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The paper presents explicit interval multistep methods of Milne type, which may be considered as alternative methods to other known explicit interval multistep methods (of Adams-Bashforth and Nyström). It is proved that enclosures of solutions (in the form of intervals) obtained by these methods contain the exact solutions of the initial value problem. Numerical examples show that the widths of intervals obtained by proposed methods are smaller than those obtained by explicit interval multistep methods known so far. [ABSTRACT FROM AUTHOR]

This paper derives physically meaningful boundary conditions for fractional diffusion equations, using a mass balance approach. Numerical solutions are presented, and theoretical properties are reviewed, including well-posedness and steady state solutions. Absorbing and reflecting boundary conditions are considered, and illustrated through several examples. Reflecting boundary conditions involve fractional derivatives. The Caputo fractional derivative is shown to be unsuitable for modeling fractional diffusion, since the resulting boundary value problem is not positivity preserving. [ABSTRACT FROM AUTHOR]

Gaussian elimination with no pivoting and block Gaussian elimination are attractive alternatives to the customary but communication intensive Gaussian elimination with partial pivoting 1 provided that the computations proceed safely and numerically safely , that is, run into neither division by 0 nor numerical problems. Empirically, safety and numerical safety of GENP have been consistently observed in a number of papers where an input matrix was pre-processed with various structured multipliers chosen ad hoc. Our present paper provides missing formal support for this empirical observation and explains why it was elusive so far. Namely we prove that GENP is numerically unsafe for a specific class of input matrices in spite of its pre-processing with some well-known and well-tested structured multipliers, but we also prove that GENP and BGE are safe and numerically safe for the average input matrix pre-processed with any nonsingular and well-conditioned multiplier. This should embolden search for sparse and structured multipliers, and we list and test some new classes of them. We also seek randomized pre-processing that universally (that is, for all input matrices) supports (i) safe GENP and BGE with probability 1 and/or (ii) numerically safe GENP and BGE with a probability close to 1. We achieve goal (i) with a Gaussian structured multiplier and goal (ii) with a Gaussian unstructured multiplier and alternatively with Gaussian structured augmentation. We consistently confirm all these formal results with our tests of GENP for benchmark inputs. We have extended our approach to other fundamental matrix computations and keep working on further extensions. 1 Hereafter we use the acronyms GENP , BGE , and GEPP . [ABSTRACT FROM AUTHOR]

Inspired by the theoretical results on optimal preconditioning stated by Ng, Chan, and Tang in the framework of Reflective boundary conditions (BCs), in this paper we present analogous results for Anti-Reflective BCs. Here a key technical difficulty is represented by the non-orthogonal character of the Anti-Reflective transform and indeed the proof proposed by Ng, Chan, and Tang does not work. Nevertheless, in both cases, the optimal preconditioner is the blurring matrix associated to the symmetrized Point Spread Function (PSF). The geometrical idea on which our proof is based is very simple and general, so it may be useful in the future to prove theoretical results for new proposed BCs. Numerical tests show that the optimal preconditioning strategy is effective when using both preconditioned conjugate gradient methods and recently introduced nonstationary preconditioned iterations. [ABSTRACT FROM AUTHOR]

Embedded boundary meshes may have cut cells of arbitrarily small volume which can lead to stability problems in finite volume computations with explicit time stepping. We show that time step constraints are not as strict as often believed. We prove this in one dimension for linear advection and the first order upwind scheme. Numerical examples in two dimensions demonstrate that this carries over to more complicated situations. This analysis sheds light on the choice of time step when using cell merging to stabilize the arbitrarily small cells that arise in embedded boundary schemes. [ABSTRACT FROM AUTHOR]

