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This paper introduces a novel approach for the construction of bulk–surface splitting schemes for semilinear parabolic partial differential equations with dynamic boundary conditions. The proposed construction is based on a reformulation of the system as a partial differential–algebraic equation and the inclusion of certain delay terms for the decoupling. To obtain a fully discrete scheme, the splitting approach is combined with finite elements in space and a backward differentiation formula in time. Within this paper, we focus on the second-order case, resulting in a |$3$| -step scheme. We prove second-order convergence under the assumption of a weak CFL-type condition and confirm the theoretical findings by numerical experiments. Moreover, we illustrate the potential for higher-order splitting schemes numerically. [ABSTRACT FROM AUTHOR]

Let |$({\mathcal{M}},d,\mu)$| be the metric measure space with a Dirichlet form |$\mathscr{E}$|⁠. In this paper, we obtain that the Campanato function and the Lipschitz function do always coincide. Our approach is based on the harmonic extension technology, which extends a function u on |${\mathcal{M}}$| to its Poisson integral P t u on |${\mathcal{M}}\times\mathbb{R}_+$|⁠. With this tool in hand, we can utilize the same Carleson measure condition of the Poisson integral to characterize its Campanato/Lipschitz trace, and hence, they are equivalent to each other. This equivalence was previously obtained by Macías–Segovia [Adv. Math. 1979]. However, we provide a new proof, via the boundary value problem for the elliptic equation. This result indicates the famous saying of Stein–Weiss at the beginning of Chapter II in their book [Princeton Mathematical Series, No. 32, 1971]. [ABSTRACT FROM AUTHOR]

SUMMARY: When the conventional finite-element method is used to simulate the 3-D direct-current (DC) resistivity response over a conductive earth with large complex structures, it requires finely discretized mesh to accurately represent the underground structures. Directly solving the current-conduction problem on the fine mesh will lead to a huge amount of calculation. In this paper, we develop a fast 3-D forward modelling method for DC resistivity method based on smooth multiscale finite-element algorithm. Instead of using the conventional polynomial basis functions, we construct the multiscale basis functions by solving the local boundary-value problems of partial differential equation in parallel on the multiscale meshes. The multiscale basis functions can capture the small-scale heterogeneous information in coarse cells and reflect it to the large scale by assembling macro matrix of coarse mesh. Thus, it enables us to quickly obtain the accurate solution by solving the original current-conduction problem with complex structures on coarse mesh. We further adopt the oversampling technology to improve the forward modelling accuracy. Besides, by combining with the gradient smoothing technology, we avoid establishing the continuous form of multiscale basis functions and their spatial derivative integral operation to rapidly assemble the macro matrix. Finally, the reliability of the proposed algorithm is verified by applying our code to the 3-D complex models and comparing it with the conventional finite-element method. [ABSTRACT FROM AUTHOR]

In this paper, we present a numerical strategy to check the strong stability (or GKS-stability) of one-step explicit totally upwind schemes in 1D with numerical boundary conditions. The underlying approximated continuous problem is the one-dimensional advection equation. The strong stability is studied using the Kreiss–Lopatinskii theory. We introduce a new tool, the intrinsic Kreiss–Lopatinskii determinant, which possesses remarkable regularity properties. By applying standard results of complex analysis, we are able to relate the strong stability of numerical schemes to the computation of a winding number, which is robust and cheap. The study is illustrated with the Beam–Warming scheme together with the simplified inverse Lax–Wendroff procedure at the boundary. [ABSTRACT FROM AUTHOR]

This paper studies bulk–surface splitting methods of first order for (semilinear) parabolic partial differential equations with dynamic boundary conditions. The proposed Lie splitting scheme is based on a reformulation of the problem as a coupled partial differential–algebraic equation system, i.e. the boundary conditions are considered as a second dynamic equation that is coupled to the bulk problem. The splitting approach is combined with bulk–surface finite elements and an implicit Euler discretization of the two subsystems. We prove first-order convergence of the resulting fully discrete scheme in the presence of a weak CFL condition of the form |$\tau \leqslant c h$| for some constant |$c>0$|⁠. The convergence is also illustrated numerically using dynamic boundary conditions of Allen–Cahn type. [ABSTRACT FROM AUTHOR]

We show that inverse square singularities can be treated as boundary conditions containing rational Herglotz–Nevanlinna functions of the eigenvalue parameter with 'a negative number of poles'. More precisely, we treat in a unified manner one-dimensional Schrödinger operators with either an inverse square singularity or a boundary condition containing a rational Herglotz–Nevanlinna function of the eigenvalue parameter at each endpoint and define Darboux-type transformations between such operators. These transformations allow one, in particular, to transfer almost any spectral result from boundary value problems with eigenparameter-dependent boundary conditions to those with inverse square singularities and vice versa. [ABSTRACT FROM AUTHOR]

