Boundary Layer Function Research Articles

On the asymptotics of solution of one problem of optimal control of the small-parameter parabolic equation

For the singularly perturbed parabolic problem, a regularized asymptotics of the solution of the problem of optimal control was constructed. The solution asymptotics involves parabolic boundary-layer functions obeying a special function called the "complementary probability integral."

Effective Approximation for Elliptic Partial Differential Equation with Periodical Boundary Value Problem

Elliptic partial differential equation with periodical boundary value problem was considered. The equation would degenerate to parabolic partial differential equation when small parameter tends to zero. This is a multi-scale problem. Firstly, the property of boundary layer was discussed. Secondly, the boundary layer function was presented. The smooth component was constructed according to the boundary layer function. Thirdly, finite difference scheme for the smooth component is proposed according to transition point in time direction. Finally, experiment was proposed to illustrate that our presented method is an effective computational method.

Asymptotic Analysis for Hyperbolic Equation with Neumann Condition

The second-order hyperbolic equation with small parameter and Neumann con- dition is considered. This kind of problem loses the boundary conditions both in x = 0 and x = 1, while it also loses two initial boundary conditions in t = 0. The solution changes rapidly near two boundary layers and one initial layer. Firstly, the asymptotic solution was studied. The analytical solution was approximated by the degenerate solution and two boundary layer functions and one initial layer function. Secondly, three transition points were presented ac- cording to Shishkin’s idea. Non-equidistant mesh partitions both in x direction and t direction were introduced. An effective computational method is given according to non-equidistant mesh partitions. Finally, numerical experiment was given.

Levy-type solution for buckling analysis of thick functionally graded rectangular plates based on the higher-order shear deformation plate theory

In this article, an analytical approach for buckling analysis of thick functionally graded rectangular plates is presented. The equilibrium and stability equations are derived according to the higher-order shear deformation plate theory. Introducing an analytical method, the coupled governing stability equations of functionally graded plate are converted into two uncoupled partial differential equations in terms of transverse displacement and a new function, called boundary layer function. Using Levy-type solution these equations are solved for the functionally graded rectangular plate with two opposite edges simply supported under different types of loading conditions. The excellent accuracy of the present analytical solution is confirmed by making some comparisons of the present results with those available in the literature. Furthermore, the effects of power of functionally graded material, plate thickness, aspect ratio, loading types and boundary conditions on the critical buckling load of the functionally graded rectangular plate are studied and discussed in details. The critical buckling loads of thick functionally graded rectangular plates with various boundary conditions are reported for the first time and can be used as benchmark.

On the stress singularities and boundary layer in moderately thick functionally graded sectorial plates

In this paper, the stress analysis of moderately thick functionally graded sector plate is developed for studying the singularities in vicinity of the vertex and effects of boundary layer. Based on the first-order shear deformation plate theory, the governing partial differential equations are obtained. Using an analytical method, the stretching and bending equilibrium equations are decoupled. Also, introducing a function, called boundary layer function, the three bending equations are converted into two decoupled equations. These equations are solved analytically and the effects of boundary layer function on stress components are shown. Also, the singularities of shear force, moment resultants and boundary layer function are discussed for both salient ( α ⩽ 180 ) and re-entrant ( α > 180 ) sectorial plates. In order to verify the accuracy of the results, the governing equations are also solved using differential quadrature method (DQM). By comparing the results of exact method with DQM, a good agreement can be seen.

Bending Analysis of Thick Laminated Rectangular Plates Using a Boundary Layer Function

In this article, the governing bending equations of thick laminated transversely isotropic rectangular plates are derived based on third-order shear deformation theory (TSDT). Using a new function, called the boundary layer function, the three coupled governing equations are converted to two decoupled equations. These equations are in terms of the deflection of the plate and the mentioned boundary layer function, which are written in invariant form. By solving the decoupled equations, a Levy-type analytical solution is presented for bending of a transversely isotropic plate. Finally, numerical results are presented for boundary layer phenomenon and its effects in TSDT. It is shown that all of the boundary layer effects in Mindlin—Reissner theory appear in this theory. However, it is shown that the intensity of the boundary layer effects in TSDT exceeds that of the Mindlin—Reissner theory.

