General Boundary Value Problems Research Articles

The generalized method of separation of variables for diffusion-influenced reactions: Irreducible Cartesian tensor technique.

Motivated by the various applications of the trapping diffusion-influenced reaction theory in physics, chemistry, and biology, this paper deals with irreducible Cartesian tensor (ICT) technique within the scope of the generalized method of separation of variables (GMSV). We provide a survey from the basic concepts of the theory and highlight the distinctive features of our approach in contrast to similar techniques documented in the literature. The solution to the stationary diffusion equation under appropriate boundary conditions is represented as a series in terms of ICT. By means of proved translational addition theorem, we straightforwardly reduce the general boundary value diffusion problem for N spherical sinks to the corresponding resolving infinite set of linear algebraic equations with respect to the unknown tensor coefficients. These coefficients exhibit an explicit dependence on the arbitrary three-dimensional configurations of N sinks with different radii and surface reactivities. Our research contains all relevant mathematical details such as terminology, definitions, and geometrical structure, along with a step by step description of the GMSV algorithm with the ICT technique to solve the general diffusion boundary value problem within the scope of Smoluchowski's trapping model.

General Boundary-Value Problem for Nonuniformly Parabolic Equations with Power Singularities

General Boundary-Value Problem for Nonuniformly Parabolic Equations with Power Singularities

Learning based numerical methods for acoustic frequency-domain simulation with high frequency

Acoustic simulation in frequency-domain is related to solving Helmholtz equations, which is still highly challenging at high frequency with complex geometries. In this paper, a learning based numerical method (LbNM) is proposed for general boundary value problems of Helmholtz equation. By using Tikhonov regularization, the solution operator is stably learned from various data solutions especially fundamental solutions. The reconstructed solution operator is reusable and enables fast solution updating with new boundary inputs. The present method is featured by strong interpretability and generalizability. Numerical results demonstrate advantages in precision and efficiency. It can be applied to various engineering problems such as cabin acoustic simulations flexibly.

A new constitutive relation to describe the response of bones

Trabecular bone, a solid that has a heterogeneous porous structure, demonstrates nonlinear stress–strain relationship, even within the small strain region, when subject to stresses. It also exhibits different responses when subject to tension and compression. This study presents the development of an implicit constitutive relation between the stress and the linearized strain specifically tailored for trabecular bone-like materials. The structure of the constitutive relation requires the solution of the balance of linear momentum and the constitutive relations simultaneously, and in view of this, a two-field mixed finite element model capable of solving general boundary value problems governed by a system of coupled equations is proposed. We investigate the effects of nonlinearity and heterogeneity in a dogbone-shaped sample. Our study is able to capture the significant nonlinear characteristics of the response of the trabecular bone undergoing small strains in experiments, in both tension and compression, very well.

Retraction: Asymptotic behavior of the solution of a singularly perturbed general boundary value problem with boundary jumps

Retraction: Nurgabyl DN, Uaissov AB. Asymptotic behavior of the solution of a singularly perturbed general boundary value problem with boundary jumps. Numer Methods Partial Differential Eq. 2021; 37: 2375–2392. https://doi.org/10.1002/num.22719The above article published online on 12 December 2020 in Wiley Online Library (wileyonlinelibrary.com) has been retracted by agreement between the Editor, Clayton G. Webster, and Wiley Periodicals, LLC. The retraction has been agreed following concerns raised by a third party regarding the peer review process. Further investigation by the publisher has found manipulation of the peer review process. As a result, the conclusions reported in the article are not considered reliable. Corresponding author Duisebek Nurgabyl disagrees with the retraction. Alpamys Uaissov was contacted but did not respond.

A Study of an IBVP of Fractional Differential Equations in Banach Space via the Measure of Noncompactness

In this article, we are concerned with a very general integral boundary value problem of Riemann–Liouville derivatives. We will study the problem in Banach space. To be more specific, we are interested in proving the existence of a solution to our problem via the measure of noncompactness and Mönch fixed-point theorem. Our study in Banach space contains two nonlinear terms and two different orders of derivatives, ς and τ, such that ς∈1,2 and τ∈0,ς. Our paper ends with a conclusion.

