First-order Evolution Equations Research Articles

Macroscopic description of quasi-static and quasi-stationary processes in unsaturated porous materials

The main aim of this work is to derive equations describing quasi-static and quasi-stationary processes of capillary transport of liquid and gas in unsaturated porous materials. Considerations are based on the new macroscopic model of the capillary transport in porous media, proposed in the paper (Cieszko, 2016). This model has been formulated in the spirit of the theory of interacting continua and shows that the capillary processes of transport in porous materials take place in five-dimensional pressure-time-space. This means that the equations for quasi-static and quasi-stationary processes are not a simple consequence of reduction of the general system of equations. An extended definition of quasi-static processes is proposed, allowing such processes to be analysed as a special case of the general description. A nonlinear first-order evolution equation is obtained for pore saturation with mobile liquid in quasi-static processes; it simultaneously describes the non-stationary processes of saturation evolution taking place in the pressure-space continuum. A new class of processes of capillary transport in unsaturated porous materials is proposed, called quasi-stationary, and a system of strongly coupled nonlinear equations describing these processes is derived. It is shown that even the geometrically very simple boundary value problem of non-wetting liquid flow through an unsaturated porous layer is described by a system of three nonlinear coupled equations for the saturation, pressure and velocity of the liquid.

Alternating Triangular Schemes for Second-Order Evolution Equations

Schemes of the Samarskii alternating triangular method are based on splitting the problem operator into two operators that are conjugate to each other. When the Cauchy problem for a first-order evolution equation is solved approximately, this makes it possible to construct unconditionally stable two-component factorized splitting schemes. Explicit schemes are constructed for parabolic problems based on the alternating triangular method. The approximation properties can be improved by using three-level schemes. The main possibilities are indicated for constructing alternating triangular schemes for second-order evolution equations. New schemes are constructed based on the regularization of the standard alternating triangular schemes. The features of constructing alternating triangular schemes are pointed out for problems with many operator terms and for second-order evolution equations involving operator terms for the first time derivative. The study is based on the general stability (well-posedness) theory for operator-difference schemes.

Unified error analysis for nonconforming space discretizations of wave-type equations

AbstractThis paper provides a unified error analysis for nonconforming space discretizations of linear wave equations in the time domain. We propose a framework that studies wave equations as first-order evolution equations in Hilbert spaces and their space discretizations as differential equations in finite-dimensional Hilbert spaces. A lift operator maps the semidiscrete solution from the approximation space to the continuous space. Our main results are a priori error bounds in terms of interpolation, data and conformity errors of the method. Such error bounds are the key to the systematic derivation of convergence rates for a large class of problems. To show that this approach significantly eases the proof of new convergence rates, we apply it to an isoparametric finite element discretization of the wave equation with acoustic boundary conditions in a smooth domain. Moreover, our results reproduce known convergence rates for already investigated conforming and nonconforming space discretizations in a concise and unified way. The examples discussed in this paper comprise discontinuous Galerkin discretizations of Maxwell’s equations and finite elements with mass lumping for the acoustic wave equation.

Numerical analysis of a thermal contact problem with adhesion

In this paper, we study a class of dynamic thermal problems involving a frictional normal compliance adhesive contact model and a non-clamped condition for visco-elastic materials. The variational formulation of the problem leads to a general system defined by a second-order quasi-variational evolution inequality on the displacement field coupled with the two first-order evolution equations describing the evolution of the temperature and adhesion. We establish an existence and uniqueness result of the solution on displacement, temperature and adhesion. Then we provide a fully discrete numerical scheme of approximation and derive an error estimate. Finally, various numerical computations are included.

Controllability for a class of integro-differential evolution equations involving non-local initial conditions

ABSTRACTThis paper deals with the exact controllability for a class of first-order integro-differential evolution equations involving non-local initial conditions. By using Sadovskii fixed point theorem, exact controllability results are obtained without assuming the compactness and Lipschitz conditions for non-local functions. An example is given to illustrate the effectiveness of the abstract results.

Parabolic First and Second Order Differential Equations

We are concerned with an inverse problem for a first-order linear evolution equation. Moreover, a complete second-order evolution equation will be considered, too. We indicate sufficient conditions for existence and uniqueness of a solution. All the results apply well to inverse problems for equations from mathematical physics. As a possible application of the abstract theorems, some examples of partial differential equations are given.