Heteroclinic-induced spiral waves may arise in systems of partial differential equations that exhibit robust heteroclinic cycles between spatially uniform equilibria. Robust heteroclinic cycles arise naturally in systems with invariant subspaces, and their robustness is considered with respect to perturbations that preserve these invariances. We make use of particular symmetries in the system to formulate a relatively low-dimensional spatial two-point boundary-value problem in Fourier space that can be solved efficiently in conjunction with numerical continuation. The standard numerical set-up is formulated on an annulus with small inner radius, and Neumann boundary conditions are used on both inner and outer radial boundaries. We derive and implement alternative boundary conditions that allow for continuing the inner radius to zero and so compute spiral waves on a full disk. As our primary example, we investigate the formation of heteroclinic-induced spiral waves in a reaction–diffusion model that describes the spatiotemporal evolution of three competing populations in a 2D spatial domain—much like the Rock–Paper–Scissors game. We further illustrate the efficiency of our method with the computation of spiral waves in a larger network of cyclic dominance between five competing species, which describes the so-called Rock–Paper–Scissors–Lizard–Spock game. [ABSTRACT FROM AUTHOR]

We consider the diffraction of an electromagnetic plane wave by a biperiodic structure. This paper is concerned with a numerical solution of the diffraction grating problem for three-dimensional Maxwell's equations. Based on the Dirichlet-to-Neumann (DtN) operator, an equivalent boundary value problem is formulated in a bounded domain by using a transparent boundary condition. An a posteriori error estimate-based adaptive edge finite element method is developed for the variational problem with the truncated DtN operator. The estimate takes account of both the finite element approximation error and the truncation error of the DtN operator, where the former is used for local mesh refinements and the latter is shown to decay exponentially with respect to the truncation parameter. Numerical experiments are presented to demonstrate the competitive behaviour of the proposed method. [ABSTRACT FROM AUTHOR]

The basins of attraction of periodic orbits or focus equilibria of a given vector field are foliated by forward-time isochrons, defined as all initial conditions that synchronize under the flow with a given phase. Similarly, backward-time isochrons of repelling periodic orbits or focus equilibria foliate their respective basins of repulsion. We present a case study of a planar system that features a sequence of bifurcations, including a saddle-node bifurcation of periodic orbits, a homoclinic bifurcation and Hopf bifurcations, that change the nature and existence of periodic orbits. We explain how the basins and isochron foliations change throughout the sequence of bifurcations. In particular, we identify structurally stable tangencies between the foliations by forward-time and backward-time isochrons, which are curves in the plane, in regions of phase space where they exist simultaneously. Such tangencies are generically quadratic and associated with sharp turns of isochrons and phase sensitivity of the system. In contrast to the earlier reported cubic isochron foliation tangency (CIFT) mechanism, which generates a pair of tangency orbits, we find isochron foliation tangencies that occur along single specific orbits in the respective basin of attraction or repulsion. Moreover, the foliation tangencies we report arise from actual bifurcations of the system, while a CIFT is not associated with a topological change of the underlying phase portrait. The properties and interactions of isochron foliations are determined and illustrated by computing a representative number of forward-time and backward-time isochrons as arclength-parametrized curves with a boundary value problem set-up. Our algorithm for computing isochrons has been further refined and implemented in the Matlab package CoCo; it is made available as Matlab code in the supplementary material of this paper, together with a guide that walks the user through the computation of two specific isochron foliations. [ABSTRACT FROM AUTHOR]

We correct Figures 6 and 8 in our paper On the stability of totally upwind schemes for the hyperbolic initial boundary value problem (https://doi.org/10.1093/imanum/drad040).In the originally published version of this manuscript, there were errors in Figures 6 and 8.Figure 6 should read:Graphinstead of:GraphFigure 8 should read:Graphinstead of:GraphThese emendations have been made in the article.By Benjamin Boutin; Pierre Le Barbenchon and Nicolas SeguinReported by Author; Author; Author [Extracted from the article]

In a series of papers published since 2017 the author introduced a simple alternative of the |$n$| -simplex type, to enhance the accuracy of approximations of second-order boundary value problems subject to Dirichlet boundary conditions, posed on smooth curved domains. This technique is based upon trial functions consisting of piecewise polynomials defined on straight-edged triangular or tetrahedral meshes, interpolating the Dirichlet boundary conditions at points of the true boundary. In contrast, the test functions are defined by the standard degrees of freedom associated with the underlying method for polytopic domains. While the mathematical analysis of the method for Lagrange and Hermite methods for two-dimensional second- and fourth-order problems was carried out in earlier paper by the author this paper is devoted to the study of the three-dimensional case. Well-posedness, uniform stability and optimal a priori error estimates in the energy norm are proved for a tetrahedron-based Lagrange family of finite elements. Novel error estimates in the |$L^2$| -norm, for the class of problems considered in this work, are also proved. A series of numerical examples illustrates the potential of the new technique. In particular, its superior accuracy at equivalent cost, as compared to the isoparametric technique, is highlighted. [ABSTRACT FROM AUTHOR]

In this paper, we propose a two-level additive Schwarz domain decomposition preconditioner for the symmetric interior penalty Galerkin method for a second-order elliptic boundary value problem with highly heterogeneous coefficients. A specific feature of this preconditioner is that it is based on adaptively enriching its coarse space with functions created by solving generalized eigenvalue problems on thin patches covering the subdomain interfaces. It is shown that the condition number of the underlined preconditioned system is independent of the contrast if an adequate number of functions are used to enrich the coarse space. Numerical results are provided to confirm this claim. [ABSTRACT FROM AUTHOR]

