General Linear Boundary Conditions Research Articles

Analytical approach to solving linear diffusion–advection–reaction equations with local and nonlocal boundary conditions

Initial–boundary value problems for a linear diffusion–advection–reaction equation are considered, with general nonhomogeneous linear boundary conditions and general linear nonlocal boundary conditions. Analytical solutions are obtained using an embedding method. The solutions are expressed in terms of time‐varying functions that satisfy coupled linear Volterra integral equations of the first kind. A boundary element method is applied to numerically solve the integral equations. Three examples are given to demonstrate the accuracy of the numerical solutions when compared with the analytical solutions. The embedding method is applicable to problems with bounded and unbounded spatial domains.

Telegraph systems on networks and port-Hamiltonians. Ⅲ. Explicit representation and long-term behaviour

<p style='text-indent:20px;'>In this paper we present an explicit formula for the semigroup governing the solution to hyperbolic systems on a metric graph, satisfying general linear Kirchhoff's type boundary conditions. Further, we use this representation to establish the long term behaviour of the solutions. The crucial role is played by the spectral decomposition of the boundary matrix.</p>

Telegraph systems on networks and port-Hamiltonians. Ⅱ. Network realizability

<p style='text-indent:20px;'>Hyperbolic systems on networks often can be written as systems of first order equations on an interval, coupled by transmission conditions at the endpoints, also called port-Hamiltonians. However, general results for the latter have been difficult to interpret in the network language. The aim of this paper is to derive conditions under which a port-Hamiltonian with general linear Kirchhoff's boundary conditions can be written as a system of <inline-formula><tex-math id="M1">\begin{document}$ 2\times 2 $\end{document}</tex-math></inline-formula> hyperbolic equations on a metric graph <inline-formula><tex-math id="M2">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>. This is achieved by interpreting the matrix of the boundary conditions as a potential map of vertex connections of <inline-formula><tex-math id="M3">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula> and then showing that, under the derived assumptions, that matrix can be used to determine the adjacency matrix of <inline-formula><tex-math id="M4">\begin{document}$ \Gamma $\end{document}</tex-math></inline-formula>.</p>

Existence-uniqueness and fixed-point iterative method for general nonlinear fourth-order boundary value problems

Existence and uniqueness results are established for the general nonlinear fourth-order two-point boundary value problem with general linear homogeneous boundary conditions. The proofs are based on the Banach fixed-point theorem under the conditions that the nonlinear term is bounded and Lipschitz in a specific region. A fixed-point iterative method for finding the solution of the problem is also proposed with error estimates. The resulting iterative solutions automatically satisfy the boundary conditions provided that the initial iterate does so. Two examples are given to illustrate the applicability of the proposed approach and the efficiency of the iterative method.

Unsteady anisotropic heat conduction in heterogeneous composite conical shells with temperature-dependent thermal conductivities: an analytical study

The present study proposes an exact analysis for unsteady anisotropic conduction heat transfer in heterogeneous composite conical shells. The fibers in the composite conical shell are wound in arbitrary directions so that a defined fiber angle changes between 0° and 90°. The cone’s base is considered to have a general linear boundary condition, while the effects of internal heat generation, thermal convection, and external radiation heat flux on the temperature distribution are investigated. The heterogeneity of the heat transfer problem is caused by a linear dependence of conduction coefficients on the temperature. To solve the governing equations, first, the Kirchhoff transformation followed by an appropriate finite integral transform is applied. The resulting nonhomogeneous equation is then separated into two equations, i.e., steady equation and unsteady equation. The solution of the steady case is derived using Green’s function, while the separation of variables method is utilized to solve the unsteady part. Eventually, the temperature distribution is achieved by applying the inverse transforms on the subsequent solutions. The verification of this solution is based on the comparison between the present analytical results and numerical computations with a second-order finite difference code. Applicability of the present solution is evaluated for resolving an industrial case problem.

