# Asymptotic Solutions of Singular Perturbed Problems with an Instable Spectrum of the Limiting Operator

Authors: Burkhan T. Kalimbetov; Marat A. Temirbekov; Zhanibek O. Khabibullayev
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The regularization method is applied for the construction of algorithm for an asymptotical solution for linear singular perturbed systems with the irreversible limit operator. The main idea of this method is based on the analysis of dual singular points of investigated equations and passage in the space of the larger dimension, what reduces to study of systems of first-order partial differential equations with incomplete initial data.


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## Body

## 1. Introduction

The investigation of singular perturbed systems for ordinary and partial differential equations occurring in systems with slow and fast variables, chemical kinetics, the mathematical theory of boundary layer, control with application of geoinformational technologies, quantum mechanics, and plasma physics (the Samarsky-Ionkin problem) has been studied by many researchers (see, e.g., [1–19]).In this work, the algorithm for construction of an asymptotical solution for linear singular perturbed systems with the irreversible limit operator is given—the regularization method [1]. The main idea of this method is based on the analysis of dual singular points of investigated equations and passage in the space of the larger dimension, what reduces to the study of systems of first-order partial differential equations with incomplete (more exactly, point) initial data.In this paper, we consider linear singular perturbed systems in the form(1.1)εẏ=A(t)y+h(t),y(0,ε)=y0,t∈[0,T],
where y={y1,…,yn},A(t) is a matrix of order (n×n),h(t)={h1,…,hn} is a known function, y0∈Cn is a constant vector, and ε>0 is a small parameter, in the case of violation of stability of a spectrum {λj(t)} of the limiting operator A(t).Difference of such type problems from similar problems with a stable spectrum (i.e., in the case ofλi(t)≠0,λi(t)≠λj(t),i≠j,i,j=1,n¯forallt∈[0,T]) is that the limiting system 0=A(t)y̅+h(t)at violation of stability of the spectrum can have either no solutions or uncountable set of them. In the last case, presence of discontinuous on the segment [0,T] solutions y̅(t)of the limiting system is not excluded. Under conditions, one can prove (see, e.g., [1, 6]) that the exact solution y(t,ε) of problem (1.1) tends (at ε→+0) to a smooth solution of the limiting system. However, there is a problematic problem about construction of an asymptotic solution of problem (1.1). When the spectrum is instable, essentially special singularities are arising in the solution of system (1.1). These singularities are not selected by the spectrum {λj(t)} of the limiting operator A(t). As it was shown in [3–7], they were induced by instability points tj of the spectrum.In the present work, the algorithm of regularization method [1] is generalized on singular perturbed systems of the form (1.1), the limiting operator of which has some instable points of the spectrum. In order to construct the spectrum, we use the new algorithm requiring more constructive theory of solvability of iterative problems. These problems arose in application of the algorithm.We will consider the problem (1.1) at the following conditions. Assume that(i)
A(t)∈C∞([0,T],Cn),h(t)∈C∞[0,T]; for any t∈[0,T], the spectrum {λj(t)} of the operator A(t) satisfies the conditions:(ii)
λi(t)=-(t-ti)siki(t),ki(t)≠0,ti∈[0,T],i=1,m¯,m<n (here si- are even natural numbers),(iii)
λi(t)≠0,j=m+1,n¯,(iv)
λi(t)≠λj(t),i≠j,i,j=1,n¯,(v)
Reλj(t)⩽0,j=1,n¯.


