Open access peer-reviewed chapter

Singularly Perturbed Parabolic Problems

By Asan Omuraliev and Ella Abylaeva

Submitted: September 19th 2018Reviewed: January 11th 2019Published: September 27th 2019

DOI: 10.5772/intechopen.84339

Downloaded: 351


The aim of this work is to construct regularized asymptotic of the solution of a singularly perturbed parabolic problems. Namely, in the first paragraph, we consider the case when the scalar equation contains a free term consisting of a finite sum of the rapidly oscillating functions. In the first paragraph, it is shown that the asymptotic solution of the problem contains parabolic, power, rapidly oscillating, and angular boundary layer functions. Angular boundary layer functions have two components: the first one is described by the product of a parabolic boundary layer function and a boundary layer function, which has a rapidly oscillating change. The second section is devoted to a two-dimensional equation of parabolic type. Asymptotic of the scalar equation contains a rapidly oscillating power, parabolic boundary layer functions, and their product; then, the multidimensional equation additionally contains a multidimensional composite layer function.


  • singularly perturbed parabolic problem
  • asymptotic
  • stationary phase
  • power boundary layer
  • parabolic boundary layer
  • angular boundary layer

1. Asymptotics of the solution of the parabolic problem with a stationary phase and an additive-free member

1.1 Introduction

Singularly perturbed problems with rapidly oscillating free terms were studied in [1, 2, 3]. Ordinary differential equations with a rapidly oscillating free term whose phase does not have stationary points are studied in [1]. Using the regularization method for singularly perturbed problems [4], differential equations of parabolic type with a small parameter were studied in [2, 3] when fast-oscillating functions as a free member. The asymptotic solutions constructed in [1, 2, 3] contain a boundary layer function having a rapidly oscillating character of change. In addition to such a boundary layer function, ordinary differential equations contain an exponential [1], and parabolic equations - parabolic [2, 3] and angular boundary layer [2, 5] functions. If the phase of the free term has stationary points, then boundary layers arise additionally, having a power character of change. In this case, the asymptotic solution consists of regular and boundary layer terms. The boundary layer members are parabolic, power, rapidly oscillating boundary layer functions, and their products, which are called angular boundary layer functions [4]. In this chapter we used the methods of [4, 5].

1.2 Statement of the problem

In this chapter we study the following problem:


where ε>0is a small parameter and Ω = {(x, t): x01,t0T}.

The problem is solved under the following assumptions:

  1. ax>0,axС01,bxt,fxtСΩ¯.

  2. x01functionax>0.

  3. θktt=0=0is the phase function.

1.3 Regularization of the problem

For the regularization of problem (Eq. (1)), we introduce regularizing independent variables using methods [5, 6]:


Instead of the desired function uxtε, we will study the extended function


such that its restriction by regularizing variables coincides with the desired solution:


Taking into account (Eqs. (21)) and ((3)), we find the derivatives

On the basis of (Eqs. (1)(4)) for the extended function uˇMε, we set the problem:

LεuˇMε1ε2T0uˇMε+k=1NiθktεrkuˇMε+T1uˇMε=k=1Nfкxtexprk+iθк0ε+LζuˇMε+εLξuˇMε+ε2LxuˇMε     uˇMεt=rk=η=0=uˇMεx=0,ξ1=ζ1=0=uˇMεx=1,ξ2=ζ2=0=0,T1ην=12ζv2,T2tν=12ξv2bxt+k=1Nexprkσk,Lξaxv=12Dξ,v,Lζaxv=12Dζ,v,Lxaxx2.E5

The problem (Eq. (5)) is regular in ε as ε → 0:


1.4 Solution of iterative problems

The solution of problem (Eq. (5)) will be determined in the form of a series:


For the coefficients of this series, we obtain the following iterative problems:


The solution of this problem contains parabolic boundary layer functions; internal power boundary layer functions which are connected with a rapidly oscillating free term in a phase which are vanished at t=tl,l=0,1,,nin addition; and the asymptotic also contain angular boundary layer functions. We introduce a class of functions in which the iterative problems will be solved:



From these spaces we construct a new space:


The element uMϵGhas the form:


1.5 Solvability of intermediate tasks

The iterative problems (Eq. (9)) in general form will be written:


Theorem 1. Suppose that the conditions (1)(3) and HMϵG3are satisfied. Then, equation (Eq. (10)) is solvable in G.

