# Solution of the First Boundary-Value Problem for a System of Autonomous Second-Order Linear Partial Differential Equations of Parabolic Type with a Single Delay

Authors: Josef Diblík; Denis Khusainov; Oleksandra Kukharenko; Zdeněk Svoboda
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The first boundary-value problem for an autonomous second-order system of linear partial differential equations of parabolic type with a single delay is considered. Assuming that a decomposition of the given system into a system of independent scalar second-order linear partial differential equations of parabolic type with a single delay is possible, an analytical
solution to the problem is given in the form of formal series and the character of their convergence is discussed. A delayed exponential function is used in order to analytically solve auxiliary initial problems (arising when Fourier method is applied) for ordinary linear differential equations of the first order with a single delay.


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## Body

## 1. Introduction

In this paper, we deal with an autonomous second-order system of linear partial differential equations of the parabolic type with a single delay(1.1)∂u(x,t)∂t=a11∂2u(x,t-τ)∂x2+a12∂2v(x,t-τ)∂x2+b11∂2u(x,t)∂x2+b12∂2v(x,t)∂x2,∂u(x,t)∂t=a21∂2u(x,t-τ)∂x2+a22∂2v(x,t-τ)∂x2+b21∂2u(x,t)∂x2+b22∂2v(x,t)∂x2,
where the matrices of coefficients
(1.2)A=(a11a12a21a22),B=(b11b12b21b22)
are constant and τ>0, τ=const.Usually, when systems of differential equations are investigated, the main attention is paid to systems of ordinary differential equations or systems of partial differential equations [1–5]. The analysis of systems of partial differential equations with delay is rather neglected. This investigation is extremely rare.The first boundary-value problem for (1.1) is solved for A having real eigenvalues λ1, λ2, and B having real eigenvalues σ1>0, σ2>0. Throughout the paper, we assume that there exists a real constant regular matrix
(1.3)S=(s11s12s21s22),
simultaneously reducing both matrices A and B into diagonal forms
(1.4)Λ=(λ100λ2),Σ=(σ100σ2),
that is,
(1.5)S-1AS=Λ,S-1BS=Σ,
where
(1.6)S-1=1Δ(s22-s12-s21s11),Δ=s11s22-s12s22.
For some classes of matrices, suitable transformations are known. Let us mention one of such results [6, Theorem 11′, page 291]. First, we recall that a complex square matrix 𝒜 is a normal matrix if 𝒜*𝒜=𝒜𝒜* where 𝒜* is the conjugate transpose of 𝒜. If 𝒜 is a real matrix, then 𝒜*=𝒜T, that is, the real matrix is normal if 𝒜T𝒜=𝒜𝒜T. A square matrix 𝒰 is called unitary if 𝒰𝒰*=E where E is the identity matrix.Theorem 1.1.
If a finite or infinite set of pairwise commuting normal matrices is given, then all these matrices can be carried by one and the same unitary transformation into a diagonal form.Letl be a positive constant, and let
(1.7)μi:[-τ,∞)→ℝ,i=1,2,θi:[-τ,∞)⟶R,i=1,2,φ:[0,l]×[-τ,0]→ℝ,ψ:[0,l]×[-τ,0]→ℝ
be continuously differentiable functions such that
(1.8)μ1(t)=φ(0,t),t∈[-τ,0],μ2(t)=φ(l,t),t∈[-τ,0],θ1(t)=ψ(0,t),t∈[-τ,0],θ2(t)=ψ(l,t),t∈[-τ,0].
Together with system (1.1), we consider the first boundary-value problem, that is, the boundary conditions
(1.9)u(0,t)=μ1(t),t∈[-τ,∞),(1.10)u(l,t)=μ2(t),t∈[-τ,∞),(1.11)v(0,t)=θ1(t),t∈[-τ,∞),(1.12)v(l,t)=θ2(t),t∈[-τ,∞)
and the initial conditions
(1.13)u(x,t)=φ(x,t),(x,t)∈[0,l]×[-τ,0],(1.14)v(x,t)=ψ(x,t),(x,t)∈[0,l]×[-τ,0].
A solution to the first boundary-value problem (1.1), (1.9)–(1.14) is defined as a pair of functions
(1.15)u,v:[0,l]×[-τ,∞)⟶R,
continuously differentiable with respect to variable t if (x,t)∈[0,l]×[0,∞), twice continuously differentiable with respect to x if (x,t)∈[0,l]×[0,∞), satisfying the system (1.1) for (x,t)∈[0,l]×[0,∞), the boundary conditions (1.9)–(1.12), and the initial conditions (1.13), (1.14). If necessary, we restrict the above definition of the solution to (x,t)∈[0,l]×[-τ,kτ] where k is a positive integer.The purpose of the paper is to describe a method of constructing a solution of the above boundary-initial problem. Assuming that a decomposition of system (1.1) into a system of independent scalar second-order linear partial differential equations of parabolic type with a single delay is possible, an analytical solution to the problem (1.9)–(1.14) is given in the form of formal series in part 3. Their uniform convergence as well as uniform convergence of the partial derivatives of a formal solution is discussed in part 4. A delayed exponential function (defined in part 2 together with the description of its main properties) is used in order to analytically solve auxiliary initial problems (arising when Fourier method is applied) for ordinary linear differential equations of the first-order with a single delay.To demonstrate this method, we will use systems of two equations only, although it can simply be extended to systems ofn equations.