## 2. Regularization of the Problem

We introduce basic regularized variables by the spectrum of the limiting operator(2.1)τj=ε-1∫0tλj(s)ds≡φj(t)ε,j=1,n̅.
Instable points ti∈[0,T] of the spectrum {λj(t)} induce additional regularized variables described by the formulas(2.2)σiqi=eφi(t)/ε∫0te-φi(s)/ε(s-ti)qiqids≡ψiqi(t,ε),i=1,m̅,qi=0,si-1̅.
We consider a vector function ỹ(t,τ,σ,ε) instead of the solution y(t,ε) to be found for problem (1.1). This vector function is such that(2.3)ỹ(t,τ,σ,ε)∣τ=φ,σ=ψ≡y(t,ε).Forỹ(t,τ,σ,ε), it is natural to set the following problem:(2.4)Lεỹ(t,τ,σ,ε)≡ε∂ỹ∂t+∑j=1nλj(t)∂ỹ∂τj+∑i=1m∑qi=0si-1[λi(t)σiqi+ε(t-ti)qiqi!]∂ỹ∂σiqi-A(t)ỹ=h(t),ỹ(0,0,0,ε)=y0.
We determine the solution of problem (2.4) in the form of a series(2.5)ỹ(t,τ,σ,ε)=∑k=-1∞εkyk(t,τ,σ),
with coefficients yk(t,τ,σ)∈C∞[0,T].If we substitute (2.5) in (2.4) and equate coefficients at identical degrees of ε, we obtain the systems for coefficients yk(t,τ,σ):(ε-1)Ly-1(t,τ,σ)≡∑j=1nλj(t)∂y-1∂τj+∑i=1m∑qi=0si-1λi(t)σiqi∂y-1∂σiqi-A(t)y-1=0,y-1(0,0,0)=0,(ε0)Ly0(t,τ,σ)=-∂y-1∂t-∑i=1m∑qi=0si-1(t-ti)qiqi!∂y-1∂σiqi+h(t),y0(0,0,0)=y0,⋮(εk+1)Lyk+1(t,τ,σ)=-∂yk∂t-∑i=1m∑qi=0si-1(t-ti)qiqi!∂yk∂σiqi,k⩾1,yk+1(0,0,0)=0,⋮