New Analytical Solutions of Mathematical Models of Heat Shock of Local Non-Equilibrium Heat Transfer

The article is devoted to practically new model representations of locally nonequilibrium heat transfer in terms of nonstationary heat conduction for hyperbolic type equations (wave equations), as well as dynamic models of heat shock based on wave equations. The results presented in the article practically open up an independent scientific direction in analytical thermal physics and in the theory of thermal shock, namely: the study of the thermal response of solids of a canonical form of finite sizes to intense heating and cooling under conditions of a locally nonequilibrium heat transfer process. This direction required the development of a special apparatus of operational calculus due to the appearance in analytical solutions of model problems in the image space according to Laplace of non-standard operational images, the originals of which are unknown and are not available in reference books on operational calculus. The presented images are typical for operational solutions of a wide class of generalized boundary value problems for equations of hyperbolic type in the theory of heat conduction, diffusion, hydrodynamics, vibrations, propagation of electricity, thermomechanics and other areas of science and technology. Illustrative examples of analytical solutions of specific model problems of locally nonequilibrium heat transfer and the theory of thermal shock for a finite region are given in both classical and generalized formulations (the latter taking into account the finite rate of heat propagation). The characteristic features of functional structures as analytical solutions of the considered mathematical models are revealed.

Discontinuous Galerkin methods for an elliptic optimal control problem with a general state equation and pointwise state constraints

We investigate discontinuous Galerkin methods for an elliptic optimal control problem with a general state equation and pointwise state constraints on general polygonal domains. We show that discontinuous Galerkin methods for general second-order elliptic boundary value problems can be used to solve the elliptic optimal control problems with pointwise state constraints. We establish concrete error estimates and numerical experiments are shown to support the theoretical results.

Приближенное определение потерь характеристической скорости и приращения высоты полета верхних ступеней многоступенчатых ракет-носителей

An important role is assigned to design-ballistic calculations at the initial stage of developing a new type of the launch vehicle. Results of such calculations are a necessary condition in solving a more general multidimensional boundary value problem of the ballistic design. It should be noted that ballistic conditions determine the design-ballistic parameters ratio to solve the boundary value problem, and its algorithms could be used as components of the higher-level optimization problems. In order to save computer time, development of the so-called fast algorithms implemented using the analytical approaches to solving a ballistic problem becomes expedient. In the logical chain of calculation dependencies making part of the ballistic calculation algorithms, the main are those relations that connect losses in the characteristic velocity with the design and ballistic parameters. They are understood as the minimum set of parameters that uniquely determine the rocket trajectory. Analytical calculated dependences of the characteristic velocity losses and of the flight altitude increments on the design-ballistic parameters of the second and third stages of three-stage launch vehicles were derived.

Approximate solutions for a fractional thermostat model boundary value problem via Bernstein's collocation method with Legendre polynomials

Numerous classical models of thermostats have been analyzed in the literature, and while existence and uniqueness results have been established, the investigation of other properties, such as stability, can present significant challenges. Therefore, there is a need for fractional mathematical models that are nonlocal problems and allow for the straightforward study of the asymptotic behavior of their solutions. However, solving these problems is often exceedingly complex or even impossible, necessitating the use of numerical methods to obtain approximate solutions. The focus of this research is a generalized boundary value problem that is based on a classical thermostat model. Our initial step involves determining the associated integral equation for the problem at hand. The properties of existence and uniqueness are proved by exploiting the classical contraction principle of Banach together with another kind called ‐ ‐contraction. Afterwards, thanks to the Bernstein polynomials and fractional matrix of derivative, the approximate solutions of our nonlinear fractional thermostat model problem are obtained. This approach utilizes the Legendre operational matrix of fractional derivatives in combination with a truncated Legendre series for the numerical integration of fractional differential equations. By doing so, the problem can be reduced to solving a system of algebraic equations, resulting in significant simplification. This method has been successfully used to solve both linear and nonlinear fractional differential equations. Finally, an example on this type of problem have been studied and the approximate solution are plotted.