Proof. Let the free term HMϵG3be representable in the form:


Then, by directly substituting function uMϵGfrom (Eq. (9)) in (Eq. (10)), we see that this function is a solution if and only if the function YklNlwill be a solution of equation:


With the corresponding boundary conditions, this equation has a solution which have the estimate:


The theorem is proven.

Theorem 2. Suppose that the conditions of Theorem 1 are satisfied. Then, under additional conditions:

  1. uMt=η=0=0uMx=l1,ξl=0,ςl=0=0,l=1,2.

  • LςuM=0,LξuM=0.

  • ik=1NθktrkuvM+T2uv1M+hMG3.

  • Eq. (10) is uniquely solvable.

    Proof. By Theorem 1 equation (Eq. (10)) has a solution that is representable in the form (Eq. (9)). With satisfying condition (1), we obtain




    Due to the fact that the function erfcθ2tis zero at θ=0, the values for wlxtt=0qklxtt=0are chosen arbitrarily.

    We calculate


    Condition (3) of the theorem will be ensured, if we choose arbitrarily (Eq. (9)) as the solutions of the following equations:


    After this choice of arbitrariness, expression (Eq. (13)) is rewritten:


    In (Eq. (14)), transition was made from ξl/2tto variable ςl/2η. The function YklNlis defined as the solution of equation (Eq. (30)) under the boundary conditions from (Eq. (12)) in the form:


    We substitute this function in the corresponding equation from (Eq. (14)); then with respect to dklxt, we obtain a differential equation, which is solving under the initial condition dklxtt=0=d¯klx, and we find


    where Pklxtis known as the function.

    By substituting the obtained function into condition for dklxtx=l1from (Eq. (12)), we define the value of d¯klxx=l1. The obtained value is used as an initial condition for a differential equation with respect to d¯klx, which is obtained after substitution dklxtinto the first condition of (2). With that we ensure fulfillment of this condition and uniqueness of the function YklNl.The last equation from (Eq. (14)) due to the fact that θktk=0is solvable if


    The obtained ratio is used as the initial condition for the differential equation with respect to zk,v1lxtfrom (Eq. (14)).

    The equation with respect to vv1xtunder the initial condition from (12) determines this function uniquely. Equations with respect to wk,v1lxt,qk,v1lxtunder the corresponding condition from (Eq. (12)) have solutions representable in the form:


    where H1,v1lxt,H2,v1lxt- are known functions.

    With substituting (Eq. (16)) into the conditions under x=l1from (Eq. (12)), we define values of w¯k,v1lxx=l1q¯k,v1lxx=l1. These conditions are used in solving differential equations which are obtained from the second condition of (Eq. (21)):


    Thus, function uMis determined uniquely. The theorem is proven.

    1.6 Solution of iterative problems

    Eq. (8) is homogeneous for k = 0; therefore, by Theorem 1, it has a solution in G, representable in the form:


    If the function Yk,0lNlis the solution of the equation ηYk,0lNl=ςl2Yk,0lNlwhich is satisfying that


    from the last problem, we define


    d¯k,0lxis the arbitrary function. In the next step, equation (Eq. (8)) for k = 1 takes the form:


    According to Theorem 1, this equation is solvable in U, if ck,0xt=0; the function Yk,0lNlis the solution of the differential equation ηYk,0lNl=ςl2Yk,0lNl+Hk,0lNl, and its solution is representable in the form (Eq. (14)), where Hkl0=ktYk,0lNl.Satisfying condition (1)(3) of Theorem 1, we obtain (see (Eq. (14)))


    When the equation is obtained with respect to dk,0lxtin the qk,0lxterfcξl2t, a transition ξl2t=ςl2ηoccurs.

    The initial conditions for equation (Eq. (18)) are determined from (Eq. (12)). Functions w0lxt,dk,0lxt,qk,0lxtare expressed through arbitrary functions w¯0lx,d¯k,0lx,q¯k,0lx. These arbitrary functions provide the condition:


    ensuring the solvability of the equation with respect to ck,1lxt.Suppose that


    This relation is used by the initial condition for determining Zk,0xtfrom the equation entering into (Eq. (18)).

    Further repeating this process, we can determine all the coefficients of ukmof the partial sum:


    In each iteration with respect to vixt,wilxt,dk,ilxt,zk,ixt,qk,ilxt, we obtain inhomogeneous equations.