In this paper we study the local regularity of almost minimizers of the functional where $$q_\pm \in L^\infty (\Omega )$$ . Almost minimizers do not satisfy a PDE or a monotonicity formula like minimizers do (see Alt and Caffarelli, in J Reine Angew Math, 325:105-144, ; Alt et al., in Trans Am Math Soc 282:431-461, ; Caffarelli et al., in Global energy minimizers for free boundary problems and full regularity in three dimensions. In: Non-compact Problems at the Intersection of Geometry, Analysis, and Topology, vol. 8397. Contemporary Mathematics, vol. 350. American Mathematical Society, Providence, ; DeSilva and Jerison, in J Reine Angew Math 635:121, ). Nevertheless we succeed in proving that they are locally Lipschitz, which is the optimal regularity for minimizers. [ABSTRACT FROM AUTHOR]

The inverse scattering problem of an interior cavity with three different boundary conditions is considered. Bayesian method is used to reconstruct the shape of the cavity from scattered fields incited by point source(s) and measured on a closed curve inside the cavity. We prove the well-posedness in Bayesian perspective and present numerical examples to show the viability of the method. [ABSTRACT FROM AUTHOR]

This paper deals with numerical methods providing semi-analytic solutions to a wide class of macroscopic traffic flow models for piecewise affine initial and boundary conditions. In a very recent paper, a variational principle has been proved for models of the Generic Second Order Modeling (GSOM) family, yielding an adequate framework for effective numerical methods. Any model of the GSOM family can be recast into its Lagrangian form as a Hamilton–Jacobi equation (HJ) for which the solution is interpreted as the position of vehicles. This solution can be computed thanks to Lax–Hopf like formulas and a generalization of the inf-morphism property. The efficiency of this computational method is illustrated through a numerical example and finally a discussion about future developments is provided. [ABSTRACT FROM AUTHOR]

- Local absorbing boundary conditions with fifth order of absorption to approximate the solution of an initial value problem, for the spatially discretized wave equation, are considered. For the one dimensional case, it is proved necessary conditions for well posedness. Numerical experiments confirming well posedness and showing good results of absorption are included. [ABSTRACT FROM AUTHOR]

Using the properties of the polynomial, Cheung and Li provided an alternative proof of Wu’s result on the circular symmetry of the classical numerical range. In this article, we further extend the idea and generalize Wu’s result to higher rank numerical ranges. [ABSTRACT FROM PUBLISHER]

Abstract: Most existing damage characterization exercises require explicitly defined material properties, geometric parameters and boundary conditions of the object to be evaluated. However, satisfying such a rudimental prerequisite can be a challenging task in reality. It may be somewhat difficult or even impossible to exactly acquire these parameters beforehand in some occasions, especially when the object comprises various structural components of distinct material/geometric properties and different boundary conditions. A damage characterization strategy, residing on a multi-scale pseudo-force damage model, was developed with an aim of achieving robust damage evaluation by tolerating ignorance of the knowledge on partial material/structural properties and boundary conditions of the object a priori. Proof-of-concept validation was carried out numerically and experimentally by characterizing a fine crack in an Euler–Bernoulli beam, a structural component included in a system with irregular boundaries, without knowing its material and structural parameters beforehand. Exact characterization results demonstrated the effectiveness and accuracy of the proposed approach. [Copyright &y& Elsevier]

We consider an evolution free-boundary problem for a stationary linear system of the theory of elasticity encountered in the investigation of solid thin films in microelectronic devices. Its solvability is proved on an arbitrary time interval under the condition that the initial data are sufficiently close to the stationary solution. [ABSTRACT FROM AUTHOR]

A certain one-phase problem with free boundary is studied. The local (in time) solvability of this problem is proved; moreover, the general method elaborated is applied in a more concrete case. For this purpose, a new change of variables and the parametrization of the boundary are introduced, and the problem studied is reduced to a problem in a constant domain. [ABSTRACT FROM AUTHOR]

Abstract: We consider a variable-coefficient wave equation with nonlinear damped acoustic boundary conditions. Well-posedness in the Hadamard sense for strong and weak solutions is proved by using the theory of nonlinear semigroups. [Copyright &y& Elsevier]

