Approximation of slow moving interface phase change problems using a generalized Fourier series and the CVBEM

The inlet region of a plane-parallel pressure flow is the subject of this study. The patterns of variations in the hydrodynamic inlet region under unsteady plane-parallel pressure flow of a viscous fluid are examined in this research. Based on the boundary layer equation and flow characteristics, the boundary conditions of the problem were determined and a boundary value problem was formulated. The problem's boundary conditions were established and a boundary value problem was developed based on the boundary layer equation and flow characteristics. In order to find patterns of velocity change over time and over the length of the inlet region under general boundary conditions, a method for integrating the boundary value conditions was created. Solutions for scenarios with a constant and parabolic velocity distribution in the inlet region were derived from the general solutions. Regularities of pressure and velocity change were found along the entire hydrodynamic inlet region. Using computer analysis, graphs of velocity changes over time at various points along the entire length of the inlet region are constructed. The patterns of velocity distribution along the entire length of the inlet region depending on time can be seen using graphs. This allows one to estimate the length of the hydrodynamic inlet region and calculate the fluid flow velocity at any point in this region. The findings enable revealing the essence of the processes running in an hydropneumatic automation system's transition sections. Based on the revealed regularities of the hydrodynamic parameters of viscous incompressible liquid during unsteady flows, it is possible to correctly design of the automatic systems' channels of regulating units ensuring their smooth and accurate operation

A C1-conforming hp finite element method for fourth order singularly perturbed boundary value problems

A branch-price-and-cut algorithm is developed for a complex maritime inventory-routing problem with varying storage capacities and production/consumption rates at facilities. The resulting mixed-integer pricing problem is solved exactly and efficiently using a dynamic program that exploits certain “extremal” characteristics of the pricing problem. The formulation is tightened by using the problem's boundary conditions in preprocessing and to restrict the set of columns that are produced by the pricing problem. Branching schemes and cuts are introduced that can be implemented efficiently and that preserve the structure of the pricing problem. Some of the cuts are inspired by the capacity cuts known for the vehicle-routing problem, whereas others specifically target fractional solutions brought about by individual vessels “competing” for limited inventory at load ports and limited storage capacity at discharge ports. The branch-price-and-cut approach solves practically sized problems motivated by the operations of an oil company to optimality, and it provides reasonable bounds for larger instances.

A newly modified Trefftz method is developed to solve the exterior and interior Dirichlet problems for two-dimensional Laplace equation, which takes the characteristic length of problem domain into account. After introducing a circular artificial boundary which is uniquely determined by the physical problem domain, we can derive a Dirichlet to Dirichlet mapping equa- tion, which is an exact boundary condition. By truncating the Fourier series expansion one can match the physical boundary condition as accu- rate as one desired. Then, we use the colloca- tion method and the Galerkin method to derive linear equations system to determine the Fourier coefficients. Here, the factor of characteristic length ensures that the modified Trefftz method is stable. We use a numerical example to ex- plore why the conventional Trefftz method is fail- ure and the modified onestill survives. Numerical examples with smooth boundaries reveal that the present method can offer very accurate numeri- cal results with absoluteerrors about in the orders from 10 −10 to 10 −16 . The new method is pow- erful even for problems with complex boundary shapes, with discontinuous boundary conditions or with corners on boundary. Keyword: Laplace equation, Artificial bound- ary condition, Modified Trefftz method, Char- acteristic length, Collocation method, Galerkin method, DtD mapping

Boundary value problems with nonhomogeneous Neumann conditions on a fractal boundary

Hydrodynamic interactions in Brownian dynamics: I. Periodic boundary conditions for computer simulations

05/01712 Utilizing Philippine Calatrava coal-diesel oilmixture (CDOM) as alternative fuel for industrial steam generator

A minor modification of the Clifford Cauchy kernel is used to build up an operator calculus. This calculus works extremely well in investigating boundary value problems of elliptic partial differential equations using methods of Clifford analysis. As examples the Laplace equation and the Stokes’ system are considered here.

O-13-194 - Intact intracranial aneurysms: A comparison between microsurgical and endovascular treatment