## 2. Preliminaries—Representation of Solutions of Linear Differential Equations with a Single Delay

A solution of the systems (1.1) satisfying all boundary and initial conditions (1.9)–(1.14) will be constructed by the classical method of separation of variables (Fourier method). Nevertheless, due to delayed arguments, complications arise in solving analytically auxiliary initial Cauchy problems for first-order linear differential equations with a single delay. We overcome this circumstance by using a special function called a delayed exponential, which is a particular case of the delayed matrix exponential (as defined, e.g., in [7–10]). Here we give a definition of the delayed exponential, its basic properties needed, and a solution of the initial problem for first-order homogeneous and nonhomogeneous linear differential equations with a single delay.Definition 2.1.
Letb∈ℝ. The delayed exponential function expτ{b,t}:ℝ→ℝ is a function continuous on ℝ∖{-τ} defined as
(2.1)exp⁡τ{b,t}={0if-∞<t<-τ,1if-τ≤t<0,1+bt1!if0≤t<τ,⋯1+bt1!+b2(t-τ)22!+⋯+bk(t-(k-1)τ)kk!if(k-1)τ≤t<kτ,⋯
where k=0,1,2,….Lemma 2.2.
For the differentiation of a delayed exponential function, the formula(2.2)ddtexp⁡τ{b,t}=bexp⁡τ{b,t-τ}
holds within every interval (k-1)τ≤t<kτ, k=0,1,2,….Proof.
Within the intervals(k-1)τ≤t<kτ, k=0,1,2,…, the delayed exponential function is expressed as
(2.3)exp⁡τ{b,t}=1+bt1!+b2(t-τ)22!+b3(t-2τ)33!+⋯+bk(t-(k-1)τ)kk!.
Differentiating this expression, we obtain
(2.4)ddtexp⁡τ{b,t}=b+b2t-τ1!+b3(t-2τ)22!+⋯+bk(t-(k-1)τ)k-1(k-1)!=b[1+bt-τ1!+b2(t-2τ)22!+⋯+bk-1[t-(k-1)τ]k-1(k-1)!]=bexp⁡τ{b,t-τ}.


