Mixed Type Boundary Conditions Research Articles

A singular perturbation result for a class of periodic-parabolic BVPs

Abstract In this article, we obtain a very sharp version of some singular perturbation results going back to Dancer and Hess [Behaviour of a semilinear periodic-parabolic problem when a parameter is small, Lecture Notes in Mathematics, Vol. 1450, Springer-Verlag, Berlin, 1990, pp. 12–19] and Daners and López-Gómez [The singular perturbation problem for the periodic-parabolic logistic equation with indefinite weight functions, J. Dynam. Differential Equations 6 (1994), 659–670] valid for a general class of semilinear periodic-parabolic problems of logistic type under general boundary conditions of mixed type. The results of Dancer and Hess [Behaviour of a semilinear periodic-parabolic problem when a parameter is small, Lecture Notes in Mathematics, Vol. 1450, Springer-Verlag, Berlin, 1990, pp. 12–19] and [The singular perturbation problem for the periodic-parabolic logistic equation with indefinite weight functions, J. Dynam. Differential Equations 6 (1994), 659–670] were found, respectively, for Neumann and Dirichlet boundary conditions with L = − Δ {\mathfrak{L}}=-\Delta . In this article, L {\mathfrak{L}} stands for a general second-order elliptic operator.

Error estimates for completely discrete FEM in energy‐type and weaker norms

AbstractThe paper presents error estimates within a unified abstract framework for the analysis of FEM for boundary value problems with linear diffusion‐convection‐reaction equations and boundary conditions of mixed type. Since neither conformity nor consistency properties are assumed, the method is called completely discrete. We investigate two different stabilized discretizations and obtain stability and optimal error estimates in energy‐type norms and, by generalizing the Aubin‐Nitsche technique, optimal error estimates in weaker norms.

Axisymmetric torsion of rigid disc parallel to penny shaped crack at the interface of layer bonded by two orthotropic half spaces

Two half-spaces and a layer of finite height of linear elastic orthotropic medium containing a penny-shaped crack and a circular rigid disk are considered such that the crack surface is parallel to the rigid disk, where both are situated at two different interfaces of layer and half-space and perpendicular to the axis of symmetry. The rigid disk is agitated by an axisymmetric torsion. The equations of equilibrium are solved with the use of Hankel’s integral transform method. Then, the tangential displacement components and shear stress components are determined. With the help of mixed-type boundary conditions, a system of integral equations is derived. After that, by utilizing a trial solution, these integral equations are transformed into a pair of simultaneous Fredholm integral equations of the second kind. The numerical results of the Fredholm integral equations are evaluated by converting it into a system of algebraic equations to interpret the physical quantity, like the stress intensity factor (SIF), which is demonstrated graphically. The obtained results are compared to the previously published article to ensure its accuracy.

A rapidly converging domain decomposition algorithm for a time delayed parabolic problem with mixed type boundary conditions exhibiting boundary layers

A rapidly converging domain decomposition algorithm is introduced for a time delayed parabolic problem with mixed type boundary conditions exhibiting boundary layers. Firstly, a space-time decomposition of the original problem is considered. Subsequently, an iterative process is proposed, wherein the exchange of information to neighboring subdomains is accomplished through the utilization of piecewise-linear interpolation. It is shown that the algorithm provides uniformly convergent numerical approximations to the solution. Our analysis utilizes some novel auxiliary problems, barrier functions, and a new maximum principle result. More importantly, we showed that only one iteration is needed for small values of the perturbation parameter. Some numerical results supporting the theory and demonstrating the effectiveness of the algorithm are presented.

Effect of variable viscosity, porous walls and mixed thermal boundary condition on the onset of Rayleigh-Bénard convective instability

In this paper, we have studied the criteria for the onset of instability in the Rayleigh-Bénard convection problem, by considering a fluid layer confined between two infinitely extended rough boundaries, which are subjected to mixed-type thermal boundary conditions, physically representing the imperfectly conducting boundaries. The rough boundaries are assumed to be shallow porous layers with different pore size, properties, and permeabilities. Mathematically, it is given by the Saffman type boundary condition. Therefore, we have two generalized forms of boundary conditions, one for the flow field and the other for the thermal field. The two limiting cases of the mixed-type thermal boundary condition correspond to a perfectly conducting boundary and to a perfectly insulating boundary, whereas the two limiting cases of the hydrodynamic boundary condition correspond to a no-slip condition and to a free-surface condition, depending upon the limiting values of the parameters involved in these two generalized boundary conditions. The linear and nonlinear energy stability analyses are performed, and the existence of the region of subcritical instability is checked. Through the principle of exchange of stability analysis, it is found that the instability occurs only in the stationary mode. The adiabatic boundary condition is found to be more restrictive than the isothermal boundary condition. Under given conditions, the instability is observed to be occurring in the infinite wavelength mode for the case of adiabatic boundaries. In the present work, the effect of temperature and pressure dependent viscosity on the stability of the system has been shown and found to be of destabilizing nature. However, the roughness of the boundaries is found to be of stabilizing nature.