This paper deals with a priori pointwise error estimates for the finite element solution of boundary value problems with Neumann boundary conditions in polygonal domains. Due to the corners of the domain, the convergence rate of the numerical solutions can be lower than in the case of smooth domains. As a remedy, the use of local mesh refinement near the corners is considered. In order to prove quasi-optimal a priori error estimates, regularity results in weighted Sobolev spaces are exploited. This is the first work on the Neumann boundary value problem where both the regularity of the data is exactly specified and the sharp convergence order |$h^{2} \lvert \ln h \rvert $| in the case of piecewise linear finite element approximations is obtained. As an extension we show the same rate for the approximate solution of a semilinear boundary value problem. The proof relies in this case on the supercloseness between the Ritz projection to the continuous solution and the finite element solution. [ABSTRACT FROM AUTHOR]

The general three-component nonlinear Schrödinger (gtc-NLS) equations are completely integrable and contain the self-focusing, defocusing and mixed cases, which are applied in many physical fields. In this paper, we would like to use the Fokas method to explore the initial-boundary value (IBV) problem for the gtc-NLS equations with a |$4\times 4$| matrix Lax pair on a finite interval based on the inverse scattering transform. The solutions of the gtc-NLS equations can be expressed using the solution of a |$4\times 4$| matrix Riemann–Hilbert (RH) problem constructed in the complex |$k$| -plane. The jump matrices of the RH problem can be explicitly found in terms of three spectral functions related to the initial data, and the Dirichlet–Neumann boundary data, respectively. The global relation between the distinct spectral functions is also proposed to derive two distinct but equivalent types of representations of the Dirichlet–Neumann boundary value problems. Particularly, the relevant formulae for the boundary value problems on the finite interval can generate ones on the half-line as the length of the interval closes to infinity. Finally, we also analyse the linearizable boundary conditions for the Gel'fand–Levitan–Marchenko representation. These results will be useful to further study the solution properties of the IBV problem of the gtc-NLS system by using the Deift–Zhou's nonlinear steepest descent method and some numerical methods. [ABSTRACT FROM AUTHOR]

The multiscale hybrid-mixed (MHM) method consists of a multi-level strategy to approximate the solution of boundary value problems with heterogeneous coefficients. In this context, we propose a new family of finite elements for the linear elasticity equation defined on coarse polytopal partitions of the domain. The finite elements rely on face degrees of freedom associated with multiscale bases obtained from local Neumann problems with piecewise polynomial interpolations on faces. We establish sufficient conditions on the fine-scale interpolations such that the MHM method is well-posed and optimally convergent under local regularity conditions. Also, a multi-level error analysis demonstrates that the MHM method achieves convergence without refining the coarse partition. The upshot is that the Poincaré and Korn's inequalities do not degenerate, and then the convergence arises on general meshes. Two- and three-dimensional numerical tests assess theoretical results and verify the robustness of the method on a multi-layer media case. [ABSTRACT FROM AUTHOR]

We consider a space–time fractional parabolic problem. Combining a sinc quadrature-based method for discretizing the Riesz–Dunford integral with |$hp$| -FEM in space yields an exponentially convergent scheme for the initial boundary value problem with homogeneous right-hand side. For the inhomogeneous problem, an |$hp$| -quadrature scheme is implemented. We rigorously prove exponential convergence with focus on small times |$t$|⁠ , proving robustness with respect to startup singularities due to data incompatibilities. [ABSTRACT FROM AUTHOR]

In this work, we present a boundary oriented graph embedding (BOGE) approach for the graph neural network to assist in rapid design and digital prototyping. The cantilever beam problem has been solved as an example to validate its potential of providing physical field results and optimized designs using only 10 ms. Providing shortcuts for both boundary elements and local neighbor elements, the BOGE approach can embed unstructured mesh elements into the graph and performs an efficient regression on large-scale triangular-mesh-based finite element analysis (FEA) results, which cannot be realized by other machine-learning-based surrogate methods. It has the potential to serve as a surrogate model for other boundary value problems. Focusing on the cantilever beam problem, the BOGE approach with 3-layer DeepGCN model achieves the regression with mean square error (MSE) of0.011706 (2.41% mean absolute percentage error) for stress field prediction and 0.002 735 MSE (with 1.58% elements having error larger than 0.01) for topological optimization. The overall concept of the BOGE approach paves the way for a general and efficient deep-learning-based FEA simulator that will benefit both industry and Computer Aided Design (CAD) design-related areas. [ABSTRACT FROM AUTHOR]

We present N -soliton solutions for the classical (1+1)-dimensional Gross–Neveu model which satisfy non-zero boundary conditions. These solutions are obtained by a direct method using some properties of the soliton matrices that appear in the framework of the Cauchy matrix approach. [ABSTRACT FROM AUTHOR]

We give a short proof of local well-posedness for the initial boundary value problem in general relativity with sole boundary condition the requirement that the boundary is umbilic. This includes as a special case the totally geodesic boundary condition that we had previously addressed in [ 8 ]. The proof is based on wave coordinates and the key observation that the momentum constraint is always valid for umbilic boundaries. This allows for a greater freedom in the choice of boundary conditions, since imposing the umbilic condition also provides Neumann boundary conditions for three of the four wave coordinates conditions. Moreover, the umbilic condition, being geometric, implies that geometric uniqueness in the sense of Friedrich holds in this specific case. [ABSTRACT FROM AUTHOR]

