Biharmonic Steklov problem in half-space

Abstract: In this paper, a half space adaptive fast multipole boundary facemethod (FMBFM) is presented for solving the three-dimensional half space ex-terior acoustic problems. In the presented method, the Burton-Miller equationbased on the conventional boundary integral equation (CBIE) and its hyper-singularboundary integral equation (HBIE) is used to deal with the fictitious eigenfrequen-cies problem. The half space Green’s function is employed, thus the tree structurein the fast multipole method can be used only for the real domain. The higher orderelements and an adaptive tree structure are used to improve the efficiency of theFMBFM. This half space adaptive FMBFM for half space acoustic problems is anextension of the adaptive FMBFM for full space acoustic problems developed bythe authors. Numerical examples for half space acoustic problems in this paperdemonstrate the efficiency and validity of this method.Keywords: fast multipole boundary face method, Burton-Miller equation, acous-tic problems, half space, modified hyper-singular boundary integral equation1 IntroductionAcoustic problems in half space are a major subject in engineering. Many researchworks have been done on this aspect (Seybert et al., 1988; Secgin et al., 2010; Caoet al., 2010). The half space Green’s function (Seybert et al., 1988) can be appliedby adding the source solution at the image point to the original full-space Green’sfunction for dealing with half space problems. Sound source localization and iden-tification of a refrigerator were accomplished by using the half space boundaryelement method (BEM) (Secgin et al., 2010). Using the half space Green’s func-tion, the discretization of the rigid infinite plane is removed and only the boundaries

In design, fixation is a recurring problem with novices where they choose the first solution that they think of for a given problem rather than explore a range of possible solutions. Fixation can result in barriers in design process, restricted search and premature commitment to solutions that may be sub-optimal. Among undergraduate engineering students, design fixation is due to limited prior conceptual knowledge, insufficient knowledge of design heuristics, and disconnected and evolving knowledge base, skills and strategies. One mechanism to overcome design fixation is expansion of problem and solution space where learners explore a variety of aspects related to the design problem such as constraints, assumptions, concepts and principles, before deciding on the solution to be followed. To be able to expand problem and solution space, novice designers require support. We have devised a pedagogical strategy flare-fork which scaffolds students’ problem space and solution space expansion in the flare phase and generation of multiple constraint optimized designs in the fork phase. Flare-fork is a systematic process that supports students to build on their opportunistic ideas to expand design boundaries and subsequently identify trade-offs to generate a variety of solutions tailored to fulfill a combination of constraints and assumptions. The pedagogy brings together design strategies, tools and practices such as rapid ideation, shared visual representation, categorization and techniques to restructure thinking patterns in a collaborative environment. In this paper, we first describe the conceptual design of our proposed pedagogy. We then describe a study to examine how the intervention operationalizing the flare phase of the pedagogy supports students in expanding their problem and solution space.

The reaction-diffusion system for an $SIR$ epidemic model with a free boundary is studied. This model describes a transmission of diseases. The existence, uniqueness and estimates of the global solution are discussed first. Then some sufficient conditions for the disease vanishing are given. With the help of investigating the long time behavior of solution to the initial and boundary value problem in half space, the long time behavior of the susceptible population $S$ is obtained for the disease vanishing case.

Symmetry of positive solutions to semilinear elliptic problems in half space

In the machine learning based on kernel tricks, people often put one variable of a kernel function on the given samples to produce the basic functions of a solution space of learning problem. If the collection of the given samples deviates from the data distribution, the solution space spanned by these basic functions will also deviate from the real solution space of learning problem. In this paper a multikernel-like learning algorithm based on data probability distribution (MKDPD) is proposed, in which the parameters of a kernel function are locally adjusted according to the data probability distribution, and thus produces different kernel functions. These different kernel functions will generate different Reproducing Kernel Hilbert Spaces (RKHS). The direct sum of the subspaces of these RKHS constitutes the solution space of learning problem. Furthermore, based on the proposed MKDPD algorithm, a new algorithm for labeling new coming data is proposed, in which the basic functions are retrained according to the new coming data, while the coefficients of the retrained basic functions remained unchanged to label the new coming data. The experimental results presented in this paper show the effectiveness of the proposed algorithms.

Energy method in the partial Fourier space and application to stability problems in the half space

A nonlinear integrodifferential initial value problem, which is induced from a “simple transport model,” is investigated. The underlying equation contains two parameters c and α. Here c (c≥0) denotes the fraction of the scattering per collision and α (0≤α≤1) is an angular shift. In this paper, we exploit the relationship between the solution in the half space and that in slab geometry. We are thus able to show that the problem has a unique, positive, uniformly bounded and globally defined solution for 0≤c≤ 1 and 0≤α≤1. Moreover, it is shown that such global solution converges to the minimal positive solution of the half space problem as the spatial variable approaches ∞ (i.e. the slab becomes thicker).