## 3. Resolvability of Iterative Problems

We solve each of the iterative problems(εk) in the following space of functions:(3.1)U={y(t,τ,σ):y=∑k=1n∑j=1nykj(t)ck(t)eτj+∑k=1n∑i=1m∑qi=0si-1ykiqi(t)ck(t)σiqi+∑k=1nyk(t)ck(t),ykj(t),ykiqi(t),yk(t)∈C∞([0,T],C1)},
where ck(t) are eigenvectors of the operator A(t) corresponding eigenvalues λk(t),k=1,n¯. We represent U in the form of U(1)⊕U(0) where(3.2)U(0)={y(0)(t):y(0)=∑j=1nyj(0)(t)cj(t),yj(0)(t)∈C∞([0,T],C1)},U(1)=UU(0).
It is easy to note that each of the systems (εk+1) can be written in the form(3.3)Ly(t,τ,σ)=h(t,τ,σ),
where h(t,τ,σ) are the corresponding right hand side. Using representations of space U, we can write system (3.3) in the equivalent form(3.4)Ly(1)(t,τ,σ)=h(1)(t,τ,σ),(3.5)-A(t)y(0)(t)=h(0)(t),
where y(1)(t,τ,σ),h(1)(t,τ,σ)∈U(1),y(0)(t),h(0)(t)∈U(0).We have the following result.Theorem 3.1.
Leth(1)(t,τ,σ)∈U(1) and satisfy conditions (i)–(iv). Then, system (3.4) is solvable in the U(1) if and only if
(3.6)〈h(1)(t,τ,σ),νj(t,τ,σ)〉≡0∀t∈[0,T],j=1,n¯,〈h(1)(t,τ,σ),νiqi(t,τ,σ)〉≡0,i=1,m¯,qi=0,si-1¯,
where νj(t,τ,σ),νiqi(t,τ,σ) are basic elements of the kernel of the operator
(3.7)L*≡∑j=1nλj(t)∂∂τj+∑i=1m∑qi=0si-1λi(t)σiqi∂∂σiqi-A*(t).Proof.
Leth(1)(t,τ,σ)=∑k=1n∑j=1nhkj(t)cj(t)eτk+∑k=1n∑i=1m∑qi=0si-1hkiqi(t)ck(t)σiqi.
Determine solutions of system (3.4) in the form
(3.8)y(1)(t,τ,σ)=∑k=1n∑j=1nykj(t)ck(t)eτj+∑k=1n∑i=1m∑qi=0si-1ykiqi(t)ck(t)σiqi.
Substituting (3.8) in (3.4) and equating separately coefficients at eτj and σiqi, we obtain the equations
(3.9)[λk(t)-λj(t)]ykj(t)=hkj(t),k,j=1,n¯,[λi(t)-λk(t)]yiqik(t)=hkiqi(t),i=1,m¯,qi=0,si-1¯,k=1,n¯.
One can see from this that obtained equations are solvable if and only if(3.10)hkk(t)≡0,k=1,n¯,hiiqi(t)≡0,i=1,m¯,qi=0,si-1¯,
and these conditions coincide with conditions (3.6). Theorem 3.1 is proved.Remark 3.2.
Equations (1.1) imply that under conditions (3.6), system (3.4) has a solution in U(1) representable in the form
(3.11)y(1)(t,τ,σ)=∑k=1n∑j=1,j≠knhkj(t)[λk(t)-λj(t)]cj(t)eτj+∑k=1nαk(t)ck(t)eτk+∑i=1m∑qi=0si-1γiqi(t)ci(t)σiqi+∑k=1n∑i=1,i≠km∑qi=0si-1hkiqi(t)[λi(t)-λk(t)]ck(t)σiqi,
where αk(t),γiqi(t)∈C∞([0,T],C1) are arbitrary functions.
Consider now system (3.5). As detA(t)≡0 in points t=ti, i=1,m¯, this system does not always have a solution in U(0). Introduce the space V(0)⊂U(0) consisting of vector functions
(3.12)z(0)(t)=∑j=1nzj(t)cj(t),zj(t)∈C∞([0,T],C1),j=1,n¯,
having the properties
(3.13)[Dli(z(0)(t),di(t))]t=ti=(Dlizi)(ti)=0,∀li=0,si-1¯,i=1,m¯,
where di(t) are eigenvectors of the operator A*(t) with regard to eigenvalues λ̅i(t),i=1,m¯. Let h(0)(t)=∑j=1nhj(t)cj(t)∈V(0), that is,
(3.14)(Dlihi)(ti)=0∀li=0,si-1¯,i=1,m¯.
Determine a solution of system (3.5) in the
(3.15)y(0)(t)=∑j=1nyj(t)cj(t).
Substituting this function in (3.5), we obtain