In this paper we derive a priori |$L^{\infty }(L^{2})$| and L 2(L 2) error estimates for a linear advection–reaction equation with inlet and outlet boundary conditions. The goal is to derive error estimates for the discontinuous Galerkin method that do not blow up exponentially with respect to time, unlike the usual case when Gronwall's inequality is used. While this is possible in special cases, such as divergence-free advection fields, we take a more general approach using exponential scaling of the exact and discrete solutions. Here we use a special scaling function, which corresponds to time taken along individual pathlines of the flow. For advection fields, where the time that massless particles carried by the flow spend inside the spatial domain is uniformly bounded from above by some |$\widehat{T}$|⁠, we derive |$\mathcal{O}$| (h p +1/2) error estimates where the constant factor depends only on |$\widehat{T}$|⁠, but not on the final time T. This can be interpreted as applying Gronwall's inequality in the error analysis along individual pathlines (Lagrangian setting), instead of physical time (Eulerian setting). [ABSTRACT FROM AUTHOR]

In this paper, we develop and study algorithms for approximately solving linear algebraic systems: |${{\mathcal{A}}}_h^\alpha u_h = f_h$|⁠ , |$ 0< \alpha <1$|⁠ , for |$u_h, f_h \in V_h$| with |$V_h$| a finite element approximation space. Such problems arise in finite element or finite difference approximations of the problem |$ {{\mathcal{A}}}^\alpha u=f$| with |${{\mathcal{A}}}$|⁠ , for example, coming from a second-order elliptic operator with homogeneous boundary conditions. The algorithms are motivated by the method of Vabishchevich (2015, Numerically solving an equation for fractional powers of elliptic operators. J. Comput. Phys. , 282 , 289–302) that relates the algebraic problem to a solution of a time-dependent initial value problem on the interval |$[0,1]$|⁠. Here we develop and study two time-stepping schemes based on diagonal Padé approximation to |$(1+x)^{-\alpha }$|⁠. The first one uses geometrically graded meshes in order to compensate for the singular behaviour of the solution for |$t$| close to |$0$|⁠. The second algorithm uses uniform time stepping, but requires smoothness of the data |$f_h$| in discrete norms. For both methods, we estimate the error in terms of the number of time steps, with the regularity of |$f_h$| playing a major role for the second method. Finally, we present numerical experiments for |${{\mathcal{A}}}_h$| coming from the finite element approximations of second-order elliptic boundary value problems in one and two spatial dimensions. [ABSTRACT FROM AUTHOR]

We provide the first significant extension of the unified transform (also known as the Fokas method) applied to elliptic boundary value problems, namely, we extend the method to curvilinear polygons and partial differential equations (PDEs) with variable coefficients. This is used to solve the generalized Dirichlet-to-Neumann map. The central component of the unified transform is the coupling of certain integral transforms of the given boundary data and of the unknown boundary values. This has become known as the global relation and, in the case of constant coefficient PDEs, simply links the Fourier transforms of the Dirichlet and Neumann boundary values. We extend the global relation to PDEs with variable coefficients and to domains with curved boundaries. Furthermore, we provide a natural choice of global relations for separable PDEs. These generalizations are numerically implemented using a method based on Chebyshev interpolation for efficient and accurate computation of the integral transforms that appear in the global relation. Extensive numerical examples are provided, demonstrating that the method presented in this paper is both accurate and fast, yielding exponential convergence for sufficiently smooth solutions. Furthermore, the method is straightforward to use, involving just the construction of a simple linear system from the integral transforms, and does not require knowledge of Green's functions of the PDE. Details on the implementation are discussed at length. [ABSTRACT FROM AUTHOR]

A correction to the article "L∞(L∞)-boundedness and convergence of DG(p)-solutions for nonlinear conservation laws with boundary conditions," that was published in the previous issue is presented.

In the context of integrable difference equations on quad-graphs, we introduce the method of open boundary reductions, as an alternative to the well-known periodic reductions, for constructing discrete integrable mappings and their invariants. The mappings are obtained from well-posed initial value problems for quad and boundary equations restricted to strips on |${{\mathbb{Z}}}^2$| -lattices. The invariants are constructed using Sklyanin's double-row monodromy matrix. To establish its properties, we use the discrete zero curvature condition and boundary zero curvature condition, showing how the latter derives from the boundary consistency condition. We focus on the Adler–Bobenko–Suris classification and associated integrable boundary equations. Examples are given for the H1 and Q1(⁠|$\delta =0$|⁠) equations, leading to novel maps of the plane. [ABSTRACT FROM AUTHOR]

This research merges insights from the communications literature with that on the self-brand connection to examine a novel question: how does speaking versus writing about a liked brand influence the communicator’s own later reactions to that brand? Our conceptualization argues that because oral communication involves a greater focus on social interaction with the communication recipient than does written communication, oral communicators are more likely to express self-related thoughts than are writers, thereby increasing their self-brand connection (SBC). We also assess the implications of this conceptualization, including the identification of theoretically derived boundary conditions for the speech/writing difference, and the downstream effects of heightened SBC. Results from five studies provide support for our predictions, informing both the basic literature on communications, and the body of work on consumer word of mouth. [ABSTRACT FROM AUTHOR]