First-order inverse evolution equations

We are concerned with an inverse problem for first order linear evolution equations. We indicate sufficient conditions for existence and uniqueness of a solution to these problems. All the results apply well to inverse problems for equations from mathematical physics. Indeed, as a possible application of the abstract theorems, some examples of partial differential equations are given.

Almost periodic viscosity solutions of nonlinear evolution equations in Carnot groups

The aim of this paper is to establish existence and uniqueness of time almost periodic viscosity solutions to first-order evolution equations in Carnot groups.

Operator-difference schemes for a class of systems of evolution equations

For a special system of evolution equations of first order, discrete time approximations for the approximate solution of the Cauchy problem are considered. Such problems arise after the spatial approximation in the Schrodinger equation and the subsequent separation of the imaginary and real parts and in nonstationary problems of acoustics and electrodynamics. Unconditionally stable two time level operator-difference weighted schemes are constructed. The second class of difference schemes is based on the formal passage to explicit operator-difference schemes for evolution equations of second order when explicit-implicit approximation is used for isolated equations of the system. The regularization of such schemes in order to obtain unconditionally stable operator difference schemes is discussed. Splitting schemes involving the solution of simplest problems at each time step are constructed.

Generalized Yosida Approximations Based on RelativelyA-Maximalm-Relaxed Monotonicity Frameworks

We introduce and study a new notion of relativelyA-maximalm-relaxed monotonicity framework and discuss some properties of a new class of generalized relatively resolvent operator associated with the relativelyA-maximalm-relaxed monotone operator and the new generalized Yosida approximations based on relativelyA-maximalm-relaxed monotonicity framework. Furthermore, we give some remarks to show that the theory of the new generalized relatively resolvent operator and Yosida approximations associated with relativelyA-maximalm-relaxed monotone operators generalizes most of the existing notions on (relatively) maximal monotone mappings in Hilbert as well as Banach space and can be applied to study variational inclusion problems and first-order evolution equations as well as evolution inclusions.

Hybrid inexact proximal point algorithms based on RMM frameworks with applications to variational inclusion problems

A new application-oriented notion of relatively A-maximal monotonicity (RMM) framework is introduced, and then it is applied to the approximation solvability of a general class of inclusion problems, while generalizing other existing results on linear convergence, including Rockafellar’s theorem (1976) on linear convergence using the proximal point algorithm in a real Hilbert space setting. The obtained results not only generalize most of the existing investigations, but also reduce smoothly to the case of the results on maximal monotone mappings and corresponding classical resolvent operators. Furthermore, our proof approach differs significantly to that of Rockafellar’s celebrated work, where the Lipschitz continuity of M−1, the inverse of M:X→2X, at zero is assumed to achieve a linear convergence of the proximal point algorithm. Note that the notion of relatively A-maximal monotonicity framework seems to be used to generalize the classical Yosida approximation (which is applied and studied mostly based on the classical resolvent operator in the literature) that in turn can be applied to first-order evolution equations as well as evolution inclusions.

On nonclassical problems for first-order evolution equations

Abstract The present paper deals with nonclassical initial-boundary value problems for parabolic equations and systems and their generalizations in abstract spaces. Nonclassical problems with nonlocal initial conditions for an abstract first-order evolution equation with time-dependent operator are considered, the existence and uniqueness results are proved and the algorithm of approximation of nonlocal problems by a sequence of classical problems is constructed. Applications of the obtained general results to initial-boundary value problems for parabolic equations and systems are considered.

Time discretisation of monotone nonlinear evolution problems by the discontinuous Galerkin method

A class of discontinuous Galerkin methods is studied for the time discretisation of the initial-value problem for a nonlinear first-order evolution equation that is governed by a monotone, coercive, and hemicontinuous operator. The numerical solution is shown to converge towards the weak solution of the original problem. Furthermore, well-posedness of the time-discrete problem as well as a priori error estimates for sufficiently smooth exact solutions are studied.

CONSERVATION LAWS FOR SELF-ADJOINT FIRST-ORDER EVOLUTION EQUATION

We consider the problem on group classification and conservation laws for first-order evolution equations. Subclasses of these general equations which are quasi-self-adjoint and self-adjoint are obtained. By using the recent new conservation theorem due to Ibragimov, conservation laws for equations admiting self-adjoint equations are established. The results are illustrated applying them to the inviscid Burgers equation. In particular an infinite number of new symmetries of this equation are found.