Electromagnetic Scattering Analysis of SHDB Objects Using Surface Integral Equation Method

A surface integral equation (SIE) method is applied in order to analyze electromagnetic scattering by bounded arbitrarily shaped three-dimensional objects with the SHDB boundary condition. SHDB is a generalization of SH (Soft-and-Hard) and DB boundary conditions (at the DB boundary, the normal components of the D and B flux densities vanish). The SHDB boundary condition is a general linear boundary condition that contains two scalar equations that involve both the tangential and normal components of the electromagnetic fields. The multiplication of these scalar equations with two orthogonal vectors transforms them into a vector form that can be combined with the tangential field integral equations. The resulting equations are discretized and converted to a matrix equation with standard method of moments (MoM). As an example of use of the method, we investigate scattering by an SHDB circular disk and demonstrate that the SHDB boundary allows for an efficient way to control the polarization of the wave that is reflected from the surface. We also discuss perspectives into different levels of materialization and realization of SHDB boundaries.

Formulation of Problems for Stationary Dispersive Equations of Higher Orders on Bounded Intervals with General Boundary Conditions

Boundary value problems for linear stationary dispersive equations of order 2l + 1, l ∈ N with general linear boundary conditions have been considered on finite intervals (0, L). The existence and uniqueness of regular solutions have been established.

Coupled longitudinal and transverse vibrations of tensioned Euler-Bernoulli beams with general linear boundary conditions

The paper concerns coupling between longitudinal and transverse vibrations of tensioned Euler-Bernoulli beams with general linear boundary conditions of arbitrary elastic stiffness. The beams are assumed to be perfectly straight with symmetric boundary conditions, and excitation aligned with the longitudinal axis. Two types of excitation, harmonic and impact, are considered. A relatively simple approach is proposed to predict and qualitatively describe the coupling, allowing approximate explicit analytical expressions for the key parameters affecting the phenomenon. The beam model considered, with both ends supported by longitudinal, translational and rotational springs, can be used in particular to model tensioned bolts used in engineering. The effects of beam tension on the coupling phenomenon are revealed. A series of physical experiments is carried out to illustrate application of the obtained theoretical predictions. Numerical solution of the full nonlinear equations governing coupled transverse-longitudinal vibrations has been conducted to validate the theoretical predictions.

Asymmetric Conduction in an Infinite Functionally Graded Cylinder: Two-Dimensional Exact Analytical Solution Under General Boundary Conditions

Abstract This paper focuses on using an analytical method to obtain an exact solution for a long cylindrical vessel made of functionally graded materials (FGMs). Heat conduction equations are assumed to be in both radial and circumferential directions. The conduction coefficients are considered as different power-law functions of the radius. The general linear boundary conditions are adopted to make the solution applicable to the full range of problems. The obtained solution is successfully validated. Through solving illustrative test examples, the effects of material constants and boundary conditions on temperature distribution are studied. The obtained formulation can be utilized for tailoring of FGM based on the actual sophisticated thermal boundary conditions in the production process. The current analytical findings can help to manage the temperature distribution in FGMs which is an essential parameter in controlling the thermal stresses.

GLOBAL ANALYSIS OF THE SHADOW GIERER-MEINHARDT SYSTEM WITH GENERAL LINEAR BOUNDARY CONDITIONS IN A RANDOM ENVIRONMENT

The global analysis of the shadow Gierer-Meinhardt system with multiplicative white noise and general linear boundary conditions is investigated in this paper. For this reaction-diffusion system, we employ a fixed point argument to prove local existence and uniqueness. Our results on global existence are based on a priori estimates of solutions.

К оценке значений линейных функционалов на решениях систем с последействием

For a wide class of linear functional differential systems with Volterra operators, a constructive technique is proposed to obtain estimates of linear functionals values over solutions in conditions of uncertainty of external perturbations. It can be applied to solutions of boundary value problems with arbitrary number of boundary conditions as well as to description of attainability sets in control problems with respect to given on-target functionals. External perturbations are constrained by a given linear inequalities system on the main time segment. The technique is based on the results of general theory of functional differential equations about the solvability of boundary value problems with general linear boundary conditions and the representation of solutions. The problem under consideration is reduced to the generalized moment problem. Therewith the results on the properties of the Cauchy matrix to systems with aftereffect are of essential importance. The general form of functionals allows one to cover many cases being topical in applications such as multipoint, integral ones, as well as hybrids of those.

