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.
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 formεẏ=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 of λi(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
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:
λi(t)=-(t-ti)siki(t),ki(t)≠0,ti∈[0,T],i=1,m¯,m<n (here si- are even natural numbers),
λi(t)≠0,j=m+1,n¯,
λi(t)≠λj(t),i≠j,i,j=1,n¯,
Reλj(t)⩽0,j=1,n¯.
2. Regularization of the Problem
We introduce basic regularized variables by the spectrum of the limiting operatorτj=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 formulasσiqi=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 thatỹ(t,τ,σ,ε)∣τ=φ,σ=ψ≡y(t,ε).
For ỹ(t,τ,σ,ε), it is natural to set the following problem: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 seriesỹ(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):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,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,⋮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: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) whereU(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 formLy(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 formLy(1)(t,τ,σ)=h(1)(t,τ,σ),-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.
Let h(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
〈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
L*≡∑j=1nλj(t)∂∂τj+∑i=1m∑qi=0si-1λi(t)σiqi∂∂σiqi-A*(t).
Proof.
Let h(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
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
[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
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
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
z(0)(t)=∑j=1nzj(t)cj(t),zj(t)∈C∞([0,T],C1),j=1,n¯,
having the properties
[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,
(Dlihi)(ti)=0∀li=0,si-1¯,i=1,m¯.
Determine a solution of system (3.5) in the
y(0)(t)=∑j=1nyj(t)cj(t).
Substituting this function in (3.5), we obtain
-∑j=1nyj(t)λj(t)cj(t)=∑j=1nhj(t)cj(t).
Since {cj(t)} is a basis in Cn, we get
-λi(t)yi(t)=hi(t),i=1,m¯,-λj(t)yj(t)=hj(t),j=m+1,n¯.
It is easy to see that (3.18) has the unique solution
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
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
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
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
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 operator A(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 functions αj(t),γiqi(t), consider system (3.4) with additional conditions:y(1)(0,0,0)=y*,〈-∂y(1)∂t,νj(t,τ,σ)〉≡0∀t∈[0,T],j=1,n¯,〈-∂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 Theorem 3.1 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
γiqi(t)=γiqi0⋅ePiqi(t)+fiqi(t),
where Piqi(t),fiqi(t) are known functions, and γiqi0arbitrary constants.
Proof.
Denote in (3.11) that
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
∑k=1n∑j=1ngkj(0)cj(0)+∑k=1nαk(0)ck(0)=y*.
Multiplying this equality scalarly by ds(0), we get
αs(0)=(y*,ds(0))-∑k=1,k≠sngks(0)≡αs0,s=1,n¯.
By (3.11) and conditions (3.25), we have
-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
-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
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
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.
Let h(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
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
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
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
∑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
[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,
[Dli((t-ti)qiqi!ePiqi(t))]t=ti=0,
for 0≤li≤qi.
Therefore, previous equalities are written in the form of
∑qi=0si-1γiqi0Cliqi(ePiqi(t))t=ti(li-qi)=rili0(i=1,m¯,li=0,si-1¯),
where
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 numbers γi00,…,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, if h(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
Let y-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):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 small ε(0≤ε≤ε0), the estimates
‖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 restriction yεn(t) of series (2.4) satisfies the initial condition yεn(0)=y0 and system (1.1) up to terms containing εn+1, that is,
εdyεn(t)dt=A(t)yεn(t)+εn+1Rn(t,ε)+h(t),
where Rn(t,s) is a known function satisfying the estimate
‖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
‖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
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
Δ(t,ε)=-εn∫0tY(x,s,ε)Rn(s,ε)ds,
whence we get the estimate
‖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
‖y(t,ε)-yεn(t)‖C[0,T]≤εnR̅n,n=-1,0,1,….
Taking instead of yεn(t) the partial sum
yε,n+1(t)≡yεn(t)+εn+1yn+1(t,φ(t)ε,ψ(t,ε)),
we get
‖(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 problemε(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,φ1=(12),φ2=(25),ψ1=(5-2),ψ2=(-21).
We get (h(t),ψ1(t))≡5t2h1(t). Therefore,(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 polynomials Kji(t).Since ψ(t)≡λ1(t)=-t2,there will be two such polynomials: K00(t)and K01(t).