Computational cost reduction for coupled system of multiple scale reaction diffusion problems with mixed type boundary conditions having boundary layers

Computational cost reduction for coupled system of multiple scale reaction diffusion problems with mixed type boundary conditions having boundary layers

On the Numerical Treatment of the Temporal Discontinuity Arising from a Time-Varying Point Mass Attachment on a Waveguide

A vibrating pylon, modeled as a waveguide, with an attached point mass that is time-varying poses a numerically challenging problem regarding the most efficient way for eigenvalue extraction. The reason is three-fold, starting with a heavy mass attachment that modifies the original eigenvalue problem for the stand-alone pylon, plus the fact that the point attachment results in a Dirac delta function in the mixed-type boundary conditions, and finally the eigenvalue problem becomes time-dependent and must be solved for a sequence of time steps until the time interval of interests is covered. An additional complication is that the eigenvalues are now complex quantities. Following the formulation of the eigenvalue problem as a system of first-order, time-dependent matrix differential equations, two eigenvalue extraction methods are implemented and critically examined, namely the Laguerre and the QR algorithms. The aim of the analysis is to identify the most efficient technique for interpreting time signals registered at a given pylon as a means for detecting damage, a procedure which finds application in structural health monitoring of civil engineering infrastructure.

Analytical and Numerical Solutions to Static Analysis of Moderately Thick Cross-Ply Plates and Shells

This study aimed to provide a static solution to the boundary value problem presented by symmetric (0°/90°/0°) and antisymmetric (0°/90°) cross-ply composite, moderately thick shallow shells and plates (a special case of the shells) subjected to mixed-type unsolved boundary conditions. The boundary-discontinuous double Fourier series (BDM) method, in which displacements are expressed in trigonometric functions, is employed in a well-established framework. The analytical solution obtained using the BDM is compared with the successful integration of the generalized differential quadrature (GDQ) method for the static analysis of composite shells with a roller skate-type boundary condition prescribed on two opposite edges, while the remaining two edges are subjected to simply supported constraints. Comprehensive results are presented in order to show the effects of curvature on the deflections and stresses of moderately thick shallow shells made up of symmetric and antisymmetric cross-ply laminated composite materials. The validity of the proposed model is authenticated through the available HSDT-based literature review, and the convergence characteristics are demonstrated. The changing trends of displacements and stresses are explained in detail by investigating the effect of various parameters such as lamination, material properties, the effect of curvature, etc. Based on the results obtained using the proposed static solution, analytical BDM results were found to be in very close agreement with the numerical GDQ method, especially for symmetric lamination. However, the results obtained using the BDM and GDQ methods for antisymmetric lamination show differences, possibly due to the presence of a discontinuity in the derivatives originating from the bending–stretching matrix in antisymmetric lamination. Important numerical results presented include the sensitivity of the predicted response quantities of interest to material properties, lamination, and thickness effects, as well as their interactions. The results presented here may also serve as benchmark comparison points with numerical solutions such as finite elements, boundary elements, etc.

Using Boundary Transform Method to Solve Geotechnical Problems with Mixed-Type Boundary Conditions

Using Boundary Transform Method to Solve Geotechnical Problems with Mixed-Type Boundary Conditions

A time-fractional dual-phase-lag framework to investigate transistors with TMTC channels (TiS3, In4Se3) and size-dependent properties

In this study, a time fractional dual-phase-lag model with temperature jump boundary condition as a choice for the Fourier's law replacement in thermal modeling of transistors, is utilized. In more details, the numerical simulation of heat transfer in newly proposed TMTC field effect transistors using fractional DPL equation has been investigated. Moreover, the Caputo fractional derivative is employed to formulate the finite difference scheme for discretization of the fractional DPL model. In order to obtain more precise results for the peak temperature rise, the temperature and heat flux profiles, the size-dependent thermal properties are taken into account. Also, the temperature jump boundary condition has been also applied by means of a mixed-type boundary condition. It is obtained that considering size-dependent thermal characteristics for transistors under study, results in an increase of the peak temperature rise. As an instance, considering the film dependent thermal properties for a 1-D silicone MOSFET with uniform heat generation at the upper part, makes the maximum temperature increase up to 250%. Furthermore, considering constant bulk thermal properties for the silicon MOSFET, certain oscillations are observed in the time-variation of the peak temperature rise for α = 0.7, 0.9 and 1. This presents the so-called negative bias temperature instability appearing in electronic nano-semiconductor devices. Finally, the hotspot temperature has been researched in transistors containing two-dimensional materials with quasi one-dimensional band structure channels. It is obtained that among the studied FETs, titanium trisulfide with maximum temperature increase of 19.63 K exhibits the least peak temperature rise. This presents that TiS3 may be an acceptable silicon channel replacement as far as the thermal issues are concerned.