(3.16)-∑j=1nyj(t)λj(t)cj(t)=∑j=1nhj(t)cj(t).
Since {cj(t)} is a basis in Cn, we get
(3.17)-λi(t)yi(t)=hi(t),i=1,m¯,(3.18)-λj(t)yj(t)=hj(t),j=m+1,n¯.
It is easy to see that (3.18) has the unique solution
(3.19)yj(t)=-hj(t)λj(t),j=m+1,n¯.
By virtue of conditions (3.14), the function hi(t) can be represented in the form
(3.20)hi(t)=(t-ti)siĥi(t),i=1,m¯,
where ĥi(t)∈C∞([0,T],C1 is the certain scalar function, -(t-ti)siki(t)yi(t)=(t-ti)siĥi(t), and we see that
(3.21)yi(t)={-ĥi(t)ki(t),t≠ti,γi,t=ti,
where γi are arbitrary constants, i=1,m¯. However, the solution of system (3.5) should belong to the space U(0), and it means that yi(t)∈C∞([0,T],C1). Therefore, constants in (3.21) γi=(ĥi(t)/ki(t))∣t=ti and functions are determined uniquely in the form
(3.22)yi(t)=-ĥi(t)ki(t),∀t∈[0,T],i=1,m.¯
Thus, under conditions (3.14), system (3.5) has the solution y(0)(t) in U(0) of
(3.23)y(0)(t)=-∑i=1mĥi(t)ki(t)ci(t)-∑j=m+1nhi(t)λi(t)ci(t),
where hi(t)=ĥi(t)/(t-ti)si (in points t=ti,i=1,m¯, this equality is understood in the limiting sense). We summarize received outcome in the form of the following assertion.Theorem 3.3.
Let the operatorA(t) satisfy condition (i), and let its spectrum satisfy conditions (ii)–(iv). Then, for any vector function h(0)(t)∈V(0), system (3.5) has the unique solution y(0)(t) in space U(0).For uniquely determination of functionsαj(t),γiqi(t), consider system (3.4) with additional conditions:(3.24)y(1)(0,0,0)=y*,(3.25)〈-∂y(1)∂t,νj(t,τ,σ)〉≡0∀t∈[0,T],j=1,n¯,(3.26)〈-∂y(1)∂t,νiqi(t,τ,σ)〉≡0,i=1,m¯,qi=0,si-1¯,
where y*∈Cn is a constant vector.We have the following result.Theorem 3.4.
Let conditions of Theorem3.1 hold. Then, the system (3.4) with additional conditions (3.24)-(3.25) has solutions of the form (3.11) in which all summands are uniquely determinate except for γiqi(t)ci(t)σiqi(i=1,m¯,qi=0,si-1¯). Functions γiqi(t) in the last summand are determined by the formula
(3.27)γiqi(t)=γiqi0⋅ePiqi(t)+fiqi(t),
where Piqi(t),fiqi(t) are known functions, and γiqi0arbitrary constants.Proof.
Denote in (3.11) that
(3.28)gkj(t)=hkj(t)λj(t)-λk(t),gkiqi(t)=hkiqi(t)λi(t)-λk(t).
Using (3.11) and condition (3.24), we obtain the equality
(3.29)∑k=1n∑j=1ngkj(0)cj(0)+∑k=1nαk(0)ck(0)=y*.
Multiplying this equality scalarly by ds(0), we get
(3.30)αs(0)=(y*,ds(0))-∑k=1,k≠sngks(0)≡αs0,s=1,n¯.
By (3.11) and conditions (3.25), we have
(3.31)-α̇s(t)-(ċs(t),ds(t))αs(t)-∑j=1,j≠sngsj(t)(ċj(t),ds(t))=0,s=1,n¯.
Considering these equations with initial conditions (3.30), we can uniquely obtain functions αs(t),s=1,n¯.
Now, using (3.11) and conditions (3.26), we get
(3.32)-γ̇iqi(t)-(ċi(t),di(t))γiqi(t)-∑k=1,k≠ingkiqi(t)(ċk(t),di(t))=0,i=1,m¯,qi=0,si-1¯.
This implies that γiqi(t) have the form (3.27) where
(3.33)Piqi(t)=-∫tit(ċi(s),di(s))ds,fiqi(t)=ePiqi(t)∫tite-Piqi(s)∑k=1,k≠ingkiqi(s)(ċk(s),di(s))ds.