Take the arbitrary numbers a00(t)and a01(t),and set the interpolation conditions for the polynomial r(t),r(t)=a00,ṙ(1)=a01.
Expand r(t)onto partial fractionsr(t)ψ(t)=At2+Bt,
from wherer(t)≡A+Bt.
Use the interpolation polynomial (5.4). We get A=a00,B=a01. Hence, (5.6) takes the formr(t)≡a00+ta01.
Since numbers a00and a01 are arbitrary, basic Lagrange-Silvestre polynomials will be coefficients standing before them, that is,K00(t)≡1,K01(t)≡t.
Introduce the regularizing variablesσ00=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):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 seriesw(t,τ,σ,ε)=∑k=0∞εkwk(t,τ,σ),
we obtain the following iteration problems:
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,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 ofw(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 ofw0(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 functions αj(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 ofLw1=-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-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 equationα1(0)φ1+α2(0)φ2+5h1(0)φ1=w0.
Multiplying it (scalar) on ψ1and ψ2, we obtain the valuesα1(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):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,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 functions τ=q(t),σ=p(t),we obtain the principal term of the asymptotics for the solution of problem (5.1):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‖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).
LomovS. A.1992112Providence, RI, USAAmerican Mathematical Societyxviii+375Translations of Mathematical Monographs1195789ZBL0790.34060SkinnerL. A.Matched expansion solutions of the first-order turning point problem19942551402141110.1137/S00361410932399651289145ZBL0809.34071Vasil'evaA. B.On contrast structures of step type for a system of singularly perturbed equations1994341012151223Vasil'evaA. B.Contrast structures of step-like type for a second-order singularly perturbed quasilinear differential equation19953544114191332022Vasil'evaA. B.ButuzovV. F.KalachevL. V.199514Philadelphia, Pa, USASociety for Industrial and Applied Mathematicsxiv+2211316892EliseevA. G.LomovS. A.The theory of singular perturbations in the case of spectral singularities of a limit operator19861314544557881913AshyralyevA.On uniform difference schemes of a high order of approximation for evolution equations with a small parameter1989105-659360610.1080/016305689088163201002893ZBL0675.65055ShishkinG. I.MillerJ. J. H.O'RiordanE.1996River Edge, NJ, USAWorld Scientificxiv+1661439750AshyralyevA.FattoriniH. O.On uniform difference schemes for second-order singular perturbation problems in Banach spaces1992231295410.1137/05230031145161ZBL0745.65037AshyralyevA.SobolevskiiP. E.2004148Basel, SwitzerlandBirkhäuserx+443Operator Theory: Advances and Applications2075306AshyralyevA.SözenY.A note on the parabolic equation with an arbitrary parameter at the derivative20115411-122565257210.1016/j.mcm.2011.06.0282841801AshyralyevA.On uniform difference schemes of a higher order of approximation for elliptical equations with a small parameter1990363-421122010.1080/000368190088399331048961ZBL0663.65056AshyralyevA.FattoriniH. O.On difference schemes of the high order of accuracy for singular perturbation elliptic equations1991Ashgabat, Turkmenistan8083XuA.CenZ.Asymptotic behaviors of intermediate points in the remainder of the Euler-Maclaurin formula20102010813439210.1155/2010/1343922746010ZBL1207.65003De la SenM.Asymptotic comparison of the solutions of linear time-delay systems with point and distributed lags with those of their limiting equations200920093721674610.1155/2009/2167462506992ZBL1191.34096WangF.YukunA.Positive solutions for singular complementary Lid-stone boundary value problems201120111310.1155/2011/714728714728Vasile'vaA. B.KalachevL. V.Singularly perturbed periodic parabolic equations with alternating boundary layer type solutions200620102152856221165810.1155/AAA/2006/52856ŠamajováH.ŠpánikováE.On asymptotic behaviour of solutions to n-dimensional systems of neutral differential equations201120111979132310.1155/2011/7913232854933GavreaI.IvanM.Asymptotic behaviour of the iterates of positive linear operators201120111167050910.1155/2011/6705092802830ZBL1223.41015