Convergence of Galerkin method solutions of the integral equation for thin wire antennas

In this paper we consider the Pocklington integro–differential equation for the current induced on a straight, thin wire by an incident harmonic electromagnetic field. We show that this problem is well posed in suitable fractional order Sobolev spaces and obtain a coercive or Gårding type inequality for the associated operator. Combining this coercive inequality with a standard abstract formulation of the Galerkin method we obtain rigorous convergence results for Galerkin type numerical solutions of Pocklington's equation, and we demonstrate that certain convergence rates hold for these methods.     

Full‐discrete finite element method for the stochastic elastic equation driven by additive noise

We study the full‐discrete finite element method for the stochastic elastic equation driven by additive noise. To analyze the error estimates, we write the stochastic elastic equation as an abstract stochastic equation. Strong convergence estimates in the root mean square L₂ ‐norm are obtained by using the error estimates for the deterministic problem and the semigroup theory. Numerical experiments are carried out to verify the theoretical results. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Dual‐primal mixed finite elements for elliptic problems

In this article, a novel dual‐primal mixed formulation for second‐order elliptic problems is proposed and analyzed. The Poisson model problem is considered for simplicity. The method is a Petrov—Galerkin mixed formulation, which arises from the one‐element formulation of the problem and uses trial functions less regular than the test functions. Thus, the trial functions need not be continuous while the test functions must satisfy some regularity constraint. Existence and uniqueness of the solution are proved by using the abstract theory of Nicolaides for generalized saddle‐point problems. The Helmholtz Decomposition Principle is used to prove the inf‐sup conditions in both the continuous and the discrete cases. We propose a family of finite elements valid for any order, which employs piecewise polynomials and Raviart—Thomas elements. We show how the method, with this particular choice of the approximation spaces, is linked to the superposition principle, which holds for linear problems and to the standard primal and dual formulations, addressing how this can be employed for the solution of the final linear system. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 137–151, 2001     

Product Semigroups and Extrapolation Spaces

A new formulation of the theory of extrapolation spaces is used to recast a theorem on the resolvent of the sum of a generator of a semigroup and a Hille–Yosida operator and to prove a regularity result related to an abstract equation.     

An existence result for a two‐phase Stefan problem arising in metal casting

We present a model arising from the thermal modelling of two metal casting processes. We consider an enthalpy formulation for this two‐phase Stefan problem in a time varying three‐dimensional domain and consider convective heat transfer in the liquid phase. Then, we introduce a weak formulation in a fixed domain, by means of a suitable transformation. Existence of solution is obtained by applying an abstract theorem. The proof of this theorem is done by taking an implicit discretization in time together with a regularization. By passing to the limit in the regularization parameter and in the time step, we obtain the existence of solution of the continuous problem. Copyright © 2005 John Wiley & Sons, Ltd.     

Existence of Periodic Solutions for the Quasi-Static Thermoelastic Thermistor Problem

We deal with a new model for the thermistor problem formulated as a coupled system of PDE’s involving nonlinear energy heat equation, stationary charge conservation equation of electrical current and thermoelastic equations of displacement. We establish the existence of weak periodic solutions rewriting our system as an abstract problem in order to utilize the maximal monotone mappings theory and a fixed point argument for a suitable operator equation.      

Initial–boundary value problems for conservation laws with source terms and the Degasperis–Procesi equation

We consider conservation laws with source terms in a bounded domain with Dirichlet boundary conditions. We first prove the existence of a strong trace at the boundary in order to provide a simple formulation of the entropy boundary condition. Equipped with this formulation, we go on to establish the well-posedness of entropy solutions to the initial–boundary value problem. The proof utilizes the kinetic formulation and the averaging lemma. Finally, we make use of these results to demonstrate the well-posedness in a class of discontinuous solutions to the initial–boundary value problem for the Degasperis–Procesi shallow water equation, which is a third order nonlinear dispersive equation that can be rewritten in the form of a nonlinear conservation law with a nonlocal source term.     

Asymptotic convergence of solutions for Laplace reaction–diffusion equations

We study the initial–boundary value problem for a Laplace reaction–diffusion equation. After constructing local solutions by using the theory of abstract degenerate evolution equations of parabolic type, we show asymptotic convergence of bounded global solutions if they exist under the assumption that the reaction function is analytic in neighborhoods of their $\omega$-limit sets. Reduction of degenerate evolution equation to multivalued evolution equation enables us to use the theory of the infinite-dimensional Łojasiewicz–Simon gradient inequality.     

The coupling of boundary integral and finite element methods for the bidimensional exterior steady stokes problem

In this paper, we represent a new numerical method for solving the steady-state Stokes equations in an unbounded plane domain. The technique consists in coupling the boundary integral and the finite element methods. An artificial smooth boundary is introduced separating an interior inhomogeneous region from an exterior one. The solution in the exterior domain is represented by an integral equation over the artificial boundary. This integral equation is incorporated into a velocitypressure formulation for the interior region, and a finite element method is used to approximate the resulting variational problem. This is studied by means of an abstract framework, well adapted to the model problem, in which convergence results and optimal error estimates are derived. Computer results will be discussed in a forthcoming paper.     

Hopf Bifurcation for a Maturity Structured Population Dynamic Model

This article is devoted to investigate some dynamical properties of a structured population dynamic model with random walk on (0,+∞). This model has a nonlinear and nonlocal boundary condition. We reformulate the problem as an abstract non-densely defined Cauchy problem, and use integrated semigroup theory to study such a partial differential equation. Moreover, a Hopf bifurcation theorem is given for this model.     