Theorem 3.4 is proved.Remark 3.5.
If conditions (3.6) hold for h(1)(t,τ,σ)∈U(1)and h(0)(t)∈U(0), then system (3.3) has a solution in the space U, representable in the form of
(3.34)y(t,τ,σ)=y(1)(t,τ,σ)+y(0)(t),
where y(1)(t,τ,σ)is a function in the form of (3.11), and y(0)(t) is a function in the form of (3.23); moreover, functions αk(t)∈C∞([0,T],C1) are found uniquely in (3.11), and functions γiqi(t) are determined up to arbitrary constants γiqi0 in the form of (3.27).Let us give the following result.Theorem 3.6.
Leth(0)(t)∈U(0),h(1)(t,τ,σ)∈U(1),and conditions (i)–(iv), (3.6), (3.24)–(3.26) hold. Then, there exist unique numbers γiqi0 involved in (3.27), such that the function (3.34) satisfies the condition
(3.35)Py≡-∂y(0)∂t-∑i=1m∑qi=0si-1(t-ti)qiqi!∂y(1)∂σiqi+H(0)(t)∈V(0),
where H(0)(t)∈V(0)is a fixed vector function.Proof.
To determine functions uniquely, calculate(3.36)Py≡-∑i=1m[hi(t)ki(t)ci(t)]′-∑j=m+1n[hj(t)λj(t)cj(t)]′-∑i=1m∑qi=0si-1(t-ti)qiqi!γiqi(t)ci(t)+∑k=1n∑i=1i≠km∑qi=0si-1(t-ti)qiqi!⋅hiqi(t)λi(t)-λk(t)ck(t)+H(0)(t),(Py,di(t))≡-[hi(t)ki(t)]′-∑i=1mhi(t)ki(t)[ċi(t),di(t)]-∑j=m+1n[hj(t)kj(t)](ċj(t),di(t))-∑qi=0si-1∑i=1m(t-ti)qiqi!γiqi(t)+[H(0)(t),di(t)],i=1,m¯.
Denote by ri(t)the known function
(3.37)ri(t)≡-[hi(t)ki(t)]′-∑i=1mhi(t)ki(t)[ċi(t),di(t)]-∑j=m+1n[hj(t)kj(t)](ċj(t),di(t))+(H(0)(t),di(t)),
and write the conditions (3.13) for (Py,di(t)).Taking into account expression (3.27) for γiqi(t), we get
(3.38)∑qi=0si-1γiqi0[Dli((t-ti)qiqi!ePiqi(x))]t=ti+∑qi=0si-1[Dlifiqi(t)]t=ti=[Dliri(t)]t=ti,i=1,m,¯li=0,si-1¯.
Using the Leibnitz formula, we obtain that
(3.39)[Dli((t-ti)qiqi!ePiqi(t))]t=ti=[∑ν=0liCliν((t-ti)qiqi!)(ν)(ePiqi(t))(li-ν)]t=ti=[∑ν=0qiCliν((t-ti)qiqi!)(ν)(ePiqi(t))(li-ν)]t=ti=Cliqi(ePiqi(t))t=ti(li-qi),
for li≥qi,
(3.40)[Dli((t-ti)qiqi!ePiqi(t))]t=ti=0,
for 0≤li≤qi.
Therefore, previous equalities are written in the form of(3.41)∑qi=0si-1γiqi0Cliqi(ePiqi(t))t=ti(li-qi)=rili0(i=1,m¯,li=0,si-1¯),
where
(3.42)rili0=-∑qi=0si-1[Dlifiqi(t)]t=ti-[Dliri(t)]t=ti,forli=0,wegetγi00ePiqi(ti)=ri00;forli=1,wegetγi00c10[ePiqi(t)]t=ti′+γi10ePiqi(ti)=ri10;⋮forli=si-1,wegetγi00csi-10[ePiqi(t)]t=tisi-1+⋯+γisi-10ePiqi(ti)=risi-10.
We obtain from here sequentially the numbersγi00,…,γisi-10.Theorem 3.6 is proved.Thus, if conditions (3.24)–(3.26), (3.35) hold, all summands of solution (3.11) are defined uniquely.So, ifh(0)(t)∈U(0),h(1)(t,τ,σ)∈U(1),and conditions (3.6), (3.24)–(3.26), and (3.35) are valid, then the systems (3.4), (3.5) (and (3.3) together with them) are solvable uniquely in the class U=U(1)⊕U(0).Two sequential problems (εk) and (εk+1) are connected uniquely by conditions (3.23)–(3.25), (3.30); therefore, by virtue of Theorems 3.1–3.6, they are solvable uniquely in the space U.