Green's functions of recurrence relations with reflection

In this work we develop an algebraic theory of linear recurrence equations and systems with constant coefficients and reflection. We obtain explicit solutions and the Green's functions associated to different problems under general linear boundary conditions. Furthermore, we establish different relations between the algebras of recurrence and differential operators, showing the similarities and differences between them.

Advanced analysis of arbitrarily shaped axially loaded beams including axial warping and distortion

In this paper, a higher order beam theory is developed for the analysis of beams of homogeneous cross-section, taking into account warping and distortional phenomena due to axial, shear, flexural and torsional behavior. The beam can be subjected to arbitrary axial, transverse and/or torsional concentrated or distributed load, while its edges are restrained by the most general linear boundary conditions. The analysis consists of two stages. In the first stage, where the Boundary Element Method is employed, a cross sectional analysis is performed based on the so-called sequential equilibrium scheme establishing the possible in-plane (distortion) and out-of-plane (warping) deformation patterns of the cross-section. In the second stage, where the Finite Element Method is employed, the extracted deformation patterns are included in the beam analysis multiplied by respective independent parameters expressing their contribution to the beam deformation. The four rigid body displacements of the cross-section together with the aforementioned independent parameters consist the degrees of freedom of the beam. The finite element equations are formulated with respect to the displacements and the independent warping and distortional parameters. Numerical examples of axially loaded beams are solved to emphasize the importance of axial mode. In addition, numerical examples of various loading combinations are presented to demonstrate the range of application of the proposed method.

Higher order beam theory for linear local buckling analysis

In this paper, a higher order beam theory is employed for linear local buckling analysis of beams of homogeneous cross-section, taking into account warping and distortional phenomena due to axial, shear, flexural, and torsional behavior. The beam is subjected to arbitrary concentrated or distributed loading, while its edges are restrained by the most general linear boundary conditions. The analysis consists of two stages. In the first stage, where the Boundary Element Method is employed, a cross-sectional analysis is performed based on the so-called sequential equilibrium scheme establishing the possible in-plane (distortion) and out-of-plane (warping) deformation patterns of the cross-section. In the second stage, where the Finite Element Method is employed, the extracted deformation patterns are included in the buckling analysis multiplied by respective independent parameters expressing their contribution to the beam deformation. The four rigid body displacements of the cross-section together with the aforementioned independent parameters constitute the degrees of freedom of the beam. The finite element equations are formulated with respect to the displacements and the independent warping and distortional parameters. The buckling load is calculated and is compared with beam and 3d solid finite elements analysis results in order to validate the method and demonstrate its efficiency and accuracy.

General Boundary Conditions for a Majorana Single-Particle in a Box in (1 + 1) Dimensions

We consider the problem of a Majorana single-particle in a box in (1 + 1) dimensions. We show that the most general set of boundary conditions for the equation that models this particle is composed of two families of boundary conditions, each one with a real parameter. Within this set, we only have four confining boundary conditions—but infinite not confining boundary conditions. Our results are also valid when we include a Lorentz scalar potential in this equation. No other Lorentz potential can be added. We also show that the four confining boundary conditions for the Majorana particle are precisely the four boundary conditions that mathematically can arise from the general linear boundary condition used in the MIT bag model. Certainly, the four boundary conditions for the Majorana particle are also subject to the Majorana condition.

Linear Sturm-Liouville problems with Riemann-Stieltjes integral boundary conditions

We study second-order linear Sturm-Liouville problems involving general homogeneous linear Riemann-Stieltjes integral boundary conditions. Conditions are obtained for the existence of a sequence of positive eigenvalues with consecutive zero counts of the eigenfunctions. Additionally, we find interlacing relationships between the eigenvalues of such Sturm-Liouville problems and those of Sturm-Liouville problems with certain two-point separated boundary conditions.