A transformation approach for the derivation of boundary conditions between a curved porous medium and a free fluid

The behaviour of a free fluid flow above a porous medium, both separated by a curved interface, is investigated. By carrying out a coordinate transformation, we obtain the description of the flow in a domain with a straight interface. Using periodic homogenisation, the effective behaviour of the transformed partial differential equations in the porous part is given by a Darcy law with non-constant permeability matrix. Then the fluid behaviour at the porous-liquid interface is obtained with the help of generalised boundary-layer functions: Whereas the velocity in normal direction is continuous across the interface, a jump appears in tangential direction. Its magnitude seems to be related to the slope of the interface. Therefore the results indicate a generalised law of Beavers and Joseph.

A NOVEL STABILITY CONDITION AND ITS APPLICATION TO GA-BASED FUZZY CONTROL FOR NONLINEAR SYSTEMS WITH UNCERTAINTY

In this study, we strive to combine the advantages of fuzzy logic control (FLC), genetic algorithms (GA), H(subscript ∞) tracking control schemes, smooth control and adaptive laws to design an adaptive fuzzy sliding model controller for the rapid and efficient stabilization of complex and nonlinear systems. First, we utilize a reference model and a fuzzy model (both involving FLC rules) to describe and well-approximate an uncertain, nonlinear plant. The FLC rules and the consequent parameter are decided on via GA. A boundary-layer function is introduced into these updated laws to cover modeling errors and to guarantee that the state errors converge into a specified error bound. After this, a H(subscript ∞) tracking problem is characterized. We solve an eigenvalue problem (EVP), and derive a modified adaptive neural network controller (MANNC) to simultaneously stabilize and control the system and achieve H(subscript ∞) control performance. Furthermore, a stability criterion is derived utilizing Lyapunov's direct method to ensure the stability of the nonlinear system. Finally, the control methodology is demonstrated via a numerical simulation.

A singularly perturbed semilinear reaction–diffusion problem in a polygonal domain

The semilinear reaction–diffusion equation − ε 2 Δ u + b ( x , u ) = 0 with Dirichlet boundary conditions is considered in a convex polygonal domain. The singular perturbation parameter ε is arbitrarily small, and the “reduced equation” b ( x , u 0 ( x ) ) = 0 may have multiple solutions. An asymptotic expansion for u is constructed that involves boundary and corner layer functions. By perturbing this asymptotic expansion, we obtain certain sub- and super-solutions and thus show the existence of a solution u that is close to the constructed asymptotic expansion. The polygonal boundary forces the study of the nonlinear autonomous elliptic equation − Δ z + f ( z ) = 0 posed in an infinite sector, and then well-posedness of the corresponding linearized problem.

On the boundary layer phenomenon in bending of thick annular sector plates using third-order shear deformation theory

In this article, the bending equations of thick annular sector plates are extracted based on the third-order shear deformation theory. Using a function, called boundary layer function, the coupled system of equations is converted into two decoupled equations. These equations are used to find a closed form solution for bending of thick transversely isotropic annular sector plates. It is shown that the solution of one of the decoupled equations has a boundary layer behavior like that of Mindlin plate theory. It is seen that the value of the boundary layer function for third order shear deformation theory is higher than that of the Mindlin theory. Thus, variations of stress components in the edge zone of the plate are more significant. Also, as in the Mindlin plate theory, there exist no boundary layer, a weak boundary layer, and a strong boundary layer effect for simply supported, clamped, and free edges, respectively.

Forced vibrations of orthotropic shells: nonclassical boundary-value problems

The forced vibrations of a cylindrical orthotropic shell are studied. Two types of boundary conditions on the outer surface are examined considering that the displacement vector prescribed on the inner surface varies harmonically with time. Asymptotic solutions of associated dynamic equations of three-dimensional elasticity are found. Amplitudes of forced vibrations are determined and conditions under which resonance occurs are established. Boundary-layer functions are defined. The rate of their decrease with distance from the ends inside the shell is determined. A procedure of joining solutions for the internal boundary-layer problem is outlined in the case for the, if clamping boundary conditions are prescribed at the ends

Accurate natural frequencies of transversely isotropic moderately thick annular sector plates