Two mechanisms widely used in the Software Product Lines (SPL) Engineering are the feature model and the conditional compilation. The former models the variability in the problem space and the latter realizes it in the solution space. Even though the research community know that the feature model imposes a number of constraints to the product line implementation, there is a lack of support to co-evolve problem space and solution space. In this paper, we present an exploratory study whether problem space constraints are considered at source code level of #ifdef-based SPL implementations. In order to accomplish our goal, we developed a preliminary approach to check problem and solution spaces in a prototype tool (fclcheck). The results show a lack of realization of feature model constraints while implementing variation points with that mechanism. We also evaluated the scalability of the approach and the recoverability of the tool.

This paper deals with the stress concentration problem of an elastic infinite thick plate with a hemispherical pit subjected to uniform pressure. The problem of an elastic half space with a hemispherical pit has been already inbestigeted. The present results for the thick plate are compared with the results for the half space. A method of solution is presented for the problem of a thick plate with a hemispherical pit by applying Green's functions for axisymmetric body force problems of an elastic infinite thick plate. The Green's functions are defined as solutions to the problem of an elastic infinite thick plate subjected to axisymmetric body forces acting along a circle in the interior of the plate.

In this manuscript, the generalized thermoelastic diffusion theory is modified by using fractional derivatives. The revised equation is employed to study a two‐dimensional problem of the axisymmetric temperature distribution during a half space with a permeating substance in touch with the bounding plane. The surface of the half space is taken into account to be traction‐free and subject to an axisymmetric temperature distribution. Laplace and Hankel transformation techniques are used. The analytical solution within the transform domain is obtained by an immediate method. By employing a numerical method supported the Fourier expansion technique, the inverse of the double transforms is often obtained. Numerical results for the temperature, displacement, stress, concentration, and chemical potential are administered and represented graphically. The figures show some comparisons to estimate the impact of fractional parameter on all research areas.

We study an initial boundary value problem for the Broadwellmodel with a supersonic physical boundary. The Green's function for an initialvalue problem is constructed and its detailed pointwise structure is obtainedthrough the novel decompositions introduced in [8]. With the Green's functionfor initial value problem and energy estimates together, a new approach toconvert a priori $L^2$-boundary data into $L^\infty$ boundary data is established for the Broadwell model. The Green's function for an initial boundary value problemis obtained. Finally, a nonlinearly time-asymptotic stability of an equilibriumstate is proved.

This paper deals with mathematical properties of the semiconductor Boltzmann equation. We first investigate the question of existence of solutions and some basic properties of the collision operator. Then we study the connection with the drift diffusion model by using a perturbation approach, the so-called diffusion approximation. For justifying rigorously this theory, we have to deal with boundary layers. A half space problem (Milne problem) arises in the study of the boundary corrector. We prove existence and we give the asymptotic behaviour of its solutions.

ABSTRACTWe study the unique (non-unique) solvability of the Neumann problem for the polyharmonic equation in unbounded domains under the assumption that a generalized solution of this problem has a bounded Dirichlet integral with weight . Depending on the value of the parameter a, we prove uniqueness (or non-uniqueness) theorems or present exact formulas for the dimension of the solution space of the Neumann problem in the exterior of a compact set and in a half space.

A of is a set of solutions arriving at the same local optimal solution by Best-improvement Local Search. By utilizing the concept of basin of attraction in the solution space of combinatorial optimization problem, the solution space is interpreted as a higher structure, which is a set of basin of attraction, and a lower structure, which is a set of solutions, in this paper. Based on the hierarchical interpretation of the solution space, which consists of the higher structure and the lower structure, and basic strategy in metaheuristics, an optimization method with search strategy to find a basin of attraction to which superior local optimal solution belongs is proposed. The search performance of this method was evaluated through numerical experiments using benchmark problems. In addition, the search situation and influence on search by parameter of this method are considered.

Generative AI, with its rapidly developing technologies, has changed how designers look to solve problems relating to the scene’s animation: eliminating inefficiencies, reducing high production costs, and lifting the cognitive load involved in the conception of an animation scene. Therefore, a systematic conceptual design model based on the design of animation scenes, which integrates generative AI tools—ChatGPT and Midjourney—is proposed in this paper. Based on the Co-evolution Model of Design and Divergent-Convergent Thinking, this model enables iterative co-evolution between the problem space and solution space, further provisionally refining a designer’s capacity to explicate goals while exploring various visual solutions effectively. The following is a series of design iterations in which such a model has been applied practically in line with the Research through Design approach. The results thus demonstrated that the proposed framework reduces the cognitive load on designers, increases efficiency, and improves creativity and coherence in the scenes created. Hence, this contributes to AIGen-assisted design-valid methodologies from both theoretical and practical perspectives, appealing to both practitioners and academics with the quick turn of generative AI into creative practice.