This article presents a |$P_0$| finite element method for boundary value problems for linear elasticity equations. The new method makes use of piecewise constant approximating functions on the boundary of each polytopal element and is devised by simplifying and modifying the weak Galerkin finite element method based on |$P_1/P_0$| approximations for the displacement. This new scheme includes a tangential stability term on top of the simplified weak Galerkin to ensure the necessary stability due to the rigid motion. The new method involves a small number of unknowns on each element, it is user friendly in computer implementation and the element stiffness matrix can be easily computed for general polytopal elements. The numerical method is of second-order accurate, locking-free in the nearly incompressible limit, and ease polytopal partitions in practical computation. Error estimates in |$H^1$|⁠ , |$L^2$| and some negative norms are established for the corresponding numerical displacement. Numerical results are reported for several two-dimensional and three-dimensional test problems, including the classical benchmark Cook's membrane problem in two dimensions as well as some three-dimensional problems involving shear-loaded phenomena. The numerical results show clearly the simplicity, stability, accuracy and efficiency of the new method. [ABSTRACT FROM AUTHOR]

We consider an inverse boundary value problem for a semilinear wave equation on a time-dependent Lorentzian manifold with time-like boundary. The time-dependent coefficients of the nonlinear terms can be recovered in the interior from the knowledge of the Neumann-to-Dirichlet map. Either distorted plane waves or Gaussian beams can be used to derive uniqueness. [ABSTRACT FROM AUTHOR]

This paper presents a combined field and boundary integral equation method for solving the time-dependent scattering problem of a thermoelastic body immersed in a compressible, inviscid and homogeneous fluid. The approach here is a generalization of the coupling procedure employed by the authors for the treatment of the time-dependent fluid–structure interaction problem. Using an integral representation of the solution in the infinite exterior domain occupied by the fluid, the problem is reduced to one defined only over the finite region occupied by the solid, with nonlocal boundary conditions. The nonlocal boundary problem is analysed with Lubich's approach for time-dependent boundary integral equations. Existence and uniqueness results are established in terms of time-domain data with the aid of Laplace domain techniques. Galerkin semidiscretization approximations are derived and error estimates are obtained. A full discretization based on the convolution quadrature method is also outlined. Some numerical experiments in two dimensions are also included in order to demonstrate the accuracy and efficiency of the procedure. [ABSTRACT FROM AUTHOR]

In fluid dynamical simulations in astrophysics, large deformations are common and surface tracking is sometimes necessary. The smoothed particle hydrodynamics (SPH) method has been used in many such simulations. Recently, however, it has been shown that SPH cannot handle contact discontinuities or free surfaces accurately. There are several reasons for this problem. The first one is that SPH requires that the density is continuous and differentiable. The second one is that SPH does not have consistency, and thus the accuracy is of the zeroth-order in space. In addition, we cannot express accurate boundary conditions with SPH. In this paper, we propose a novel, high-order scheme for particle-based hydrodynamics of compressible fluid. Our method is based on a kernelweighted high-order fitting polynomial for intensive variables. With this approach, we can construct a scheme which solves all of the three problems described above. For shock capturing, we use a tensor form of von Neumann-Richtmyer artificial viscosity. We have applied our method to many test problems and obtained excellent results. Our method is not conservative, since particles do not have mass or energy, but only their densities. However, because of the Lagrangian nature of our scheme, the violation of the conservation laws turned out to be small. We name this method Consistent Particle Hydrodynamics in Strong Form (CPHSF). [ABSTRACT FROM AUTHOR]

In this paper, we analyse initial-boundary problems for the vector derivative nonlinear Schrödinger equation on the semi-infinite strip (x, t) ∊ (0,ϕ)×(0, T) via the unified transform method of Fokas. Even though additional technical complications arise in the vector case compared with scalar ones, we show that it also can be expressed in terms of the solution of a matrix Riemann-Hilbert problem. The Riemann- Hilbert problem involves a jump matrix, uniquely defined in terms of four matrix functions called spectral functions and denoted by {a(λ), b(λ), A(λ), B(λ)} that depend on the initial data and all boundary values, respectively. A key role is played by the so-called global relation which involves the known and unknown boundary values. By analysing the global relation, we present an effective characterization of the latter two spectral functions in terms of the given initial and boundary data. [ABSTRACT FROM AUTHOR]

In the previous paper, we have examined how the self-trapping of g-mode oscillations in geometrically thin relativistic discs is affected by the presence of uniform vertical magnetic fields. Discs considered are isothermal in the vertical direction and are truncated at a certain height by presence of hot corona. After the correction of simple analytical error, we showed that the self-trapping of axisymmetric g-mode oscillations in non-magnetized discs is destroyed by weak magnetic fields as Fu and Lai showed. In this paper, however, we re-examine the same problem by imposing a different, more relevant boundary condition on the disc-corona surface and find that the self-trapping of axisymmetric g-mode oscillations seems to still exist like in the case of non-magnetized discs. [ABSTRACT FROM AUTHOR]

We consider the Dirichlet boundary value problem for graphical maximal submanifolds inside Lorentzian-type ambient spaces and obtain general existence and uniqueness results that apply to any codimension. [ABSTRACT FROM AUTHOR]