Variable time-step ϑ-scheme for nonlinear evolution equations governed by a monotone operator

The single-step ϑ-scheme on a variable time grid is employed for the approximate solution of the initial-value problem for a nonlinear first-order evolution equation. The evolution equation is supposed to be governed by a possibly time-dependent hemicontinuous operator that is (up to a shift) monotone and coercive, and fulfills a certain growth condition. A piecewise constant as well as piecewise linear prolongation of the time-discrete solution is shown to converge towards the exact solution if ϑ≥1/2 (including the Crank-Nicolson scheme). In the appearance of a strongly continuous perturbation of the monotone main part, the method is still convergent if ϑ>1/2 and if the ratio of adjacent step sizes is bounded from above by a power of ϑ/(1−ϑ). Besides convergence also well-posedness of the time-discrete problem as well as a priori error estimates are studied.

Maximal regularity for evolution problems on the line

Let $A$ be a hyperbolic bisectorial operator on a Banach space. In this paper we study the optimal regularity of the solutions of the abstract first-order evolution equation $u' (t) = Au(t) + f (t) $ on the whole line, depending on the regularity of the inhomogeneity $f.$

Convergence of the variable two-step BDF time discretisation of nonlinear evolution problems governed by a monotone potential operator

The initial-value problem for a first-order evolution equation is discretised in time by means of the two-step backward differentiation formula (BDF) on a variable time grid. The evolution equation is governed by a monotone and coercive potential operator. On a suitable sequence of time grids, the piecewise constant interpolation and a piecewise linear prolongation of the time discrete solution are shown to converge towards the weak solution if the ratios of adjacent step sizes are close to 1 and do not vary too much.

Relatively Inexact Proximal Point Algorithm and Linear Convergence Analysis

Based on a notion ofrelatively maximal(m)-relaxed monotonicity, the approximation solvability of a general class of inclusion problems is discussed, while generalizing Rockafellar's theorem (1976) on linear convergence using the proximal point algorithm in a real Hilbert space setting. Convergence analysis, based on this new model, is simpler and compact than that of the celebrated technique of Rockafellar in which the Lipschitz continuity at 0 of the inverse of the set-valued mapping is applied. Furthermore, it can be used to generalize the Yosida approximation, which, in turn, can be applied to first-order evolution equations as well as evolution inclusions.

Singular Perturbation of Kolmogorov Equation in Gauss–Sobolev Space

This article is concerned with the Kolmogorov equation associated to a stochastic partial differential equation with an additive noise depending on a small parameter ϵ > 0. As ϵ vanishes, the parabolic equation degenerates into a first-order evolution equation. In a Gauss–Sobolev space setting, we prove that, as ϵ ↓ 0, the solution of the Cauchy problem for the Kolmogorov equation converges in L 2(μ, H) to that of the reduced evolution equation of first-order, where μ is a reference Gaussian measure on the Hilbert space H.

Evolution of Close White Dwarf Binaries

We describe the evolution of double degenerate binary systems, consisting of components obeying the zero-temperature mass-radius relationship for white dwarf stars, from the onset of mass transfer to one of several possible outcomes, including merger, tidal disruption of the donor, or survival as a semidetached AM CVn system. We use a combination of analytic solutions and numerical integrations of the standard orbit-averaged first-order evolution equations, including direct-impact accretion and the evolution of the components due to mass exchange. We include also the effects of mass loss during supercritical (super-Eddington) mass transfer and the tidal and advective exchanges of angular momentum between the binary components. With the caveat that our formalism does not include an explicit treatment of common-envelope phases, our results suggest that a larger fraction of detached double white dwarfs survive the onset of mass transfer than has been hitherto assumed, even if this mass transfer is initially unstable and rises to super-Eddington levels. In addition, as a consequence of the tidal coupling, systems that come into contact near the mass transfer instability boundary undergo a phase of oscillation cycles in their orbital period (and other system parameters). Unless the donor star has a finite entropy such that the effective mass-radius relationship deviates significantly from that of a zero-temperature white dwarf, we expect our results to be valid. Much of the formalism developed here would also apply to other mass-transferring binaries, and in particular to cataclysmic variables and Algol systems.