## 4. Asymptotical Character of Formal Solutions

Lety-1(t,τ,σ),…,yk(t,τ,σ) be solutions of formal problems (ε-1), …,(εk) in the space U,respectively. Compose the partial sum for series (2.4):(4.1)Sn(t,τ,σ)=∑k=-1nεkyk(t,τ,σ),
and take its restriction yεn(t)=Sn(t,φ(t)/ε,ψ(t,ε)).We have the following result.Theorem 4.1.
Let conditions (i)–(v) hold. Then, for sufficiently smallε(0≤ε≤ε0), the estimates
(4.2)‖y(t,τ)-yεn(t)‖C[0,T]≤Cnεn+1,n=-1,0,1,…,
hold. Here, y(t,ε) is the exact solution of problem (1.1), and yεn(t) is the states above restriction of the nth partial sum of series (2.4).Proof.
The restrictionyεn(t) of series (2.4) satisfies the initial condition yεn(0)=y0 and system (1.1) up to terms containing εn+1, that is,
(4.3)εdyεn(t)dt=A(t)yεn(t)+εn+1Rn(t,ε)+h(t),
where Rn(t,s) is a known function satisfying the estimate
(4.4)‖R(t,ε)‖C[0,T]≤R̅n,R̅n—const.
Under conditions of Theorem 4.1 on the spectrum of the operator A(t) for the fundamental matrix Y(t,s,ε)≡Y(t,ε)Y-1(t,ε) of the system εẎ=A(t)Y, the estimate
(4.5)‖Y(t,s,ε)‖≤const∀(t,ε)∈Q≡{0≤s≤t≤T},∀ε>0∈[0,ε0],
is valid. Here, ε0>0-is sufficiently small. Now, write the equation
(4.6)εdΔ(t,ε)dt=A(t)Δ(t,ε)-εn+1Rn(t,ε),Δ(0,ε)=0,
for the remainder term Δ(t,ε)≡y(t,ε)-yεn(t).We obtain from this equation that
(4.7)Δ(t,ε)=-εn∫0tY(x,s,ε)Rn(s,ε)ds,
whence we get the estimate
(4.8)‖Δ(t,ε)‖C[0,T]≤-εnR̅n,
where R̅n=max(t,s)∈Q∥Y(t,s,ε)∥·∥R̅n(t,s)∥·T. So, we obtain the estimate
(4.9)‖y(t,ε)-yεn(t)‖C[0,T]≤εnR̅n,n=-1,0,1,….
Taking instead of yεn(t) the partial sum
(4.10)yε,n+1(t)≡yεn(t)+εn+1yn+1(t,φ(t)ε,ψ(t,ε)),
we get
(4.11)‖(y(t,ε)-yεn(t))-εn+1yn+1(t,φ(t)ε,ψ(t,ε))‖≤εn+1R̅n+1,
which implies the estimates (4.2). Theorem 4.1 is proved.