On the resolution of the heat equation in unbounded non-regular domains of R³

We will prove well posedness and regularity results for the bidimensional heat equation, subject to mixed Dirichlet-Neumann type boundary conditions on the parabolic boundary of an unbounded (in one space variable direction) time-dependent domain. Our results are proved in anisotropic Hilbertian Sobolev spaces by using the domain decomposition method. This work complements the results obtained in [13] in the one-space variable case.

Groundwater flow near a recirculation well with mixed-type boundary condition along wellbore in confined, unconfined, leaky, or reservoir-covered aquifers

Vertical recirculation wells used for remediation or heat recovery induce drawdown distribution that varies for confined, unconfined, leaky, and reservoir-covered aquifers. A recirculation well with arbitrary arrangement of extraction and injection screen intervals is simulated with the general well function of Perina and Lee (2006) and Perina (2014) after a minor modification. Distribution of radial flux along the screen intervals is determined as part of the solution for the mixed boundary condition along the well-aquifer interface. Drawdown distribution reaches steady state, with the transient period lasting over one year for unconfined aquifer and less than one hour for the other three aquifer types in the example considered. The drawdown distribution varies by aquifer type and shows sensitivity to aquifer hydraulic properties and well construction, suggesting that a recirculation well offers an effective and economical alternative to pumping tests for characterization of aquifer hydraulic properties.

A mathematical study of diffusive logistic equations with mixed type boundary conditions

<p style='text-indent:20px;'>The purpose of this paper is to provide a careful and accessible exposition of static bifurcation theory for a class of mixed type boundary value problems for diffusive logistic equations with indefinite weights, which model population dynamics in environments with spatial heterogeneity. We discuss the changes that occur in the structure of the positive solutions as a parameter varies near the first eigenvalue of the linearized problem, and prove that the most favorable situations will occur if there is a relatively large favorable region (with good resources and without crowding effects) located some distance away from the boundary of the environment. A biological interpretation of main theorem is that an initial population will grow exponentially until limited by lack of available resources if the diffusion rate is below some critical value; this idea is generally credited to the English economist T. R. Malthus. On the other hand, if the diffusion rate is above this critical value, then the model obeys the logistic equation introduced by the Belgian mathematical biologist P. F. Verhulst. The approach in this paper is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in partial differential equations.</p>

Pointwise estimates of the solution to one dimensional compressible Naiver-Stokes equations in half space

<p style='text-indent:20px;'>In this paper, we study the global existence and pointwise behavior of classical solution to one dimensional isentropic Navier-Stokes equations with mixed type boundary condition in half space. Based on classical energy method for half space problem, the global existence of classical solution is established firstly. Through analyzing the quantitative relationships of Green's function between Cauchy problem and initial boundary value problem, we observe that the leading part of Green's function for the initial boundary value problem is composed of three items: delta function, diffusive heat kernel, and reflected term from the boundary. Then applying Duhamel's principle yields the explicit expression of solution. With the help of accurate estimates for nonlinear wave coupling and the elliptic structure of velocity, the pointwise behavior of the solution is obtained under some appropriate assumptions on the initial data. Our results prove that the solution converges to the equilibrium state at the optimal decay rate <inline-formula><tex-math id="M1">\begin{document}$ (1+t)^{-\frac{1}{2}} $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M2">\begin{document}$ L^\infty $\end{document}</tex-math></inline-formula> norm.</p>

A Mathematical Model and Simulations of Low Temperature Nitriding

Low-temperature nitriding of steel or iron can produce an expanded austenite phase, which is a solid solution of a large amount of nitrogen dissolved interstitially in fcc lattice. It is characteristic that the nitogen depth profiles in expanded austenite exhibit plateau-type shapes. Such behavior cannot be considered with a standard analytic solution for diffusion in a semi-infinite solid and a new approach is necessary. We formulate a model of interdiffusion in viscoelastic solid (Maxwell model) during the nitriding process. It combines the mass conservation and Vegard’s rule with the Darken <i>bi</i>-velocity method. The model is formulated in any dimension, <i>i.e.</i>, a mixture is included in , <i>n</i> = 1, 2, 3. For the system in one dimension, <i>n</i> = 1, we transform a differential-algebraic system of 5 equations to a differential system of 2 equations only, which is better to study numerically and analytically. Such modification allows the formulation of effective mixed-type boundary conditions. The resulting nonlinear strongly coupled parabolic-elliptic differential initial-boundary Stefan type problem is solved numerically and a series of simulations is made.