We consider a numerical solution of the stochastic moving boundary value problem, whose existence and uniqueness of solution are proved in [16]. Numerical approximations are based on the transformation, which transforms the stochastic moving boundary problem whose spatial domain is a priori unknown to a nonlinear stochastic partial differential equation which has a fixed spatial domain. We construct a numerical solution of the nonlinear stochastic partial differential equation and investigate the convergence theory. [ABSTRACT FROM AUTHOR]

The spectral methods of solving boundary value problems for mixed-type differential equations of second order in a 3D domain are studied. Existence and uniqueness of a solution of the Lavrent'ev–Bitsadze problem is proved. [ABSTRACT FROM AUTHOR]

This paper is devoted to the convergence analysis of the upwind finite volume scheme for the initial-and boundary-value problems associated with the linear transport equation in any dimension, on general unstructured meshes. We are particularly interested in the case where the initial and boundary data are in L∞ and the advection vector field v has low regularity properties, namely v ∈ L¹ (]0, T[, (W¹, ¹ (Q))d), with suitable assumptions on its divergence. In this general framework, we prove uniform in time strong convergence in LP(Q), with p < +∞, of the approximate solution towards the unique weak solu-tion of the problem as well as the strong convergence of its trace. The proof relies, in particular, on the Friedrichs' commutator argument, which is classical in the renormalized solutions theory. Note that this result remains valid if the data are suitably approximated in L¹. This is nothing but the discrete coun-terpart of the nice compactness properties deduced from the renormalized solution theory. We conclude with some numerical experiments showing that the convergence rate seems to be 1/2, as in the case of smoother advection fields, but this is still an open question up on up [ABSTRACT FROM AUTHOR]

In this paper the stability of the upwind scheme for second order, linear ordinary differential equations with general boundary conditions of the third kind is proved. The proof is carried out via compactness arguments and is rather short. Moreover, no information on the behaviour of the solution of the differential equation itself is needed. This approach seems not to be present in the existing literature. [ABSTRACT FROM AUTHOR]

In this paper we prove the existence of nonnegative renormalized solutions for the initial-boundary value problem of a non-uniformly parabolic equation. Some well-known parabolic equations are the special cases of this equation. [ABSTRACT FROM AUTHOR]

For the numerical solution of boundary-value problems on the semiaxis for second-order nonlinear ordinary differential equations, an exact three-point difference scheme is constructed and substantiated. Under the conditions of existence and uniqueness of solution of a boundary-value problem, we prove the existence and uniqueness of solution of the exact three-point difference scheme and convergence of the method of successive approximations for its solution. [ABSTRACT FROM AUTHOR]

We consider the time-harmonic Maxwell equations with constant coefficients in a bounded, uniformly star-shaped polyhedron. We prove wavenumber-explicit norm bounds for weak solutions. This result is pivotal for convergence proofs in numerical analysis and may be a tool in the analysis of electromagnetic boundary integral operators. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we prove the existence and uniqueness of solutions for the boundary value problem of nonlinear impulsive differential equations of fractional order . Our results are based on Altman’s fixed point theorem and Leray–Schauder’s fixed point theorem. [Copyright &y& Elsevier]

Abstract: Time-domain impedance boundary conditions are reviewed and similarities between the models are shown. The extended Helmholtz resonator model is used with a time-domain impedance boundary condition for impedance eduction and simulation of the noise radiation from a lined aeroengine nozzle. Model parameters obtained by impedance eduction from flow duct measurements and then used to predict the noise radiation for realistic aeroengine nozzle flow and geometry. The prediction of sound radiation from a coaxial jet nozzle at approach conditions with the educed model parameters is performed in 2D and 3D. Various features of the numerical method that enable it to be used for realistic applications are presented. This includes the suppression of flow instabilities in the linear inviscid solution and analysis of the solution to prove the global conservation of acoustic energy in each individual result. It is found that the current approach is too time consuming for optimisation, even for a 2D simulation using parallel processing. Thus, finally the potential of porting a CAA method to the graphics processing unit (GPU) is shown in one example. The GPU based computation is about 100 times faster. The results are encouraging, even though the GPU variant needs further validation. [Copyright &y& Elsevier]

