In  this  paper,  we study  the  moving  front solution  of the  reaction-diffusion  initialboundary value problem  with a small diffusion coefficient. Problems  in such statements can be used to model physical processes associated  with the propagation of autowave  fronts, in particular, in biophysics or in combustion. The moving front solution is a function  the distinctive feature of which is the presence in the domain  of its definition  of a subdomain where the function  has a large gradient. This subdomain is called an internal  transition layer.  In the nonstationary case, the position of the transition layer varies with  time  which, as it is well known,  complicates  the  numerical  solution  of the  problem  as well as the justification of the correctness  of numerical calculations. In this case the analytical method is an essential component  of the  study.   In the  paper,  asymptotic methods  are applied  for analytical investigation of the  solution  of the  problem  posed.   In particular, an  asymptotic approximation of the  solution  as an expansion  in powers of a small parameter is constructed by the  use of the  Vasil’eva algorithm and  the existence  theorem  is carried  out using the asymptotic method  of differential  inequalities.  The methods used also make it possible to obtain  an equation  describing  the motion  of the front.  For this purpose  a transition to local coordinates  takes  place in the  region of the  front localization.   In the  present paper, in comparison  with earlier publications dealing with two-dimensional problems  with internal  transition layers the  transition to local coordinates  in the  vicinity  of the  front has been modified, that led to the simplification  of the algorithm of determining the equation  of the curve motion.

Abstract. A singularly  perturbed elliptic problem with Dirichlet boundary conditions  is considered in the case of multiple roots of the degenerate equation.  A three-zone boundary layer arises in the vicinity of the  domain  boundary with a different scale of boundary-layer variables  and  a different behaviour  of the  solution  in different zones.  The  asymptotic expansion  of the  solution  being in fractional  powers of the small parameter, boundary-layer series are constructed using a non-standard algorithm. A complete asymptotic expansion  of the solution  is constructed and justified.

A boundary value  problem  for a singularly  perturbed differential  equation  of second order  is considered  in two  cases, when one root  of the  degenerate equation  is two-tuple.   It  is proved that in the  first  case the  problem  has  a solution  with  the  transition from  the  two-tuple root  of the degenerate equation  to one-tuple  root  in the  small neighbourhood of an internal  point of the  interval, and in the second case the problem has a solution which has the spike in the interior  layer.  Such solutions are named,  correspondingly, a contrast structure of step-type and a contrast structure of spike-type.  In each case the  asymptotic expansion  of the  contrast structure is constructed.  It distinguishes from the known expansion  in the case, when all the roots of the degenerate equation  are one-tuple,  in particular, the interior  layer is multizonal.

We consider a nonstationary process of spreading  some substance in a one-dimensional spatially  inhomogeneous  system  of cells.  It  is assumed  that a change  in the  concentration of \(u_n (t)\) in a cell with  the  number  \(n\) with  time \(t\) is determined by the  difference in concentration in this  cell and in its  two  neighbors  on the  left and  on the  right,  as well as the  source  density,  which  depends  on \(n\) and  depends  on \(u_n (t)\).  Such a model leads to the  initial-boundary value  problem  for the  differentialdifference  equation  (differentiation with  respect  to  t variable,  the  difference  expression  with  respect to \(n\)).   With  a sufficiently  small  difference in concentration in each  pair  of neighboring  cells we can replace the difference expression by the second partial derivative  with respect  to the spatial  coordinate, and  describe the  propagation by the  reaction-diffusion  equation.   This  equation  belongs to the  class of quasilinear  parabolic  equations.   It is assumed  that the  density  of the  sources vanishes  (with  changing the  sign) at  three  values of the  concentration, two of which, lower and  upper,  are stable.  There  is also an intermediate unstable  state  with zero source density,  in which the sign reversal also takes place.  The peculiarity of our model is that we assume,  that two extreme  roots  of the  source density  function  are degenerate (with an integer or fractional  exponent). We intend to show analytically and by the computer simulation, that this  model leads  to  the  fact,  that the  rate  of asymptotic aspiration of concentration to equilibrium  values for a moving front becomes power-law instead  of exponential, which takes  place for standard models.   In the  paper,  we have  constructed a formal  asymptotics solution  of the  initialboundary value problem for the reaction-diffusion  equation  in a homogeneous medium with a power-law dependence  of the  source  density  on the  temperature, an  upper  and  lower solutions  are  constructed, a rigorous  justification of the  formal  asymptotics is given.   Precise  solutions  of the  diffusion reaction equation  are constructed for a wide class of source density  functions.