This paper considers the initial-boundary value problem for the heat equation with a dynamic-type boundary condition. Under some regularity, consistency and orthogonality conditions, the existence, uniqueness and continuous dependence upon the data of the classical solution are shown by using the generalized Fourier method. This paper also investigates the inverse problem of finding a time-dependent coefficient of the heat equation from the data of integral overdetermination condition. [ABSTRACT FROM AUTHOR]

In this paper, we study the first initial-boundary value problem for a class of fully nonlinear parabolic equations on Riemannian manifolds. The main part of this paper is to derive the a priori estimates for the second-order derivatives of solutions by using a new idea that works under very general structure conditions. As an application, the existence of smooth solutions is proved. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to push forward the theory of operator-valued Riemann-Hilbert problems and demonstrate their effectiveness in respect to the implementation of a non-linear steepest descent method a la Deift-Zhou. In this paper, we demonstrate that the operator-valued Riemann-Hilbert problem arising in the characterization of so-called c-shifted integrable integral operators allows one to extract the large-x asymptotics of the Fredholm determinant associated with such operators. [ABSTRACT FROM AUTHOR]

This paper concerns the asymptotic expansion of the solution of the Dirichlet-Laplace problem in a domain with small inclusions. This problem is well understood for the Neumann condition in dimension ≥2 or the Dirichlet condition in dimension ≥3. The case of two circular inclusions in a bidimensional domain was considered in [1]. In this paper, we generalize the previous result to any shape and relax the assumptions of regularity and support of the data. Our approach uses conformal mapping and suitable lifting of Dirichlet conditions.We also analyze configurations with several scales for the distance between the inclusions (when the number is larger than 2). [ABSTRACT FROM AUTHOR]

We consider the Dirichlet boundary value problem for graphical maximal submanifolds inside Lorentzian-type ambient spaces and obtain general existence and uniqueness results that apply to any codimension. [ABSTRACT FROM AUTHOR]

A new transform pair representing solutions to the complex Helmholtz equation in a convex 2D polygon is derived using the theory of Bessel's functions and Green's second identity. The derivation is a direct extension of that given by Crowdy (2015, A transform method for Laplace's equation in multiply connected circular domains. IMA J. Appl. Math. , 80 , 1902–1931) for 'Fourier–Mellin transform' pairs associated with Laplace's equation in various domain geometries. It is shown how the new transform pair fits into the collection of ideas known as the Fokas transform where the key step in solving any given boundary value problem is the analysis of a global relation. Here we contextualize those global relations from the point of view of 'reciprocal theorems', which are familiar tools in the study of the effective properties of physical systems. A survey of the many uses of this new transform approach to the complex Helmholtz equation in applications is given. This includes calculation of effective impedance in electrochemical impedance spectroscopy and in other spectroscopy methods in diffusion wave field theory, application to the 3 |$\omega $| method for measuring thermal conductivity and to unsteady Stokes flow. A theoretical connection between this analysis of the global relations and Lorentz reciprocity in mathematical physics is also pointed out. [ABSTRACT FROM AUTHOR]

This research examines the implications of consumers' belief in karma--the belief that the universe bestows rewards for doing right and exacts punishments for doing wrong--in the context of prosocial behavior. Although intuitively, believing in karma should result in greater intentions to do right by supporting a charity, karmic beliefs are found to facilitate prosocial behavior only in contexts not associated with selfgains. A series of experiments shows that those with strong (vs. weak) beliefs in karma actually respond less favorably to charitable appeals that rely on common marketing tools meant to enhance consumer responses but that also cue self-gains by offering incentives or by highlighting self-benefits. However, these effects are only obtained for donations of time, which represent a means to enhance social connections, but not for donations of money. Consistent with the proposition that prosocial behaviors motivated by self-gains do not engender karmic rewards, lower intentions to do right among those with strong karmic beliefs are driven by a shift from other-focused to self-focused attention following appeals that cue self-gains, as compared to appeals that do not. Results imply that marketers need to take into account consumers' karmic beliefs when seeking to incentivize prosocial behavior. [ABSTRACT FROM AUTHOR]