On a general discrete boundary value problem for an elliptic pseudo-differential equation in a quadrant

On a general discrete boundary value problem for an elliptic pseudo-differential equation in a quadrant

Free boundary value problem for a model of inviscid liquid‐gas two‐phase flow with radial symmetry

AbstractIn this paper, we are concerned with the free boundary value problem for a model of inviscid liquid‐gas two‐phase flow with radial symmetry. For simplicity, we assume that the gas velocity is always equal to the liquid one and the gas and liquid are both connected continuously to the outer vacuum through the same free boundary. First, we constructed two classes of global analytical solutions by using a self‐similar ansatz. Second, for the general free boundary value problem of this model, we show that the free boundary separating the two‐phase flow from vacuum will spread outward at least linearly in time, provided that the two‐phase fluids motion remains smooth and the initial flow moves outward on average to the vacuum. Third, we present a sufficient condition for the blowup of smooth solutions to the problem.

Well-Posedness of Boundary-Value Problems for Conditionally Well-Posed Integro-Differential Equations and Polynomial Approximations of Their Solutions

The this paper, we introduce a pair of Sobolev spaces with special Jacobi–Gegenbauer weights, in which the general boundary-value problem for a class of ordinary integro-differential equations characterized by the positivity of the difference of orders of the inner and outer differential operators is well-posed in the Hadamard sense. Based on this result, we justify the general polynomial projection method for solving the corresponding problem. An application of general results to the proof of the convergence of the polynomial Galerkin method for solving the Cauchy problem in the Sobolev weighted space is given. The convergence rate of the method is characterized in terms of the best polynomial approximations of an exact solution, which automatically responds to the smoothness properties of the coefficients of the equation.

Influence of flexoelectricity on interface crack problems under a dynamic load

In the present paper, the influence of flexoelectricity on behavior of the interface crack between two dissimilar dielectric materials under a dynamic mechanical load is investigated. The induced electric field affects the distribution and evolution of mechanical fields in dielectric materials. Large strain gradients induce the electric polarization in the direct flexoelectricity. Due to the large strain gradients at the crack tip vicinity it is needed to consider the strain gradient theory model. Governing equations in this theory contain higher-order derivatives than in the conventional continuum mechanics approach. The mixed finite element method (FEM) is developed here for a general boundary value problem, where the standard C0 continuous finite elements are applied for independent approximations of displacements and strains. The constraints between them are satisfied by collocation at appropriate internal points in elements. Interface cracks are observed in layered structures frequently due to a poor adhesion of layers. The incorporation of flexoelectricity and micro-inertial effects is needed into the failure analysis of interface cracks in nano-sized structures under dynamic loading. In numerical examples, we discuss the influence of flexoelectricity coefficients as well as the ratio of elastic coefficients and the geometrical size to microstructural (micro-stiffness and micro-inertia) length scale parameters of the bilayer composite on the crack opening displacement, stresses ahead the crack tip and induced electric intensity vector.

On the evolution of beaches of finite length

This article provides closed form solutions of the Diffusion Equation (DE) for beaches of finite length bounded by outcrops; the method employed relies on the sum of two Sturm-Liouville Boundary Value Problems, which allow accounting for the effects of wave angle. From the general solution, three situations of engineering interest were extrapolated and discussed in detail. The first corresponds to a stretch of coast “pinned” at the extremes; the second refers to an “insulated” beach bounded by groins of infinite length and the third corresponds to a groin compartment bypassed at one end. All those solutions were previously known in the field of thermodynamics, but they are here derived systematically from a general Boundary Value Problem (BVP). A detailed comparison with the software GENESIS is carried out to establish the limits of application with increasing wave angles. The effects of the presence of outcrops is studied comparing the solutions for finite beaches with those proposed in the literature under the hypothesis of infinite shoreline (open coast). It is shown that the new approach can be particularly useful in the case of beach nourishments placed asymmetrically to the cell centre. For application purposes, approximate solutions are given, based on curve fitting, to calculate the remaining value of sediments in a beach fill of rectangular shape.

Ensuring safe operation of monolithic structures in undermined areas

In order to ensure safe operation of buildings in undermined areas, it is necessary to take into account the influence of loads extrinsic to typical structures. The solution of the generalized boundary value problem on the evaluation of the stress-strain state (SSS) of the building–foundation–soil system with due regard for the complete geometry and physically nonlinear behavior of all the elements is inexpedient since it is complicated by the large dimensionality of the problem. This paper discusses an approach allowing the state of monolithic reinforced concrete building structures located in an undermined area to be estimated by solving boundary value problems on different scales, from modeling the whole system in an elastic formulation to modeling joints between load-bearing members (columns and floors) in a nonlinear formulation for concrete and reinforcement. In these problems, strain energy is taken as the parameter characterizing the deformation process at critical deformation stages and connecting the boundary value problems. The obtained loading diagrams for a unit and the evaluation of the SSS of the whole structure enable one to find the values of maximum permissible horizontal soil deformations in the vicinity of the foundation, at which the bearing members reach the SSS preceding the loss of bearing capacity.