A General Boundary Condition with Linear Flux for Advection-Diffusion Models

Advection-diffusion equations are widely used in modeling a diverse range of problems. These mathematical models consist in a partial differential equation or system with initial and boundary conditions, which depend on the phenomena being studied. In the modeling, boundary conditions may be neglected and unnecessarily simplified, or even misunderstood, causing a model not to reflect the reality adequately, making qualitative and/or quantitative analyses more difficult. In this work we derive a general linear flux dependent boundary condition for advection-diffusion problems and show that it generates all possible boundary conditions, according to the outward flux on the boundary. This is done through an integral formulation, analyzing the total mass of the system. We illustrate the exposed cases with applications willing to clarify their meanings. Numerical simulations, by means of the Finite Difference Method, are used in order to exemplify the different boundary conditions' impact, making it possible to quantify the flux along the boundary. With qualitative and quantitative analysis, this work can be useful to researchers and students working on mathematical models with advection-diffusion equations.

A general exact analytical solution for anisotropic non-axisymmetric heat conduction in composite cylindrical shells

In this study, a solution for anisotropic conductive heat transfer in a composite cylindrical shell is computed by an exact analytical method. The fibers are considered to be winded around the cylinder at any arbitrary angle. A general linear boundary condition is applied on both circular bases of the cylindrical shell to account for various combinations of thermal conditions. The solution considers the effects of convection of the ambient flow motion and various external radiative heat fluxes. The analytical solution describes the temperature distribution in the axial and circumferential directions. In principle, the heat conduction equation should involve a dual second-order derivation, which precludes solving the equation by the direct application of common exact methods. Therefore, an appropriate canonical mapping is selected as a solution to cancel the dual derivation of temperature in the mapped equations. The separation of variables method is then applied to the mapped equations, which are conducted to obtain an exact form of the solution. Finally, an analytical technique based on the Fourier series concept is proposed to determine unknown coefficients in the final solution. The obtained analytical results correspond well with the solved numerical results. The capability of the current solution is examined by application to a few relevant industrial cases.

A Generalized Model for Complex Wastewater Treatment with Simultaneous Bioenergy Production Using the Microbial Electrochemical Cell

The objective of this study was to construct a novel model to be applied in a general manner to simulate microbial electrochemical cells (MXCs); for both microbial fuel cell (MFC) and microbial electrolysis cell (MEC). The liquid bulk was modeled based on the organic matters degradation to acetate via the anaerobic digestion process. Biofilm simulation was established based upon one-dimensional distribution and the dynamical electron transfer was completed by means of the conduction-based mechanism. We, for the first time, introduced biofilm local potential modeling for MEC simulation with general and simplified linear boundary conditions. The MFC-related part of the model was evaluated based on the experimental results from a gluconic acid-fed MFC with various simulated variables of liquid bulk and biofilm such as, bulk concentrations, distributions of microbial volume fractions and local potential in the biofilm, and biofilm thickness. The MEC-related part of the model with simplified linear boundary condition was assessed by the MEC experimental data fed with potato wastewater as the complex substrate. The MEC performance as an energy carrier generator was characterized based on the hydrogen production rate evolution, the variations of microbial distributions, and methane production at different applied potential differences. In addition, polarization characteristics were simulated and discussed based on the experimental results. These valuations showed that the presented model is successfully able to predict both MFC and MEC performance.

Axisymmetric steady temperature field in FGM cylindrical shells with temperature-dependent heat conductivity and arbitrary linear boundary conditions

An approximate analytical solution to the axisymmetric heat conduction equation for a hollow cylinder made of functionally graded material with temperature-dependent heat conductivity is presented. General linear boundary conditions are considered. The Poincare method for regular perturbation problems is employed to obtain an analytical closed-form approximate solution for the temperature field. The hierarchical asymptotic problems are solved up to the second-order approximation. A numerical example is worked out, i.e., the one-dimensional heat conduction in the radial direction with prescribed temperatures at the boundaries. The approximate temperature profiles are compared with a numerical solution of the full nonlinear problem which provides the reference “exact” solution. A good agreement between the approximate and reference solutions is established. The convergence of the asymptotic series as well as the properties of the temperature field are studied.