The article examines the regular graph wherein the Hoffman bound for cocliques is equivalent to the Cvetcovich bound. It notes that the graph employs various parameters and eigenvalues with Krein conditions, and addresses some problems associated with regular graphs without multiple edges or loops. Moreover, it also analyzes the existence of a pseudogeometric graph containing a coclique of 49, and offers proofs for the lemma and the proposition using the assertions being derived from the process of substituting the corresponding values of the variables.

The Sturm-Liouville problem with linear discontinuities is investigated in the case where an eigenparameter appears not only in a differential equation but also in boundary conditions. Properties and the asymptotic behavior of spectral characteristics are studied for the Sturm-Liouville operators with Coulomb potential that have discontinuity conditions inside a finite interval. Moreover, the Weyl function for this problem is defined and uniqueness theorems are proved for a solution of the inverse problem with respect to this function. [ABSTRACT FROM AUTHOR]

Abstract: We discuss the inverse medium problem associated with semi-infinite domains. In particular, we attempt to image the spatial variability of shear moduli or shear wave velocities from scant surficial measurements of an arbitrarily heterogeneous semi-infinite domain’s response to prescribed dynamic excitations. We use a full waveform approach to drive the inversion process, within a PDE-constrained optimization framework. Due to the semi-infinite extent of the targeted domains, we introduce perfectly-matched-layers (PMLs) to arrive at finite computational domains. The numerical implementation is based on a mixed finite-element method that is used to resolve the ensuing state and adjoint boundary-value problems, both of which are PML-endowed. To alleviate the inherent solution multiplicity, we use Tikhonov and total variation (TV) regularization schemes, in conjunction with a regularization factor continuation scheme. To further improve the optimizer’s chances to converge, we also discuss a source-frequency continuation scheme. We report on two-dimensional numerical experiments using synthetic data. Included are layered profiles, and profiles involving inclined layers and inclusions. We also report on our methodology’s reconstruction of the highly-heterogeneous Marmousi benchmark velocity model. [ABSTRACT FROM AUTHOR]

A finite difference scheme is presented for a parabolic problem with mixed boundary conditions. We use an immersed interface technique to discretize the Neumann condition, and we use the Shortley-Weller approximation for the Dirichlet condition. The proof of a discrete maximum principle is given as well as the proof of convergence of the scheme. This convergence is also validated on numerical examples. [ABSTRACT FROM AUTHOR]

Abstract: Based on a previous publication (Savruk, 1981), a dislocation distribution layer method for the solution of interior and exterior boundary value problem (BVP) is studied in more detail. Properties of an integral operator in the resulting integral equation are studied. It is proved theoretically that the tractions applied on the outer boundary should be in equilibrium. In addition, a dislocation distribution layer method for the solution of exterior BVP is also suggested. In the exterior BVP, the tractions applied on the boundary may not be in equilibrium. In the exterior BVP, one must consider the single-valued condition of displacements. The formulation in the exterior BVP is not same as in the interior BVP. In the process of discretization, a technique for balance of the numbers of resulting algebraic equations and unknowns is suggested. Numerical examples prove that the suggested method can give sufficient accurate results. [Copyright &y& Elsevier]

This paper presents the first-known exact solutions for vibration of open circular cylindrical shells with multiple stepwise thickness variations based on the Flügge thin shell theory. An open cylindrical shell is assumed to be simply supported along the two straight edges and the remaining two opposite curved edges may have any combination of edge support conditions. The shell is subdivided into segments at the locations of thickness variations. The state-space technique is adopted to derive the homogenous differential equations for a shell segment and the domain decomposition method is employed to impose the equilibrium and compatibility requirements along the interfaces of the shell segments. The correctness of the proposed method is checked against existing results in the open literature and results generated from finite element package ANSYS and excellent agreement is achieved. Several open shells with various combinations of end boundary conditions are studied by the proposed method. [ABSTRACT FROM AUTHOR]