Bifurcation of solutions of $U(1)$-equivariant semilinear boundary value problems

Assuming that there is a known (trivial) branch of solutions of a parameterized family of equations, topological bifurcation studies the topological invariants of the linearized equations along the trivial branch whose nonvanishing entails the appearance of bifurcation from the trivial branch. We introduce here some refined topological invariants for semilinear elliptic boundary value problems equivariant with respect to the action of the circle $U(1)$ allowing to improve, in this case, some previously obtained bifurcation criteria for general nonlinear elliptic boundary value problems.

Experimental investigations of linear and nonlinear periodic travelling waves in a viscous fluid conduit

Conduits generated by the buoyant dynamics between two miscible Stokes fluids with high viscosity contrast, a type of core–annular flow, exhibit a rich nonlinear wave dynamics. However, little is known about the fundamental wave dispersion properties of the medium. In the present work, a pump is used to inject a time-periodic flow that results in the excitation of propagating small- and large-amplitude periodic travelling waves along the conduit interface. This wavemaker problem is used as a means to measure the linear and nonlinear dispersion relations and corresponding periodic travelling wave profiles. Measurements are favourably compared with predictions from a fully nonlinear, long-wave model (the conduit equation) and the analytically computed linear dispersion relation for two-Stokes flow. A critical frequency is observed, marking the threshold between propagating and non-propagating (spatially decaying) waves. Measurements of wave profiles and the wavenumber–frequency dispersion relation quantitatively agree with wave solutions of the conduit equation. An upshift from the conduit equation's predicted critical frequency is observed and is explained by incorporating a weak recirculating flow into the full two-Stokes flow model. When the boundary condition corresponds to the temporal profile of a nonlinear periodic travelling wave solution of the conduit equation, weakly nonlinear and strongly nonlinear, cnoidal-type waves are observed that quantitatively agree with the conduit nonlinear dispersion relation and wave profiles. This wavemaker problem is an important precursor to the experimental investigation of more general boundary value problems in viscous fluid conduit nonlinear wave dynamics.

The smooth extension embedding method with Chebyshev polynomials

AbstractWe propose an implementation of the smooth extension embedding method (SEEM), first described by Agress and Guidotti's study, in the setting of Chebyshev polynomials. SEEM is a hybrid fictitious domain/collocation method which solves general boundary value problems in complex domains by recasting them as constrained optimization problems in a simple encompassing set. Previously, SEEM was introduced and implemented using a periodic box (read a torus) using Fourier series; here, it is implemented on a (non‐periodic) rectangle using Chebyshev polynomial expansions. This implementation has faster convergence on smaller grids. Numerical experiments will demonstrate that the method provides a simple, robust, efficient, and high order fictitious domain method which can solve elliptic and parabolic problems in complex geometries, with non‐constant coefficients, and for general boundary conditions. We consider applications to two and three dimensional boundary value problems as well as an initial boundary value problem via a genuinely space–time discretization.

Gradient theory of thermoelasticity for interface crack problems with a quasicrystal layer

The interface crack problem between two dissimilar materials under a thermal load is analyzed by the gradient elasticity theory since the size effect is expected in the very thin quasicrystal layer. The uncoupled thermoelasticity is considered here, where the thermal fields are independent on mechanical ones. The thermal transport can’t be well described by classical Fourier’s law in nano-sized structures. Therefore, a novel gradient theory is developed in such structures, where the second derivatives of temperature in the constitutive equation for the high-order heat flux are considered. Moreover, the phonon strain gradients are incorporated into the higher-grade continuum model. Decagonal quasicrystal material properties are considered here. Governing equations and corresponding boundary conditions are derived from the principle of virtual work. The mixed finite element method (MFEM) is developed for a general 2D boundary value problem. Numerical examples are presented to illustrate the phonon, phason and thermal response to external thermal loading within the considered model.