    1.7 Assessment of the remainder term

    For the remainder term


    taking into account (Eqs. (3) and (6)), we obtain the equation


    with homogeneous boundary conditions. Using the maximum principle, like work of [7], we get the estimate:


    Theorem 3. Suppose that conditions (1)(3) are satisfied. Then, the constructed solution is an asymptotic solution of problem (Eq. (1)), i.e., n=0,1,2,; the estimate is fair (Eq. (18)).

    2. Two-dimensional parabolic problem with a rapidly oscillating free term

    2.1 Introduction

    In the case when a small parameter is also included as a multiplier with a temporal derivative, the asymptotic of the solution acquires a complex structure.

    Different classes of singularly perturbed parabolic equations are studied in [2]. There, regularized asymptotics of the solution of these equations are constructed, when a small parameter is in front of the time derivative and with one spatial derivative. It is shown that the constructed asymptotic contains exponential, parabolic, and angular products of exponential and parabolic boundary layer functions. The equations are studied when the limiting equation has a regular singularity. Such equations have a power boundary layer. If a small parameter is entering as the multiplier for all spatial derivatives, then the asymptotic solution contains a multidimensional parabolic boundary layer function. When entering into the equation, as free terms of rapidly oscillating functions, then the asymptotic of the solution additionally contains fast-oscillating boundary layer functions. If it is additionally assumed that the phase of this free term has a stationary point, in addition to the rapidly oscillating boundary layer function that arises as a power boundary layer.

    This section is devoted to a two-dimensional equation of parabolic type.

    2.2 Statement of the problem

    Consider the problem:


    where ε>0is the small parameter, x=x1x2, Ω=0<x1<1x0<x2<1,E=0<tT,al=12alxlxl2.

    The problem is solved under the following assumptions:

    1. xl01the function alxlС01,l=1,2.

    2. bxt,fxtСE.

    3. θ0=0.

    2.3 Regularization of the problem

    Following the method of regularization of singularly perturbed problems [1, 2], along with the independent variables xt, we introduce regularizing variables:


    For extended function uMε,M=xtτξηsuch that


    Find from (Eq. (22)) the derivatives based on


    Below, it is shown that the solution of the iterative problems does not contain terms depending on ξ1ξ2,ξ3ξ4,ζ1ζ2,ζ3ζ4,ξlζk,l,k=1,2.Therefore, to simplify recording, the mixed derivatives of these variables are omitted. Based on (Eq. (20)), (Eq. (22)), and (Eq. (23)) for extended function uMε, set the problem:


    In this case, the identity is satisfied:


    2.4 Solution of iterative problems

    For the solution of the extended function (Eq. (24)), we search in the form of series


    Then, for the coefficients of this series, we get the following problems:


    We introduce a class of functions:


    From these classes we will construct a new one, as a direct sum:


    Any item uMUisrepresentable in the form:


    Let’s satisfy this function to the boundary conditions:




    We compute the action of the operators T0,T1,Lη,Lξon function uMU, and we have


    We write iterative equation (8) in the form:


    Theorem 1. Let be HMU4U5and condition (1) is satisfied. Then, Eq. (31) is solvable in U, if the equations are solvable:


    Theorem 2. Let beH1NlU4. Then, the problem τ1YlNl=ηYlNl+H1Nl,YlNlτ1=0=0YlNlηl=0=dlxt,l=1,4¯(Eq. (32)) has a solutionYlNlU4.

    Theorem 3. Let be H2Nr+2,lU5,YlNlU4,and then the problem τ1Yr+2,lNr+2,l=ηYr+2,lNr+2,l+H2Nr+2,l,Yr+2,lNr+2,lηl=0=Yr+2Nr+2Yr+2,lNr+2,lηr+2=0=YlNl,r,l=1,2has a solution Yr+2,lNr+2,lU5.

    The proof of these theorems is given in [2].