Uniformly Convergent Nonpolynomial Spline Method for Singularly Perturbed Robin-Type Boundary Value Problems with Discontinuous Source Term

In this paper, a singularly perturbed second-order ordinary differential equation with discontinuous source term subject to mixed-type boundary conditions is considered. A fitted nonpolynomial spline method is suggested. The stability and parameter uniform convergence of the proposed method are proved. To validate the applicability of the scheme, two model problems are considered for numerical experimentation and solved for different values of the perturbation parameter, ε , and mesh size, h . The numerical results are tabulated in terms of maximum absolute errors and rate of convergence, and it is observed that the present method is more accurate and ε -uniformly convergent for h ≥ ε where the classical numerical methods fail to give good result and it also improves the results of the methods existing in the literature.

An Eulerian Single-Phase Transport Model for Solid Fission Products in the Molten Salt Fast Reactor: Development of an Analytical Solution for Verification Purposes

Nuclear reactor modeling has been shifting, over the last decades, towards full-core multiphysics analysis due to the ever-increasing safety requirements and complexity of the designs of innovative systems. This is particularly true for liquid-fuel reactor concepts such as the Molten Salt Fast Reactor (MSFR), given their strong intrinsic coupling between thermal-hydraulics, neutronics and fuel chemistry. In the MSFR, fission products (FPs) are originated within the liquid fuel and are carried by the fuel flow all over the reactor core and through pumping and heat exchange systems. Some of FP species, in the form of solid precipitates, can represent a major design and safety challenge, e.g., due to deposition on solid boundaries, and their distribution in the core is relevant to the design and safety analysis of the reactor. In this regard it is essential, both for the design and the safety assessment of the reactor, the capability to model the transport of solid FPs and their deposition to the boundary (e.g., wall or heat exchanger structures). To this aim, in this study, models of transport of solid FPs in the MSFR are developed and verified. An Eulerian single-phase transport model is developed and integrated in a consolidated multiphysics model of the MSFR based on the open-source CFD library OpenFOAM. In particular, general mixed-type deposition boundary conditions are considered, to possibly describe different kinds of particle-wall interaction mechanisms. For verification purposes, analytical solutions for simple case studies are derived ad hoc based on the extension of the classic Graetz problem to linear decay, distributed source terms and mixed-type boundary conditions. The results show excellent agreement between the two models, and highlight the effects of decay and deposition phenomena of various intensity. The resulting approach constitutes a computationally efficient tool to extend the capabilities of CFD-based multiphysics MSFR calculations towards the simulation of solid fission products transport.

A Uniformly Convergent Collocation Method for Singularly Perturbed Delay Parabolic Reaction-Diffusion Problem

In this article, a numerical solution is proposed for singularly perturbed delay parabolic reaction-diffusion problem with mixed-type boundary conditions. The problem is discretized by the implicit Euler method on uniform mesh in time and extended cubic B-spline collocation method on a Shishkin mesh in space. The parameter-uniform convergence of the method is given, and it is shown to be ε -uniformly convergent of O Δ t + N − 2 ln 2 N , where Δ t and N denote the step size in time and number of mesh intervals in space, respectively. The proposed method gives accurate results by choosing suitable value of the free parameter λ . Some numerical results are carried out to support the theory.

Stochastic nonlinear Schrödinger equation with brown-noise boundary conditions of mixed type

We consider the stochastic nonlinear Schrödinger equations on the half-line with mixed boundary conditions containing Brownian noise. The main novelty of this work is a convenient framework for the analysis of such problems. Our approach allows us to establish the global existence and uniqueness of a solution to initial-boundary value problem with values in H1.

Evaluation of a radial basis function node refinement algorithm applied to bioheat transfer modeling

Many bioheat transfer problems involve linear/non-linear equations with non-linear or time-dependent boundary conditions. For heat transfer problems, the presence of time and space-dependent functions under Neumann and Mixed type boundary conditions characterize trivial applications in bioengineering, such as thermotherapies, laser surgeries, and burn studies. This greatly increases the complexity of the numerical solution in several problems, requiring fast and accurate numerical solutions. This paper has a main objective evaluate an adaptive mesh refinement radial basis function method strategy for the classical Penne's bioheat transfer modeling. Our numerical results had errors of ~0.1% compared to analytical solutions. Thus, the proposed methodology is accurate and has a low computational cost. For step function heating, two RBF shape parameters were applied, again achieving excellent results. The distributions of the nodes in the solution domain show that the primary source of error in the numerical solutions came from the boundary conditions. This finding should arouse the interest of engineers and scientists in the development of new strategies for problems involving boundary conditions with periodic functions.