This  work develops  a theory  of the  asymptotic-numerical investigation of the  moving fronts  in reaction-diffusion-advection models.  By considering  the  numerical  solution  of the  singularly perturbed Burgers’s  equation  we discuss a method  of dynamically  adapted mesh  construction that is able to significantly  improve  the  numerical  solution  of this  type of equations.  For  the  construction we use a priori information that is based  on the  asymptotic analysis  of the  problem.  In  particular, we take  into account the information about  the speed of the transition layer, its width  and structure. Our algorithms  are able to reduce significantly complexity and enhance stability of the numerical  calculations in comparison  with classical approaches for solving this class of problems.  The numerical  experiment is presented to demonstrate the effectiveness of the proposed  method.

The article  is published  in the authors’  wording.

Physical  phenomena  that arise near the boundaries  of media with different characteristics, for example,  changes in temperature at  the  water-air interface,  require  the  creation  of models for their  adequate description. Therefore,  when setting  model problems  one should  take  into  account the fact  that the  environment parameters undergo  changes  at  the  interface.   In particular, experimentally obtained temperature curves at the water-air interface have a kink, that is, the derivative  of the temperature  distribution function  suffers a discontinuity at the interface.  A function  with this feature  can be a solution to the problem for the heat equation  with a discontinuous thermal diffusivity and discontinuous function  describing  heat  sources.  The  coefficient of thermal diffusivity  in the  water-air transition layer is small, so a small parameter appears  in the  equation  prior  to the  spatial  derivative, which makes the equation  singularly  perturbed.  The  solution  of the  boundary value  problem  for such an equation  can have the  form of a contrast structure, that is, a function  whose domain  contains  a subdomain, where the  function  has a large gradient. This  region is called an internal  transition layer.  The  existence  of a solution with the internal  transition layer of such a problem requires justification that can be carried out with the use of an asymptotic analysis.  In the present paper,  such an analytic  investigation was carried out,  and this  made it possible to prove the  existence  of a solution  and also to construct its asymptotic approximation.

We consider  a boundary-value problem  for a singularly  perturbed parabolic  equation with an initial function  independent of a perturbation parameter in the case where a degenerate stationary equation  has smooth possibly intersecting roots.  Before, the existence of a stable stationary solution to  this  problem  was proved  and  the  domain  of attraction of this  solution  was investigated — due  to exchange  of stabilities, the  stationary solution  approaches  the  non-smooth  (but  continuous) composite root  of the  degenerate equation  as the  perturbation parameter gets smaller,  and  its domain  of attraction contains  all initial  functions  situated strictly  on one side of the other  non-smooth  (but  continuous) composite  root of the degenerate equation.   We show that if the  initial  function  is out of the boundary of this family of initial  functions  near some point,  the problem cannot  have a solution inside the domain of the problem,  i.e. this boundary is the true  boundary of the attraction domain.  The proof uses ideas of the nonlinear  capacity  method.

In the  paper,  the  dynamics  of a class of one-dimensional  piecewise linear  displays  with one gap is studied.   Stable  conditions  of equilibrium  as well as other  attractors are found by numerical methods.  During the investigation two basic cases to which all remaining ones come down are considered. In the  space of parameters, the  areas  responding  to these  or those  phase  reorganizations are selected. In particular, it was ascertained that for this  class of functions,  under  condition  of a continuity on the considered display, there is no set of parameters of it that in case of the given restrictions on the function there  were at least two attractors. In case of the existence  of a gap is there  are infinitely many areas in which two attracting cycles coexist, and if in the area there are two attracting cycles, their periods differ exactly  by a unit,  and  there  are  no areas  where there  would be three  or more attractors.  Besides,  it was revealed  that in case of three-dimensional motion  of parameters along a straight line steady  cycles of the  various  periods  with  the  following important feature  are  watched:   each  area  supports exactly one or exactly  two attracting cycles, and  the  area  containing  \(k\) attracting cycles adjoin  to  the  areas containing  \(3 − k\) attracting cycles, and  sets of values of the  periods  of any two adjoining  areas  have a nonzero intersection.

Considered  is a mathematical model of insects  population dynamics,  and  an attempt is made  to explain  classical experimental results  of Nicholson with  its help.  In the  first section  of the paper  Nicholson’s experiment is described  and dynamic  equations  for its modeling are chosen.  A priori estimates  for model parameters can be made more precise by means of local analysis  of the  dynamical system,  that is carried  out in the second section.  For parameter values found there  the stability loss of the  problem  equilibrium  of the  leads to the  bifurcation of a stable  two-dimensional torus.   Numerical simulations  based  on the  estimates  from the  second section  allows to explain  the  classical Nicholson’s experiment, whose detailed  theoretical substantiation is given in the last section.  There for an atrractor of the  system  the  largest  Lyapunov  exponent is computed. The  nature of this  exponent change allows to additionally narrow  the area of model parameters search.  Justification of this experiment was made possible  only  due  to  the  combination of analytical and  numerical  methods  in studying  equations  of insects  population dynamics.   At the  same time,  the  analytical approach made  it possible to perform numerical  analysis  in a rather narrow  region of the  parameter space.  It is not  possible to get into this area,  based only on general considerations.