    2.5 The decision of the iterative problems

    Eq. (27) under v=0,1is homogeneous. By Theorem 1, it has a solution representable in the form u0MUif functions YlNland Yr+2,lNr+2,l– are solutions of the following equations:


    Based on the boundary conditions from (Eq. (29)), the solution is written:


    wheredlxt– is arbitrary function such as


    Due to the fact that the function dvlxtпри t=τ1=0multiplied by the function becomes as d0lxtt=0=d¯0lx, an arbitrary function is accepted, and its values under x1=l1are determined from the second relation. According to Theorems 2 and 3, the functions found by the formula (Eq. (33)) satisfy the estimates:


    Free member of equation (Eq. (27)) under v=2,3has a form


    so that equation (Eq. (27)), under v=2,3, has a solution in U; we set


    Solutions of the last equations under the boundary conditions from (Eq. (29)) have a form (Eq. (33)) for which estimates of the form (Eq. (35) are fair. Eq. (27), i=4, has a free term:

    σDtz0xt+l=14Dtq0lxterfcξl2μ+r,l=12Dtz0r+2,lMr+2,lz0xt+l=14q0lxterfcξl2μ+r,l=12z0r+2,lMr+2,lexpτ2+r=12 l=2r12rDx,ξr,lw0pxterfcξl2μ+v=12 r,l=12Dx,ηv,lY0r+2,lNr+2,l.

    By providing F4MU4U5with regard to cvxt=0,v=0,1,we set




    In the last bracket, the transition is from the variables ξl2μto the variables ηl2τ2.

    Substituting the value Y0lNl=d0lxterfcηl2τ1into equation DtY0lNl=0,with respect to d0lxt,we get the equation Dtd0lxt=0,which is solved under an arbitrary initial condition d0lxtt=0=d¯0lx. This arbitrary function provides the condition LηY0l=0;therefore, Dx,ηY0l=0.The initial condition for this equation is determined from the relation:


    which comes out from (Eq. (29)) and (Eq. (33)). The functionY0r+2,lNr+2,lexpresses through Y0lNltherefore provided that


    The same is true for functions w0r+2,lMr+2,l,z0r+2,lMr+2,l; in other words, the following relations hold: Dtw0r+2,l=0,Dtz0r+2,l=0,Dx,ξv,lw0r+2,l=0,Dx,ξv,lz0r+2,l=0.

    Solutions of equations with respect w0r+2,l,z0r+2,lunder appropriate boundary conditions from (Eq. (29) are representable as (Eq. (33)), and they are expressed through w2lxt,q2lxt.The first equation (Eq. (36)) is solvable, ifz0xtt=0=fx0exp0ε.This ratio is used by the initial condition for the equation

    Dtz0xt=0. The remaining equations from (Eq. (36)) are solvable under the initial conditions from (Eq. (29)).

    Thus, the main term of the asymptotics is uniquely determined. As can be seen from the representation (Eq. (28)) and the estimates (Eq. (35)), we note that the asymptotics of the solution have a complex structure. In addition to regular members, it contains various boundary layer functions. Parabolic boundary layer functions have an estimate:


    Multidimensional and angular parabolic boundary layer functions have an estimate:


    The boundary layer functions with rapidly oscillating exponential and power type of change:


    In addition, the asymptotic contains the product of the abovementioned boundary layer functions.

    Repeating the above process, we construct a partial sum:


    2.6 Assessment of remainder term

    Substituting the function uMε=uεnM+εn+12RεnMinto problem (Eq. (24)), then taking into account the iterative tasks of (Eq. (27)) and (Eq. (29)), we obtain the following problem for the remainder term RεnM:


    where gnMε=iθtτ2un1ε12iθtτ2unMT1un1Mε12T1unMDσLηk=03εk2un3+kM+Lηk=05εk2un5+kM+Δak=07εk2un7+kM.

    We put in both parts (Eq. (38))χ=ψxtεconsidering (Eq. (25)), with respect to


    By virtue of the above constructions, the function is gεnxtε<c,xt;

    therefore, applying the maximum principle, an estimate is established:


    Thus, we have proven the following:

    Theorem 4. Suppose that the conditions (1)(3) are satisfied. Then, using the above method for solving uxtεof the problem (Eq. (20)), a regularized series (Eq. (26)) such that n=0,1,2,can be constructed, and for small enough ε>0,inequality is fair:


    where cis independent of ε.

    © 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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    Asan Omuraliev and Ella Abylaeva (September 27th 2019). Singularly Perturbed Parabolic Problems, Boundary Layer Flows - Theory, Applications and Numerical Methods, Vallampati Ramachandra Prasad, IntechOpen, DOI: 10.5772/intechopen.84339. Available from:

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