Numerous papers deal with asymptotic methods for boundary value problems in domains depending on a small parameter in a complicated way (perforated domains, partially perforated domains, framework structures, thin domains); e.g., see [1–9]. Boundary value problems in thick singularly degenerating junctions (the number of components of such junctions grows infinitely as the perturbation parameter tends to zero) have specific difficulties and require a separate consideration. It was shown in [10] that boundary value problems in thick junctions lose their coercivity under the passage to the limit, which substantially complicates asymptotic studies. The papers [11, 12] were the first in this direction. In [13–19], a classification of thick junctions was given and rigorous mathematical methods were developed for analyzing the main boundary value problems of mathematical physics in thick singularly degenerating junctions of various types. The study of boundary value problems in thick junctions is focused on the asymptotic behavior of their solutions as e → 0, i.e., as the number of thin attached domains grows infinitely and their thickness tends to zero. In the present paper, we consider a thick junction whose thin attached cylinders are divided into two levels depending on the boundary conditions posed on the lateral surface of these cylinders (inhomogeneous Neumann boundary conditions and homogeneous Dirichlet conditions). In addition, thin cylinders of each level e-periodically alternate along the junction zone. Such thick junctions will be called thick two-level junctions. A problem on a thick plane two-level junction was considered for the first time in [20], where the asymptotic behavior of eigenvalues and eigenfunctions of a spectral boundary value problem was analyzed. Other boundary value problems in plane two-level junctions were considered in [21–23]. Asymptotic analysis of boundary value problems with a periodic change of boundary conditions (Neumann and Dirichlet conditions) on the boundary of smooth unperturbed domains was carried out in [24–26], where it was shown that the first term of the asymptotics is mainly the solution of the corresponding boundary value problem with the Dirichlet conditions. The qualitatively new result of the present paper implies that the first term of the asymptotics is a vector function whose components are solutions of two independent boundary value problems (one in the junction body and another in a domain filled with thin cylinders in the limit) with homogeneous Dirichlet conditions in the junction zone. Note also that the second problem is a problem for a second-order ordinary differential equation with a new right-hand side that “remembers” the inhomogeneity in the Neumann boundary conditions of the original problem and the specific “packing density” of thin cylinders.

The second part of this book is organized analogously to the first one. We begin with boundary value problems in domains with isolated singularities of the boundaries, after that we study the asymptotics of the solutions of boundary value problems in domains that are perturbed near such singularities. However, in contrast to the first part we consider here general elliptic boundary value problems. The reader who is interested only in concrete problems of mathematical physics may restrict himself to a superficial reading of Chapters 3–5.KeywordsElliptic OperatorElliptic BoundaryHomogeneous ProblemCanonical SystemOperator PencilThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The nonlinear propagation characteristics of surface acoustic waves on an isotropic elastic solid have been studied in this paper. The solution of the harmonic boundary value problem for Rayleigh waves is obtained as a generalized Fourier series whose coefficients are proportional to the slowly varying amplitudes of the various harmonics. The infinite set of coupled equations for the amplitudes when solved exhibit an oscillatory slow variation signifying a continuous transfer of energy back and forth among the various harmonics. A conservation relation is derived among all the harmonic amplitudes.

In this paper, we consider inverse problems of finding the time-dependent source function for the population model with population density nonlocal boundary conditions and an integral over-determination measurement. These problems arise in mathematical biology and have never been investigated in the literature in the forms proposed, although related studies do exist. The unique solvability of the inverse problems are rigorously proved using generalized Fourier series and the theory of Volterra integral equations. Continuous dependence on smooth input data also holds but, as in reality noisy errors are random and non-smooth, the inverse problems are still practically ill-posed. The degree of ill-posedness is characterised by the numerical differentiation of a noisy function. In the numerical process, the boundary element method together with either a smoothing spline regularization or the first-order Tikhonov regularization are employed with various choices of regularization parameter. One is based on the discrepancy principle and another one is the generalized cross-validation criterion. Numerical results for some benchmark test examples are presented and discussed in order to illustrate the accuracy and stability of the numerical inversion.

The free and forced vibration of a sandwich plate with functionally graded face layers and electrorheological fluid core is presented in this article. Assuming classical thin plate theory for face sheets and first order shear theory for the core layer, the equations of motions are derived using Hamilton’s principle. The simply-supported boundary condition is considered for all edges and the generalized Fourier series are employed to represent the displacements of the sandwich plate. The natural frequencies and loss factors are obtained by solving the complex eigenvalue problem. The results of free and forced responses are compared and validated with the previous articles and effects of electric field, geometric aspect ratio, functionally graded component ratios and core layer thickness on the dynamic behavior of the sandwich plate are studied.

Optimal spectral decomposition (OSD) is applied to ocean data assimilation with variable (temperature, salinity, or velocity) anomalies (relative to background or modeled values) decomposed into generalized Fourier series, such that any anomaly is represented by a linear combination of products of basis functions and corresponding spectral coefficients. It has three steps: 1) determination of the basis functions, 2) optimal mode truncation, and 3) update of the spectral coefficients from innovation (observational increment). The basis functions, depending only on the topography of the ocean basin, are the eigenvectors of the Laplacian operator with the same lateral boundary conditions as the assimilated variable anomalies. The Vapnik–Chervonkis dimension is used to determine the optimal mode truncation. After that, the model field updates due to innovation through solving a set of a linear algebraic equations of the spectral coefficients. The strength and weakness of the OSD method are demonstrated through a twin experiment using the Parallel Ocean Program (POP) model.