problems for systems of partial differential equations and vector systems. The conventional schemes of the alternating triangular method used for first-order evolutionary equations are two-level ones. The approximation properties of such splitting methods can be improved by transiting to three-level schemes. Their construction is based on a general principle for improving the properties of difference schemes, namely, on the regularization principle of A.A. Samarskii. The analysis conducted in this paper is based on the general stability (or correctness) theory of operator-difference schemes.

Application of Neumann-Ulam scheme to solving the first boundary value problem for a parabolic equation

Statistical estimates of the solutions of boundary value problems for parabolic equations with constant coefficients are constructed on paths of random walks. The phase space of these walks is a region in which the problem is solved or the boundary of the region. The simulation of the walks employs the explicit form of the fundamental solution; therefore, these algorithms cannot be directly applied to equations with variable coefficients. In the present work, unbiased and low-bias estimates of the solution of the boundary value problem for the heat equation with a variable coefficient multiplying the unknown function are constructed on the paths of a Markov chain of random walk on balloids. For studying the properties of the Markov chains and properties of the statistical estimates, the author extends von Neumann-Ulam scheme, known in the theory of Monte Carlo methods, to equations with a substochastic kernel. The algorithm is based on a new integral representation of the solution to the boundary value problem.

Well-Posedness of Nonlocal Parabolic Differential Problems with Dependent Operators

The nonlocal boundary value problem for the parabolic differential equation v'(t) + A(t)v(t) = f(t) (0 ≤ t ≤ T), v(0) = v(λ) + φ, 0 < λ ≤ T in an arbitrary Banach space E with the dependent linear positive operator A(t) is investigated. The well-posedness of this problem is established in Banach spaces C 0 β,γ(E α−β) of all E α−β-valued continuous functions φ(t) on [0, T] satisfying a Hölder condition with a weight (t + τ)γ. New Schauder type exact estimates in Hölder norms for the solution of two nonlocal boundary value problems for parabolic equations with dependent coefficients are established.

Flux-splitting schemes for parabolic equations with mixed derivatives

Difference schemes of required quality are often difficult to construct as applied to boundary value problems for parabolic equations with mixed derivatives. Specifically, difficulties arise in the design of monotone difference schemes and unconditionally stable locally one-dimensional splitting schemes. In parabolic problems, certain opportunities are offered by restating the problem in question so that the quantities to be determined are fluxes (directional derivatives). The original problem is then rewritten as a boundary value one for a system of equations in flux variables. Weighted schemes for parabolic equations in flux coordinates are examined. Unconditionally stable locally one-dimensional flux schemes that are first- and second-order accurate in time are constructed for a parabolic equation without mixed derivatives. A feature of systems in flux variables for equations with mixed derivatives is that the terms with time derivatives are coupled with each other.

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

Flux-splitting schemes for parabolic problems

To solve numerically boundary value problems for parabolic equations with mixed derivatives, the construction of difference schemes with prescribed quality faces essential difficulties. In parabolic problems, some possibilities are associated with the transition to a new formulation of the problem, where the fluxes (derivatives with respect to a spatial direction) are treated as unknown quantities. In this case, the original problem is rewritten in the form of a boundary value problem for the system of equations in the fluxes. This work deals with studying schemes with weights for parabolic equations written in the flux coordinates. Unconditionally stable flux locally one-dimensional schemes of the first and second order of approximation in time are constructed for parabolic equations without mixed derivatives. A peculiarity of the system of equations written in flux variables for equations with mixed derivatives is that there do exist coupled terms with time derivatives.

Methods of Moving Boundary Based on Artificial Boundary in Heat Conduction Direct Problem

The initial boundary value problem for parabolic equation with Neumann boundary condition is a kind of classical problem in partial differential equations. In this paper we use the artificial boundary to solve the moving boundary problem. Potential theory and difference method are discussed. Numerical results are given to support the proposed schemes and to give the compare of the two methods.

Some properties of solutions of the cauchy problem for evolution equations

We consider the Cauchy problem for a first-order operator-differential equation with singular data. The results are used to study boundary value problems for parabolic equations with operator-valued coefficients.