Ever since the pioneering work of Ihlenburg and Babuška for the one-dimensional Helmholtz equation (1995, Comput. Math. Appl., 30, 9–37), a lot of pre-asymptotic analyses, including dispersion and pollution analyses, have been carried out for finite element methods (FEMs) for the Helmholtz equation, but no rigorous pre-asymptotic error analysis of the FEM for the Helmholtz equation in two and three dimensions has been done yet. This paper addresses pre-asymptotic stability and error estimates of some continuous interior penalty finite element methods (CIP-FEM) and the FEM using piecewise linear polynomials for the Helmholtz equation in two and three dimensions. The CIP-FEM, which was first proposed by Douglas and Dupont for elliptic and parabolic problems in the 1970s and then successfully applied to convection-dominated problems as a stabilization technique, modifies the bilinear form of the FEM by adding a least squares term penalizing the jump of the gradient of the discrete solution at mesh interfaces, while the penalty parameters are chosen as complex numbers here instead of real numbers as usual. In this paper, it is proved that if the penalty parameter is a pure imaginary number i γ with 0 < γ ≤ C, then the CIP-FEM is stable (hence well posed) without any mesh constraint. Moreover, the error of the CIP-FEM in the H1 norm is bounded by C1kh + C2k3h2 when k3h2 ≤ C0 and by C1kh + C2/γ when k3h2>C0 and kh≤C3, where k is the wave number, h is the mesh size, and the C's are some positive constants independent of k, h and γ. By taking γ → 0+ in the CIP-FEM it is shown that if k3h2 ≤ C0, then the FEM attains a unique solution and satisfies the error estimate C1kh + C2k3h2 in the H1 norm. Previous analytical results for the classical linear FEM in higher dimensions require the assumption that k2h is small, but it is too strict for a medium or high wave number. Optimal order L2-error estimates are also derived. Numerical results are provided to verify the theoretical findings and illustrate the performance of the two methods. In particular, it is shown that the pollution effect may be greatly reduced by tuning the penalty parameters in the CIP-FEM. The theoretical and numerical results reveal that the CIP-FEM has advantages over the FEM and is worthy of being investigated further. The pre-asymptotic error analysis of the hp version of the CIP-FEM and the FEM is studied in Part II. [ABSTRACT FROM PUBLISHER]

In this paper, we consider the stability and convergence of numerical discretizations of the Black–Scholes partial differential equation (PDE) when complemented with the popular linear boundary condition (LBC). This condition states that the second derivative of the option value vanishes when the underlying asset price gets large and is often applied in the actual numerical solution of PDEs in finance. To our knowledge, the only theoretical stability result in the literature up to now pertinent to the LBC was obtained by Windcliff et al. (2004, Analysis of the stability of the linear boundary condition for the Black–Scholes equation, J. Comput. Finance, 8, 65–92) who showed that for a common discretization, a necessary eigenvalue condition for stability holds. In this paper, we shall present sufficient conditions for stability and convergence when the LBC is employed. We deal with finite difference discretizations in the spatial (asset) variable and a subsequent implicit discretization in time. As a main result, we prove that even though the maximum norm of etM (t≥0) can grow with the dimension of the semidiscrete matrix M, this generally does not impair the convergence behaviour of the numerical discretizations. Our theoretical results are illustrated by ample numerical experiments. [ABSTRACT FROM PUBLISHER]

Standard 1D stellar evolution models do not correctly reproduce the structure of the outermost layers of stars with convective envelopes. This has been a long-standing problem in stellar modelling affecting both the predicted evolutionary paths and the attributed oscillation frequencies, and indirectly biasing numerous quantities derived from stellar evolution calculations. We present a novel method that mostly eliminates these structural defects by appending mean 3D simulations of stellar envelopes. In contrast to previous attempts, we impose the complete structure derived from 3D simulations at each time-step during the entire evolution. For this purpose, we interpolate in grids of pre-computed 3D simulations and use the resulting structure as boundary conditions, in order to solve the stellar structure equations for the 1D interior at each time-step. Our method provides a continuous transition in many quantities from the interior to the imposed interpolated 3D surface layers. We present a solar calibration model and show that the obtained structure of the surface layers reliably mimics that of the underlying 3D simulations for the present Sun. Moreover, we perform a helioseismic analysis, showing that our method mostly eliminates the structural contribution to the discrepancy between model frequencies and observed p-mode frequencies. [ABSTRACT FROM AUTHOR]

In 1951, Pólya and Szegö expressed a lower bound for the torsional rigidity of a convex plane domain in terms of the area and inradius of the domain. On the other hand, in 1995 Avkhadiev showed that the torsional rigidity and the Euclidean moment of inertia are comparable quantities for simply connected domains. In this paper, for the p-order Euclidean moments of a convex plane domain, we prove inequalities similar to that of Pólya and Szegö. These geometrical inequalities allow us to obtain upper bounds for the torsional rigidity of a convex plane domain in terms of the p-order Euclidean moment of the domain. [ABSTRACT FROM AUTHOR]

Discrete approximations to the equation $$\begin{equation*}L_\textrm{cont}u = u^{(4)} + D(x) u^{(3)} + A(x) u^{(2)} + (A^{\prime}(x)+H(x)) u^{(1)} + B(x) u = f, \quad x\in[0,1]\end{equation*}$$ are considered. This is an extension of the Sturm–Liouville case |$D(x)\equiv H(x)\equiv 0$| (Ben-Artzi et al. (2018) Discrete fourth-order Sturm–Liouville problems. IMA J. Numer. Anal. , 38 , 1485–1522) to the non-self-adjoint setting. The 'natural' boundary conditions in the Sturm–Liouville case are the values of the function and its derivative. The inclusion of a third-order discrete derivative entails a revision of the underlying discrete functional calculus. This revision forces evaluations of accurate discrete approximations to the boundary values of the second-, third- and fourth-order derivatives. The resulting functional calculus provides the discrete analogs of the fundamental Sobolev properties of compactness and coercivity. It allows us to obtain a general convergence theorem of the discrete approximations to the exact solution. Some representative numerical examples are presented. [ABSTRACT FROM AUTHOR]