## 5. Example

Let it be required to construct the asymptotical solution for the Cauchy problem(5.1)ε(ẏż)=(-5t2+42t2-2-10t2+104t2-5)(yz)+(t2h1(t)0),y(0,ε)=y0,z(0,ε)=z0,
where h1(t)∈C∞[0,2],ε>0is a small parameter. Eigenvalues of the matrix A(t)=(-5t2+42t2-2-10t2+104t2-5)are λ1(t)=-t2,λ2(t)=-1.Eigenvectors of matrices A(t)and A*(t), are, respectively,(5.2)φ1=(12),φ2=(25),ψ1=(5-2),ψ2=(-21).
We get (h(t),ψ1(t))≡5t2h1(t). Therefore,(5.3)(h(0),ψ1(0))=0,ddt(h(0),ψ1(0))=0.Hence, we can apply to problem (5.1) the above developed algorithm of the regularization method.At first, obtain the basic Lagrange-Silvestre polynomialsKji(t).Since ψ(t)≡λ1(t)=-t2,there will be two such polynomials: K00(t)and K01(t).Take the arbitrary numbersa00(t)and a01(t),and set the interpolation conditions for the polynomial r(t),(5.4)r(t)=a00,ṙ(1)=a01.
Expand r(t)onto partial fractions(5.5)r(t)ψ(t)=At2+Bt,
from where(5.6)r(t)≡A+Bt.
Use the interpolation polynomial (5.4). We get A=a00,B=a01. Hence, (5.6) takes the form(5.7)r(t)≡a00+ta01.
Since numbers a00and a01 are arbitrary, basic Lagrange-Silvestre polynomials will be coefficients standing before them, that is,(5.8)K00(t)≡1,K01(t)≡t.Introduce the regularizing variables(5.9)σ00=e(1/ε)∫0tλ1ds∫0te-(1/ε)∫0sλ1dx⋅K00(s)ds=e-t3/3ε∫0tes3/3εds≡p00(t),σ01=e(1/ε)∫0tλ1ds∫0te-(1/ε)∫0sλ1dx⋅K01(s)ds=e-t3/3ε∫0tes3/3ε⋅sds≡p01(t),τ1=1ε∫0tλ1ds=-t33ε≡q1(t),τ2=1ε∫0tλ2ds=-tε≡q2(t).Construct the extended problem corresponding to problem (5.1):(5.10)ε∂w∂t+λ1(t)∂w∂τ1+λ2(t)∂w∂τ2+λ1(t)σ00∂w∂σ00+λ1(t)σ01∂w∂σ01+ε∂w∂σ00+εt∂w∂σ01-A(t)w=h(t),w(0,0,0,ε)=w0,
where τ≡(τ1,τ2),σ=(σ00,σ01),w=w(t,τ,σ,ε).Determining solutions of problem (5.10) in the form of a series(5.11)w(t,τ,σ,ε)=∑k=0∞εkwk(t,τ,σ),
we obtain the following iteration problems:
(5.12)Lw0≡λ1(t)[∂w0∂τ+∂w0∂σ00+t⋅σ01∂w0∂σ01]+λ2(t)∂w0∂τ2-A(t)w0=h(t),w0(0,0,0)=w0,(5.13)Lw1=-∂w0∂t-∂w0∂σ00-t∂w0∂σ01,w1(0,0,0)=0,⋮We determine solutions of iteration problems (5.12), (5.13), and so on in the space U of functions in the form of(5.14)w(t,τ,σ)=w1(t)eτ1+w2(t)eτ2+w00(t)σ00+w01(t)σ01+w0(t),w0(t),w1(t),w2(t),w00(t),w01(t)∈C∞([0,2],C2).Directly calculating, we obtain the solution of system (5.12) in the form of(5.15)w0(t,τ,σ)=α1(t)φ1eτ1+α2(t)φ2eτ2+γ00(t)φ1σ00+γ01(t)φ1σ01+5h1(t)φ1-2t2h1(t)φ2,
where αj(t),γji(t)∈C∞[0,2]are for now arbitrary functions.To calculate the functionsαj(t)and γij(t),we pass to the following iteration problem (5.13). Taking into account (5.15), it will be written in the form of(5.16)Lw1=-α̇1(t)φ1eτ1-α̇2(t)φ2eτ2-γ̇00(t)φ1σ00-γ̇01(t)φ1σ01-5ḣ1(t)φ1-(2t2h1(t))′φ2-γ00(t)φ1-tγ01(t)φ1.For solvability of problem (5.13) in the space U, it is necessary and sufficient to fulfill the conditions(5.17)-α̇1(t)=0,-α̇2(t)=0,-γ̇00(t)=0,-γ̇01(t)=0,-5ḣ1(0)-γ00(0)=0,-5ḧ1(0)-γ̇00(0)-γ01(0)=0.Using solution (5.15) and the initial condition w0(0,0,0)=w0,we obtain the equation(5.18)α1(0)φ1+α2(0)φ2+5h1(0)φ1=w0.Multiplying it (scalar) onψ1and ψ2, we obtain the values(5.19)α1(0)=(w0,ψ1)-5h1(0)≡5y0-2z0-5h1(0),α2(0)=(w0,ψ2)=z0-2y0.Using equalities (5.17), and also the initial data (5.19), we obtain uniquely the functions αj(t)and γji(t):(5.20)α1(t)=5y0-2z0-5h1(0),α2(t)=z0-2y0.γ00(t)=-5ḣ1(0),γ01(t)=-5ḧ1(0).Substituting these functions into (5.15), we obtain uniquely the solution of problem (5.12) in the space U,(5.21)w0(t,τ,σ)=(5y0-2z0-5h1(0))φ1eτ1+(z0-2y0)φ2eτ2-5ḣ1(0)φ1σ00-5ḧ1(0)φ1σ01+5h1(t)φ1-2t2h1(t)φ2.Producing here restriction on the functionsτ=q(t),σ=p(t),we obtain the principal term of the asymptotics for the solution of problem (5.1):(5.22)w0ε(t)=(5y0-2z0-5h1(0))φ1e-t3/3ε+(z0-2y0)φ2e-t/ε-5ḣ1(0)φ1e-t3/3ε∫0tes3/3εds-5ḧ1(0)φ1e-t3/3ε∫0tes3/3εsds+5h1(t)φ1-2t2h1(t)φ2.The zero-order asymptotical solution is obtained: it satisfies the estimate(5.23)‖w(t,ε)-w0ε(t)‖C[0,2]≤C1⋅ε,
where w(t,ε)is an exact solution of problem (1.1), and C1>0is a constant independent of εat sufficiently small ε(0<ε≤ε0).



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*Source: 120192-2012-07-24.xml*