The effect of small initial deviations from the ideal circular shape of a shell on the frequencies and modes of flexural eigenvibrations is studied with the use of the linear theory of thin shallow shells. It is assumed that the initial deviations are responsible for interaction between flexural and radial vibrations of the shell. The modal equations are derived by the Bubnov—Galerkin method. It is shown that the initial deviations from the ideal circular shape split the flexural vibration spectrum, and the fundamental frequency decreases compared to that of the ideal shell. [ABSTRACT FROM AUTHOR]

Recently, the Shifted Boundary Method (SBM) was proposed within the class of unfitted (or immersed, or embedded) finite element methods. By reformulating the original boundary value problem over a surrogate (approximate) computational domain, the SBM avoids integration over cut cells and the associated problematic issues regarding numerical stability and matrix conditioning. Accuracy is maintained by modifying the original boundary conditions using Taylor expansions. Hence the name of the method, that shifts the location and values of the boundary conditions. In this article, we present enhanced variational SBM formulations for the Poisson and Stokes problems with improved flexibility and robustness. These simplified variational forms allow to relax some of the assumptions required by the mathematical proofs of stability and convergence of earlier implementations. First, we show that these new SBM implementations can be proved asymptotically stable and convergent even without the rather restrictive assumption that the inner product between the normals to the true and surrogate boundaries is positive. Second, we show that it is not necessary to introduce a stabilization term involving the tangential derivatives of the solution at Dirichlet boundaries, therefore avoiding the calibration of an additional stabilization parameter. Finally, we prove enhanced L 2 -estimates without the cumbersome assumption – of earlier proofs – that the surrogate domain is convex. Instead we rely on a conventional assumption that the boundary of the true domain is smooth, which can also be replaced by requiring convexity of the true domain. The aforementioned improvements open the way to a more general and efficient implementation of the Shifted Boundary Method, particularly in complex three-dimensional geometries. We complement these theoretical developments with numerical experiments in two and three dimensions. • A novel approximate domain (unfitted) method for the Poisson and Stokes operators. • The Shifted Boundary Method (SBM) does not require cut cells. • The SBM is defined on a surrogate domain, with modified boundary conditions. • A simpler SBM, without restrictive assumptions and unnecessary stabilization terms. • Improved proofs of stability, convergence and L 2 -error estimates. [ABSTRACT FROM AUTHOR]

Abstract: In this note, a new characterization of an attained boundary point of the numerical range of an operator on a Hilbert space is given. As an application, we point out a gap in the proof of the main result in the paper [M.T. Chien, L. Yeh, On the boundary of the numerical range of a matrix, Appl. Math. Lett. 23 (2010) 725–727]. [Copyright &y& Elsevier]

The notion of rational F-contractions using α -admissibility of type-S in b-metric-like spaces is introduced and the new fixed and periodic point theorems are proved for such mappings. Numerical examples are illustrated to check the efficiency and applicability of our fresh findings. It is also observed that some of the works reported in the literature are the particular cases of the present study. [ABSTRACT FROM AUTHOR]

We propose a new multi-symplectic integrating scheme for the Korteweg-de Vries (KdV) equation. The new scheme is derived by concatenating spatial discretization of the multi-symplectic Fourier pseudospectral method with temporal discretization of the symplectic Euler scheme. The new scheme is explicit in the sense that it does not need to solve nonlinear algebraic equations. It is verified that the multi-symplectic semi-discretization of the KdV equation under periodic boundary conditions has N semi-discrete multi-symplectic conservation laws. We also prove that the full-discrete scheme has N full-discrete multi-symplectic conservation laws. Numerical experiments of the new scheme on the KdV equation are made to demonstrate the stability and other merits for long-time integration. [ABSTRACT FROM AUTHOR]