This paper is concerned with the inverse scattering problem of time-harmonic acoustic plane waves by a crack buried in a piecewise homogeneous medium. We first show the well-posedness of the solution to the direct problem by using the boundary integral method. Then a uniqueness result for the inverse problem is obtained, that is, the shape of the crack and the surface parameter can be uniquely determined by the far-field pattern. The proof of the uniqueness result is based on a generalization of the mixed reciprocity relation. In particular, a mathematical basis is given to retrieve the shape of the crack by using the linear sampling method and difficulties arise from the fact that the crack has an empty interior. [ABSTRACT FROM AUTHOR]

This paper concerns the global propagation of impact-induced tensile waves in a kind of phase transforming materials. It is well known that the governing system of partial differential equations is hyperbolic–elliptic and the initial boundary-value problem is not well posed at all levels of loading. By making use of a fully non-linear stress–strain curve to model these materials, Dai and Kong succeeded in constructing a physical solution of the above initial boundary-value problem. For impact of intermediate range, they assumed that β<3α in the stress-response function for simplicity. In this paper, we revisit the impact problem and consider the propagation of impact-induced tensile waves for all values of the parameters α and β. The physical solutions for all levels of loading are obtained completely. [ABSTRACT FROM PUBLISHER]

The empirical upper luminosity boundary L max of cool supergiants (SGs), often referred to as the Humphreys-Davidson limit, is thought to encode information on the general mass-loss behaviour of massive stars. Further, it delineates the boundary at which single stars will end their lives stripped of their hydrogen-rich envelope, which in turn is a key factor in the relative rates of Type-II to Type-Ibc supernovae from single star channels. In this paper we have revisited the issue of L max by studying the luminosity distributions of cool SGs in the Large and Small Magellanic Clouds (LMC/SMC). We assemble samples of cool SGs in each galaxy which are highly complete above log L /L⊙= 5.0, and determine their spectral energy distributions from the optical to the mid-infrared using modern multiwavelength survey data. We show that in both cases L max appears to be lower than previously quoted, and is in the region of log  L /L⊙ = 5.5. There is no evidence for L max being higher in the SMC than in the LMC, as would be expected if metallicity-dependent winds were the dominant factor in the stripping of stellar envelopes. We also show that L max aligns with the lowest luminosity of single nitrogen-rich Wolf-Rayet stars, indicating of a change in evolutionary sequence for stars above a critical mass. From population synthesis analysis we show that the Geneva evolutionary models greatly overpredict the numbers of cool SGs in the SMC. We also argue that the trend of earlier average spectral types of cool SGs in lower metallicity environments represents a genuine shift to hotter temperatures. Finally, we use our new bolometric luminosity measurements to provide updated bolometric corrections for cool SGs. [ABSTRACT FROM AUTHOR]

Current implementations of the stellar atmosphere typically derive boundary conditions for the interior model from either grey plane-parallel atmospheres or scaled solar atmospheres, neither of which can be considered to have appropriate underlying assumptions for the Thermally Pulsing Asymptotic Giant Branch (TP-AGB). This paper discusses the treatment and influence of the outer boundary condition within stellar evolution codes, and the resulting effects on the AGB evolution. The complex interaction of processes, such as the third dredge up and mass-loss, governing the TP-AGB can be affected by varying the treatment of this boundary condition. Presented here are the results from altering the geometry, opacities, and the implementation of a grid of MARCS/COMARCS model atmospheres in order to improve this treatment. Although there are changes in the TP-AGB evolution, observable quantities, such as the final core mass, are not significantly altered as a result of the change of atmospheric treatment. During the course of the investigation, a previously unseen phenomenon in the AGB models was observed and further investigated. This is believed to be physical, although arising from specific conditions which make its presence unlikely. If it were present in stars, this phenomenon would increase the carbon-star lifetime above 10Myr and increase the final core mass by ~0.1M... in the narrow initial-mass range where it was observed (~2-2.3M...). [ABSTRACT FROM AUTHOR]

In this paper, we study 3-D multiparameter full waveform inversion (FWI) in viscoelastic media based on the generalized Maxwell/Zener body including arbitrary number of attenuation mechanisms. We present a frequency-domain energy analysis to establish the stability condition of a full anisotropic viscoelastic system, according to zero-valued boundary condition and the elastic-viscoelastic correspondence principle: the real-valued stiffness matrix becomes a complex-valued one in Fourier domain when seismic attenuation is taken into account. We develop a least-squares optimization approach to linearly relate the quality factor with the anelastic coefficients by estimating a set of constants which are independent of the spatial coordinates, which supplies an explicit incorporation of the parameter Q in the general viscoelastic wave equation. By introducing the Lagrangian multipliers into the matrix expression of the wave equation with implicit time integration, we build a systematic formulation of multiparameter FWI for full anisotropic viscoelastic wave equation, while the equivalent form of the state and adjoint equation with explicit time integration is available to be resolved efficiently. In particular, this formulation lays the foundation for the inversion of the parameter Q in the time domain with full anisotropic viscoelastic properties. In the 3-D isotropic viscoelastic settings, the anelastic coefficients and the quality factors using bulk and shear moduli parametrization can be related to the counterparts using P and S velocity. Gradients with respect to any other parameter of interest can be found by chain rule. Pioneering numerical validations as well as the real applications of this most generic framework will be carried out to disclose the potential of viscoelastic FWI when adequate high-performance computing resources and the field data are available. [ABSTRACT FROM AUTHOR]