Adhesive contact problem for viscoplastic materials

We examine a mathematical model that describes a quasistatic adhesive contact between a viscoplastic body and deformable foundation. The material’s behaviour is described by the rate-type constitutive law which involves functions with a non-polynomial growth. The contact is modelled by the normal compliance condition with limited penetration and adhesion, a subdifferential friction condition also depending on adhesion, and the evolution of bonding field is governed by an ordinary differential equation. We present the variational formulation of this problem which is a system of an almost history-dependent variational–hemivariational inequality for the displacement field and an ordinary differential equation for the bonding field. The results on existence and uniqueness of solution to an abstract almost history-dependent inclusion and variational–hemivariational inequality in the reflexive Orlicz–Sobolev space are proved and applied to the adhesive contact problem.     

Formal analysis of the Cauchy problem for a system associated with the (2+1)-dimensional Krichever-Novikov equation

The singularity manifold equation of the Kadomtsev-Petviashvili equation, the so-called Krichever-Novikov equation, has an exact linearization to an overdetermined system of partial differential equations in three independent variables. We study in detail the Cauchy problem for this system as an example for the use of the formal theory of differential equations. A general existence and uniqueness theorem is established. Formal theory is then contrasted with Janet-Riquier theory in the formulation of Reid. Finally, the implications of the results for the Krichever-Novikov equation are outlined.     

A perturbation method for multiple sign-changing solutions

We develop a perturbation method for non-even functionals which produces prescribed number of sign-changing solutions. The abstract theory is applied to the perturbed subcritical elliptic equation and the perturbed Brézis–Nirenberg critical exponent problem.     

Generalized solution of non‐autonomous photon transport problem

The purpose of this paper is to show the existence of a generalized solution of a non‐autonomous transport problem. By means of the theory of equicontinuous evolution system on a sequentially complete locally convex topological vector space, we show that the perturbed abstract non‐autonomous Cauchy problem has a unique solution when the perturbation operator and forcing term function satisfy certain conditions. A consequence of the abstract result is that it can be directly applied to obtain a generalized solution of the non‐autonomous photon transport problem. Copyright © 2009 John Wiley & Sons, Ltd.     

Solution of Quasilinear Stochastic Problems in Abstract Colombeau Algebras

The main object of study is the stochastic Cauchy problem for a quasilinear equation with random disturbances in the form of a Hilbert-valued white noise process and with an operator generating an integrated semigroup in the space L²(R). We use the Colombeau theory of multiplication of distributions to introduce an abstract stochastic factor algebra and construct an approximate solution of the problem in this algebra.     

On total preservation of solvability of controlled Hammerstein-type equation with non-isotone and non-majorizable operator

We prove some nontrivial corollaries of the Schauder theorem. We use these corollaries to prove a theorem concerning the total preservation of solvability of a controlled functional operator equation of the Hammerstein type with non-isotone and non-majorizable operator component in the right-hand side. We illustrate the application of the abstract theory by the example of the Dirichlet problem associated with a semilinear elliptic equation similar to a stationary diffusionreaction equation.     

Über Anfangs-und Eigenwertprobleme aus der Neutronentransporttheorie

Based on the theory of semi-groups in Hilbert space, a proof is given for the existence of a unique solution of an abstract Cauchy problem arising in the transport theory of mono-energetic neutrons, corresponding to the time-dependent linear Boltzmann equation in the general three-dimensional geometry. The spectral properties of the Boltzmann operator are investigated, an explicit representation of the solution is obtained by the perturbation theory for semi-groups of linear operators and alternatively an expansion in a series of eigenfunctions is given.     

Error Estimates for the Approximation of a Class of Optimal Control Systems Governed by Linear PDEs

This paper deals with a class of optimal control problems in which the system is governed by a linear partial differential equation and the control is distributed and with constraints. The problem is posed in the framework of the theory of optimal control of systems. A numerical method is proposed to approximate the optimal control. In this method, the state space as well as the convex set of admissible controls are discretized. An abstract error estimate for the optimal control problem is obtained that depends on both the approximation of the state equation and the space of controls. This theoretical result is illustrated by some numerical examples from the literature.     

Inhomogeneous degenerate Sobolev type equations with delay

Under consideration is the first order linear inhomogeneous differential equation in an abstract Banach space with a degenerate operator at the derivative, a relatively p-radial operator at the unknown function, and a continuous delay operator. We obtain conditions of unique solvability of the Cauchy problem and the Showalter problem by means of degenerate semigroup theory methods. These general results are applied to the initial boundary value problems for systems of integrodifferential equations of the type of phase field equations.     

A mixed formulation for the direct approximation of <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-weighted controls for the linear heat equation

This paper deals with the numerical computation of null controls for the linear heat equation. The goal is to compute approximations of controls that drive the solution from a prescribed initial state to zero at a given positive time. In [Fernandez-Cara & Münch, Strong convergence approximations of null controls for the 1D heat equation, 2013], a so-called primal method is described leading to a strongly convergent approximation of distributed control: the controls minimize quadratic weighted functionals involving both the control and the state and are obtained by solving the corresponding optimality conditions. In this work, we adapt the method to approximate the control of minimal square integrable-weighted norm. The optimality conditions of the problem are reformulated as a mixed formulation involving both the state and its adjoint. We prove the well-posedeness of the mixed formulation (in particular the inf-sup condition) then discuss several numerical experiments. The approach covers both the boundary and the inner situation and is valid in any dimension.