We consider singularly perturbed turning point problems whose solutions exhibit an interior layer. Two suitable layer-adapted mesh-types are presented. For both types we give uniform error estimates in the ε-weighted energy norm for finite elements of higher order. Numerical experiments are used to compare the meshes and to confirm the theoretical findings.

For a singularly perturbed parabolic problem with Dirichlet conditions we prove the existence of a solution periodic in time and with boundary layers at both ends of the space interval in the case that the degenerate equation has a double root. We construct the corresponding asymptotic expansion in a small parameter. It turns out that the algorithm of the construction of the boundary layer functions and the behavior of the solution in the boundary layers essentially differ from that ones in case of a simple root. We also investigate the stability of this solution and the corresponding region of attraction.

Evolution equations are derived for the contrasting-structure-type solution of the gen-eralized Kolmogorov–Petrovskii–Piskunov (GKPP) equation with the small parameter with high order derivatives. The GKPP equation is a pseudoparabolic equation with third order derivatives. This equation describes numerous processes in physics, chemistry, biology, for example, magnetic field generation in a turbulent medium and the moving front for the carriers in semiconductors. The profile of the moving internal transitional layer (ITL) is found, and an expression for drift speed of the ITL is derived. An adaptive mesh (AM) algorithm for the numerical solution of the initial-boundary value problem for the GKPP equation is developed and rigorously substantiated. AM algorithm for the special point of the first kind is developed, in which drift speed of the ITL in the first order of the asymptotic expansion turns to zero. Sufficient conditions for ITL transitioning through the special point within finite time are formulated. AM algorithm for the special point of the second kind is developed, in which drift speed of the ITL in the first order formally turns to infinity. Substantiation of the AM method is given based on the method of differential inequalities. Upper and lower solutions are derived. The results of the numerical algorithm are presented.

In the present work the model boundary value problem for a stationary singularly perturbed reaction-diffusion-advection equation arising at the description of gas impurity transfer processes in an ecosystem ”forest – swamp” is considered. Application of a boundary functions method and an asymptotic method of differential inequalities allow to construct an asymptotics of the boundary layer type solution, to prove the existence of the solution with such an asymptotics and its asymptotic stability by Lyapunov as the stationary solution of the corresponding parabolic problem with the definition of local area of boundary layer type solution formation. The latter has a certain importance for applications, since it allows to reveal the solution describing one of the most probable conditions of the ecosystem. In the final part of the work sufficient conditions for existence of solutions with interior transitional layers (contrast structures) are discussed.

We consider a Dirichlet problem for a singularly perturbed convection-diffusion equation with constant coefficients in a rectangular domain in the case when the convection is parallel to the horizontal faces of the rectangular and directed to the right while the first derivative of the boundary function is discontinuous on the left face. Under these conditions the solution of the problem has a regular boundary layer in the neighborhood of the right face, two characteristic boundary layers near the top and bottom faces, and a horizontal interior layer due to the non-smoothness of the boundary function. We show that on the piecewise uniform Shishkin meshes refined near the regular and characteristic layers, the solution given by the classical five-point upwind difference scheme converges uniformly to the solution of the original problem with almost first-order rate in the discrete maximum norm. This is the same rate as in the case of a smooth boundary function. The numerical results presented support the theoretical estimate. They show also that in the case of the problem with a dominating interior layer the piecewise uniform Shishkin mesh refined near the layer decreases the error and gives the first-order convergence.
The article is published in the author’s wording.

The process of plastic flow localization under shear deformations of a composite material consisting from welded steel and copper is studied. A mathematical model describing this physical process is proposed. A new numerical approach based on Courant–Isaacson–Rees scheme is suggested. This algorithm was verified using three benchmark problems. Operability and effectiveness of this algorithm is confirmed. A numerical simulation of plastic flow localization in composite materials is performed. The influence on localization process of boundary conditions, of initial strain rate and materials width is studied. It is shown that at the initial stage the shear velocity of material layers oscillates. Theoretical estimates of frequency and oscillation period is given. Computational results coincide with these estimates. It is found that plastic flow localizes in the copper part of the composite. One or two areas of plastic flow localization appears depending on the width of steel and copper parts, as well as on the initial plastic strain rate and the selected type of a boundary conditions. The areas locate on characteristic distance from borders. The dependence of this distance and initial strain rate is shown and the corresponding estimates are obtained for two types of boundary conditions. When two areas of localization are formed, in one of them the temperature and the deformation increas faster than in another one.

Nonlinear convection–diffusion equations are widely used for the description of various processes and phenomena in physics, mechanics and biology. In this work we consider a family of nonlinear ordinary differential equations which is a traveling wave reduction of a nonlinear convection–diffusion equation with a polynomial source. We study a question about integrability of this family of nonlinear ordinary differential equations. We consider both stationary and non–stationary cases of this equation with and without convection. In order to construct general analytical solutions of equations from this family we use an approach based on nonlocal transformations which generalize the Sundman transformations. We show that in the stationary case without convection the general analytical solution of the considered family of equations can be constructed without any constraints on its parameters and can be expressed via the Weierstrass elliptic function. Since in the general case this solution has a cumbersome form we find some correlations on the parameters which allow us to construct the general solution in the explicit form. We show that in the non–stationary case both with and without convection we can find a general analytical solution of the considered equation only imposing some correlation on the parameters. To this aim we use criteria for the integrability of the Lienard equation which have recently been obtained. We find explicit expressions in terms of exponential and elliptic functions for the corresponding analytical solutions.