This article introduces an analytical method for free vibration analysis of transversely isotropic moderately thick annular sector plates. The plate has simply supported radial edges and arbitrary conditions along the circular edges. The displacement field of Mindlin's first-order shear deformation plate theory is considered and, based on this, the governing equations of motion are obtained. By using a function, which will be referred to as the boundary layer function, the three coupled governing equations of motion have been converted into two uncoupled equations. The general solutions of these equations contain integer and non-integer order Bessel and/or modified Bessel functions of the first and second kinds. By applying the boundary conditions at the inner and outer radii, an eigenvalue problem for finding the natural frequencies is obtained. The nine distinct cases considered involve all possible combinations of boundary conditions along the circular edges. An accurate non-dimensional frequency parameter is presented for a wide range of sector angles, some thickness—radius ratios and different inner-to-outer radius ratios. Three mode shapes of an annular sector plate with a sector angle of 60° are also presented. Finally, the effects of sector angle, thickness—radius ratio, boundary conditions, and inner-to-outer radius ratio on the frequency parameter are discussed.

Singularly perturbed first-order differential equations with integral boundary condition

In this paper, a kind of singularly perturbed first-order differential equations with integral boundary condition are considered. With the method of boundary layer function and the Banach fixed-point theorem, the uniformly valid asymptotic solution of the original problem is obtained.

Construction of asymptotic solutions to conjugation problems

A steady-state mass transfer problem related to underground storage of radioactive and chemical wastes is used to illustrate the application of an approximate method that opens up new opportunities for underground thermohydrodynamic simulation. The problem is represented as a sequence of mixed conjugation problems for the expansion coefficients, the remainder term, and the boundary-layer functions. Analytical expressions for the zero- and first-order expansion coefficients are derived.

Asymptotic solution for a micropolar flow in a curvilinear channel

AbstractThis paper is concerned with an asymptotic approach for a micropolar flow through a thin curvilinear channel. A priori estimates (which we obtain together with the existence and the uniqueness of the solution) are used to establish the error between the exact solution and the asymptotic one and to justify the asymptotic analysis. We obtain the expression of an expansion of order K and we study the general problems for the boundary layer functions. Under some additional assumptions on the data we obtain satisfactory error estimates.

Analytical solution for free vibration of transversely isotropic sector plates using a boundary layer function

The exact analytical solution based on the Mindlin's plate theory is presented for the free vibration of a transversely isotropic sector plate. The plate has simply supported radial edges and arbitrary conditions along the circular edge. By introducing a boundary layer function, the three coupled governing equations of motion have been converted into two uncoupled equations. Solving the uncoupled equations and satisfying the boundary conditions yields an eigenvalue problem for finding the natural frequencies. Non-dimensional frequency parameter is presented for over a wide range of salient ( α⩽180) and re-entrant (180< α⩽360) angles, some thickness–radius ratios and different boundary conditions along the circular edge.

Construction of “on the average exact” asymptotic solution of the problem on radial distribution of temperature field in a well

The example of the problem on radial distribution of temperature field in a well is used to illustrate the application of a modification of the asymptotic method for solving a number of problems in subterranean thermodynamics. The problem is represented in the form of a set of equations of mixed types for the respective coefficients of expansion, remainder term, and boundary-layer functions. Analytical expressions are constructed for coefficients of zero-order and first-order expansion and for boundary-layer functions. It is demonstrated that the constructed asymptotic formula provides for vanishing of the solution of the averaged problem for remainder term.

Asymptotic analysis of a non-periodic flow in a thin channel with visco-elastic wall

In this paper we continue the study of a fluid-structure interaction problem with the non periodic case. We consider the non stationary flow of a viscous fluid in a thin rectangle with an elastic membrane as the upper part of the boundary. The physical problem which corresponds to non homogeneous boundary conditions is stated. By using a boundary layer method, an asymptotic solution is proposed. The properties of the boundary layer functions are established and an error estimate is obtained.

Asymptotic study of the Rayleigh equation

This paper provides an asymptotic treatment of the Cauchy problem for the Rayleigh equation µy + ?3/3 + ay – ? = 0 modelling a harmonic oscillator. The method of boundary layer functions and our version of the Tikhonov theorem are used. The models of asymptotic approximations are derived

Asymptotics of solutions to the time-dependent Schrödinger equation with a small Planck constant

A regularized asymptotics of the solution to the time-dependent Schrodinger equation in which the spatial derivative is multiplied by a small Planck constant is constructed. It is shown that the asymptotics of the solution contains a rapidly oscillating boundary layer function.