Absorbing boundary conditions for electromagnetic wave propagation

In this paper, the theoretical perfectly absorbing boundary condition on the boundary of a half-space domain is developed for the Maxwell system by considering the system as a whole instead of considering each component of the electromagnetic fields individually. This boundary condition allows any wave motion generated within the domain to pass through the boundary of the domain without generating any reflections back into the interior. By approximating this theoretical boundary condition a class of local absorbing boundary conditions for the Maxwell system can be constructed. Well-posedness in the sense of Kreiss of the Maxwell system with each of these local absorbing boundary conditions is established, and the reflection coefficients are computed as a plane wave strikes the artificial boundary. Numerical experiments are also provided to show the performance of these local absorbing boundary conditions

     

Boundary value problems with global projection conditions

Parametrices of elliptic boundary value problems for differential operators belong to an algebra of pseudodifferential operators with the transmission property at the boundary. However, generically, smooth symbols on a manifold with boundary do not have this property, and several interesting applications require a corresponding more general calculus. We introduce here a new algebra of boundary value problems that contains Shapiro-Lopatinskij elliptic as well as global projection conditions; the latter ones are necessary, if an analogue of the Atiyah-Bott obstruction does not vanish. We show that every elliptic operator admits (up to a stabilisation) elliptic conditions of that kind. Corresponding boundary value problems are then Fredholm in adequate scales of spaces. Moreover, we construct parametrices in the calculus.     

Boundary value problems with mixed lateral conditions for parabolic operators

Summary We study a parabolic problem in a cylinder with lateral conditions of mixed type. We get an existence, uniqueness and regularity result in dissymmetric function spaces nicely fitting the geometry of the problem.Partially supported by G.N.A.F.A. of C.N.R., Italy.Partially supported by I.N.F.N., Bologna, Italy.     

Boundary element methods for potential problems with nonlinear boundary conditions

Galerkin boundary element methods for the solution of novel first kind Steklov-Poincaré and hypersingular operator boundary integral equations with nonlinear perturbations are investigated to solve potential type problems in two- and three-dimensional Lipschitz domains with nonlinear boundary conditions. For the numerical solution of the resulting Newton iterate linear boundary integral equations, we propose practical variants of the Galerkin scheme and give corresponding error estimates. We also discuss the actual implementation process with suitable preconditioners and propose an optimal hybrid solution strategy.

     

An efficient boundary element method for two-dimensional transient wave propagation problems

The boundary element method (BEM) has been recognized by its unique feature of requiring neither internal cells nor their associated domain integrals in the computation. The method preserves its elegance for transient problems by means of a certain time-stepping scheme that initiates the time integration always from the initial time. Unfortunately, this time-marching scheme becomes rather difficult to apply, because the computation time and storage requirement grow dramatically with the increasing number of time steps. This paper shows that a reduction of one half of the computation time as well as the storage requirement can be achieved by an efficient truncation scheme for two-dimensional transient wave propagation problems. In particular, a guiding parameter for the determination of the truncation limit is proposed, and the overall measure of the error with respect to the truncation guide parameter is established.     

Boundary value problems with eigenvalue depending boundary conditions

We investigate some classes of eigenvalue dependent boundary value problems of the form where A ? A⁺ is a symmetric operator or relation in a Krein space K, τ is a matrix function and Γ₀, Γ₁ are abstract boundary mappings. It is assumed that A admits a self‐adjoint extension in K which locally has the same spectral properties as a definitizable relation, and that τ is a matrix function which locally can be represented with the resolvent of a self‐adjoint definitizable relation. The strict part of τ is realized as the Weyl function of a symmetric operator T in a Krein space H, a self‐adjoint extension à of A × T in K × H with the property that the compressed resolvent P_K (à – λ)^–1|_K k yields the unique solution of the boundary value problem is constructed, and the local spectral properties of this so‐called linearization à are studied. The general results are applied to indefinite Sturm–Liouville operators with eigenvalue dependent boundary conditions (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new scheme for the solution of reaction diffusion and wave propagation problems

In this paper, a robust numerical scheme is presented for the reaction diffusion and wave propagation problems. The present method is rather simple and straightforward. The Houbolt method is applied so as to convert both types of partial differential equations into an equivalent system of modified Helmholtz equations. The method of fundamental solutions is then combined with the method of particular solution to solve these new systems of equations. Next, based on the exponential decay of the fundamental solution to the modified Helmholtz equation, the dense matrix is converted into an equivalent sparse matrix. Finally, verification studies on the sensitivity of the method’s accuracy on the degree of sparseness and on the time step magnitude of the Houbolt method are carried out for four benchmark problems.     

Parallel discontinuous Galerkin unstructured mesh solvers for the calculation of three-dimensional wave propagation problems

We discuss the parallel performances of discontinuous Galerkin solvers designed on unstructured tetrahedral meshes for the calculation of three-dimensional heterogeneous electromagnetic and aeroacoustic wave propagation problems. An explicit leap-frog time-scheme along with centered numerical fluxes are used in the proposed discontinuous Galerkin time-domain (DGTD) methods. The schemes introduced are genuinely non-dissipative, in order to achieve a discrete equivalent of the energy conservation. Parallelization of these schemes is based on a standard strategy that combines mesh partitioning and a message passing programming model. The resulting parallel solvers are applied and evaluated on several large-scale, homogeneous and heterogeneous, wave propagation problems.     

A degeneracy phenomenon and resonance phenomena in problems of wave propagation

In the work the interaction of normal waves in a neighborhood of approach or intersection of dispersion curves is studied. An asymptotic expression for a normal wave is constructed by means of a certain class of special functions. In the case of intersection, these functions are well-known standard integrals (Airy, Pirsli, etc. integrals). It is shown that by means of these special functions it is possible to describe the solution in a neighborhood of a point of approach or intersection of any number of caustics.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 104, pp. 217–227, 1981.In conclusion, the author thanks I. A. Molotkov for fruitful discussions and remarks.     

Boundary value problems for first order functional differential equations with impulsive integral conditions

In this paper, by using the method of lower and upper solutions coupled with the monotone iterative technique, we give conditions for existence of extreme solutions for first order functional differential equations with a new impulsive integral condition. Some comparison results are also formulated.     

Boundary value problems with continuous and special gluing conditions for parabolic-hyperbolic type equations

In the present paper we study the unique solvability of two non-local boundary value problems with continuous and special gluing conditions for parabolic-hyperbolic type equations. The uniqueness of the solutions of the considered problems are proven by the “abc” method. Existence theorems for the solutions of these problems are proven by the method of integral equations. The obtained results can be used for studying local and non-local boundary-value problems for mixed-hyperbolic type equations with two and three lines of changing type.      

1D wave propagation in a rod: analytic treatment for non-trivial boundary conditions

An analytic solution method for the one-dimensional wave propagation problem in a rod based on the Laplace transform is applied to a simple stamping model. While the rod is mounted at its lower end to a spring, it is hit at its upper end by an impacting mass, the separation of which can be correctly treated by the method. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite speed of propagation and local boundary conditions for wave equations with point interactions

We show that the boundary conditions entering in the definition of the self-adjoint operator describing the Laplacian plus a finite number of point interactions are local if and only if the corresponding wave equation has finite speed of propagation.

     

Boundary value problems of the theory of heat propagation in a binary mixture

In this paper we study the problems of heat propagation in a binary mixture. The fundamental solution of system of equations of steady vibrations is constructed in terms of elementary functions. The uniqueness and existence theorems of classical solution of the interior and exterior boundary value problems by means of the boundary integral method are proved. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)