Investigations of initial boundary value problems for parabolic equations solutions are an important component of mathematical modeling. In this regard of special interest for mathematical modeling are the boundary value problem solutions that undergo sharp changes in any area of space. Such areas are called internal transitional layers. In case when the position of a transitional layer changes over time, the solution of a parabolic equation behaves as a moving front. For the purpose of proving the existence of such initial boundary value problem solutions, the method of differential inequalities is very effective. According to this method the so-called upper and lower solutions are to be constructed for the initial boundary value problem. The essence of an asymptotic method of differential inequalities is in receiving the upper and lower solutions as modifications of asymptotic submissions of the solutions of boundary value problems. The existence of the upper and lower solutions is a sufficient condition of existence of a solution of a boundary value problem. While proving the differential inequalities the so-called ”quasimonotony” condition is essential. In the present work it is considered how to construct the upper and lower solutions for the system of the parabolic equations under various conditions of quasimonotony.

In this paper, we analyzed the flat non-isothermal stationary flow of abnormally viscous fluid in the channels with asymmetric boundary conditions and an unknown output boundary. The geometry of the channels in which the problem is considered, is such regions, that at the transition to bipolar a system of coordinates map into rectangles. This greatly simplifies the boundary conditions, since it is possible to use an orthogonal grid and boundary conditions are given in its nodes. Fields of this type are often found in applications. The boundary conditions are set as follows: the liquid sticks to the boundaries of the channels, which rotate at different speeds and have different radius and temperature; moreover, temperature at the entrance to deformation is known, while on the boundary with the surface the material has the surface temperature; the pressure on the enter and exit of the region becomes zero. The rheological model only takes into account the anomaly of viscosity. The material is not compressible. This process can be described by a system consisting of continuity equations, the equations of conservation of momentum and an energy equation: ∇

The main objective of the paper is to present a new analytic-numerical approach to singularly perturbed reaction-diffusion-advection models with solutions containing moving interior layers (fronts). We describe some methods to generate the dynamic adapted meshes for an efficient numerical solution of such problems. It is based on a priori information about the moving front properties provided by the asymptotic analysis. In particular, for the mesh construction we take into account a priori asymptotic evaluation of the location and speed of the moving front, its width and structure. Our algorithms significantly reduce the CPU time and enhance the stability of the numerical process compared with classical approaches.
The article is published in the authors’ wording.

Parabolic singularly perturbed problems have been actively studied in recent years in connection with a large number of practical applications: chemical kinetics, synergetics, astrophysics, biology, and so on. In this work a singularly perturbed periodic problem for a parabolic reaction-diffusion equation is studied in the two-dimensional case. The case when there is an internal transition layer under unbalanced nonlinearity is considered. The internal layer is localised near the so called transitional curve. An asymptotic expansion of the solution is constructed and an asymptotics for the transitional curve is determined. The asymptotical expansion consists of a regular part, an interior layer part and a boundary part. In this work we focus on the interior layer part. In order to describe it in the neighborhood of the transition curve the local coordinate system is introduced and the stretched variables are used. To substantiate the asymptotics thus constructed, the asymptotic method of differential inequalities is used. The upper and lower solutions are constructed by sufficiently complicated modification of the asymptotic expansion of the solution. The Lyapunov asymptotical stability of the solution was proved by using the method of contracting barriers. This method is based on the asymptotic comparison principle and uses the upper and lower solutions which are exponentially tending to the solution to the problem. As a result, the solution is locally unique.
The article is published in the authors’ wording.

We consider a singularly perturbed elliptic problem, of convection-diffusion type, posed on a circular domain. Using polar coordinates, simple upwinding and a piecewise-uniform Shishkin mesh in the radial direction, we construct a numerical method that is monotone, pointwise accurate and parameter-uniform under certain compatibility constraints. Numerical results are presented to illustrate the performance of the numerical method when these constraints are not imposed on the data.

Error estimates of finite element methods for reaction-diffusion problems are often realized in the related energy norm. In the singularly perturbed case, however, this norm is not adequate. A different scaling of the H1 seminorm leads to a balanced norm which reflects the layer behavior correctly.

We consider two-dimensional singularly perturbed fourth order problems and estimate on properly constructed layer-adapted errors of a mixed method in the associated energy norms and balanced norms. This paper is a shortened version of [4].

A two-point boundary value problem on the interval [0, 1] is considered, where the highest-order derivative is a Caputo fractional derivative of order 2 − δ with 0 < δ < 1. A necessary and sufficient condition for existence and uniqueness of a solution u is derived. For this solution the derivative uŐ is absolutely continuous on [0, 1]. It is shown that if one assumes more regularity — that u lies in C2[0, 1] — then this places a subtle restriction on the data of the problem.

Interpolation of functions on the basis of Lagrange’s polynomials is widely used. However in the case when the function has areas of large gradients, application of polynomials of Lagrange leads to essential errors. It is supposed that the function of one variable has the representation as a sum of regular and boundary layer components. It is supposed that derivatives of a regular component are bounded to a certain order, and the boundary layer component is a function, known within a multiplier; its derivatives are not uniformly bounded. A solution of a singularly perturbed boundary value problem has such a representation. Interpolation formulas, which are exact on a boundary layer component, are constructed. Interpolation error estimates, uniform in a boundary layer component and its derivatives are obtained. Application of the constructed interpolation formulas to creation of formulas of the numerical differentiation and integration of such functions is investigated.