### 2.1.   First-Order Homogeneous Linear Differential Equations with a Single Delay

Let us consider a linear homogeneous equation with a single delay(2.5)x˙(t)=bx(t-τ),
where b∈ℝ, together with the initial Cauchy condition
(2.6)x(t)=β(t),t∈[-τ,0].From (2.2), it immediately follows that the delayed exponential expτ{b,t} is a solution of the initial Cauchy problems (2.5), (2.6) with β(t)≡1,  t∈[-τ,0].Theorem 2.3.
Letβ:[-τ,0]→ℝ be a continuously differentiable function. Then the unique solution of the initial Cauchy problems (2.5), (2.6) can be represented as
(2.7)x(t)=exp⁡τ{b,t}β(-τ)+∫-τ0exp⁡τ{b,t-τ-s}β′(s)ds,
where t∈[-τ,∞).Proof.
The representation (2.7) is a linear functional of the delayed exponential function expτ{b,t} and expτ{b,t-τ-s}. Because by Lemma 2.2 the delayed exponential function is the solution of (2.5), the functional on the right-hand side of (2.7) is a solution of the homogeneous equation (2.5) for arbitrary (differentiable) β(t).
We will show that initial condition (2.6) is satisfied as well, that is, we will verify that, for -τ≤t≤0, the next identity is correct:
(2.8)β(t)≡exp⁡τ{b,t}β(-τ)+∫-τ0‍exp⁡τ{b,t-τ-s}β′(s)ds.
We rewrite (2.7) as
(2.9)x(t)=exp⁡τ{b,t}β(-τ)+∫-τt‍exp⁡τ{b,t-τ-s}β′(s)ds+∫t0‍exp⁡τ{b,t-τ-s}β′(s)ds.
From Definition 2.1, it follows:
(2.10)expτ{b,t}≡1if-τ≤t≤0,exp⁡τ{b,t-τ-s}≡1if-τ≤s≤t,expτ{b,t-τ-s}≡0ift<s<0.
Therefore,(2.11)x(t)=β(-τ)+∫-τtβ′(s)ds=β(-τ)+β(t)-β(-τ)=β(t).Remark 2.4.
Computing the integral in formula (2.7) by parts, we obtain for t≥τ:
(2.12)x(t)=exp⁡τ{b,t-τ}β(0)+b∫-τ0exp⁡τ{b,t-2τ-s}β(s)ds.
We remark that it is possible to prove this formula assuming only continuity of the function β,that is, continuous differentiability of β is, in general, not necessary when we represent x by formula (2.12).Further we will consider the linear nonhomogeneous differential equation with a single delay(2.13)x˙(t)=ax(t)+bx(t-τ),
where a,b∈ℝ, together with initial Cauchy condition (2.6).Theorem 2.5.
Let the functionβ in (2.6) be continuously differentiable. Then the unique solution of the initial Cauchy problems (2.13), (2.6) can be represented as
(2.14)x(t)=exp⁡τ{b1,t}ea(t+τ)β(-τ)+∫-τ0exp⁡τ{b1,t-τ-s}ea(t-s)[β′(s)-aβ(s)]ds,
where b1=be-aτ and t∈[-τ,∞).Proof.
Transformingx by a substitution
(2.15)x(t)=eaty(t),
where y is a new unknown function, we obtain
(2.16)aeaty(t)+eaty˙(t)=aeaty(t)+bea(t-τ)y(t-τ)
or
(2.17)y˙(t)=b1y(t-τ).
Correspondingly, the initial condition for (2.17) is
(2.18)y(t)=e-atβ(t),t∈[-τ,0].
As follows, from formula (2.7), the solution of the corresponding initial Cauchy problems (2.17), (2.18) is
(2.19)y(t)=exp⁡τ{b1,t}eaτβ(-τ)+∫-τ0exp⁡τ{b1,t-τ-s}[e-asβ′(s)-ae-asβ(s)]ds.
Using substitution (2.15), we obtain
(2.20)x(t)=exp⁡τ{b1,t}ea(t+τ)β(-τ)+∫-τ0‍exp⁡τ{b1,t-τ-s}ea(t-s)[β′(s)-aβ(s)]ds,
which is formula (2.14).


### 2.2. First-Order Nonhomogeneous Linear Differential Equations with a Single Delay

Let a linear non-homogeneous delay equation with a single delay(2.21)x˙(t)=ax(t)+bx(t-τ)+f(t)
be given, where a,b∈ℝ and f:[0,∞)→ℝ. We consider the Cauchy problem with a zero initial condition
(2.22)x(t)=0,t∈[-τ,0],
that is, we put β≡0 in (2.6).Theorem 2.6.
The unique solution of the problem (2.21), (2.22) is given by the formula
(2.23)x(t)=∫0t‍exp⁡τ{b1,t-τ-s}ea(t-s)f(s)ds,
where b1=be-aτ.Proof.
We apply substitution (2.15). Then
(2.24)aeaty(t)+eaty˙(t)=aeaty(t)+bea(t-τ)y(t-τ)+f(t),
or,  equivalently,
(2.25)y˙(t)=b1y(t-τ)+e-atf(t).
We will show that the solution of the non-homogeneous equation (2.25) satisfying a zero initial condition (deduced from (2.15) and (2.22)) is
(2.26)y(t)=∫0t‍exp⁡τ{b1,t-τ-s}e-asf(s)ds.
Substituting (2.26) in (2.25), we obtain
(2.27)exp⁡τ{b1,t-τ-s}e-asf(s)|s=t+b1∫0t‍exp⁡τ{b1,t-2τ-s}e-asf(s)ds=b1∫0t-τ‍exp⁡τ{b1,t-2τ-s}e-asf(s)ds+e-atf(t).
Since
(2.28)exp⁡τ{b1,t-τ-s}e-asf(s)|s=t=exp⁡{b1,-τ}e-atf(t)=e-atf(t),
we obtain
(2.29)e-atf(t)+b1∫0t-τ‍exp⁡τ{b1,t-2τ-s}e-asf(s)ds+b1∫t-τt‍exp⁡τ{b1,t-2τ-s}e-asf(s)ds=b1∫0t-τ‍exp⁡τ{b1,t-2τ-s}e-asf(s)ds+e-atf(t).
Hence,
(2.30)∫t-τt‍exp⁡τ{b1,t-2τ-s}e-asf(s)ds=0.
This equality is true since
(2.31)t-2τ-s≤t-2τ-(t-τ)=-τ
and, by formula (2.1) in Definition 2.1,
(2.32)exp⁡τ{b1,t-2τ-s}≡0
if t-2τ-s<-τ. In accordance with (2.15), we get
(2.33)x(t)=eaty(t)=∫0t‍exp⁡τ{b1,t-τ-s}ea(t-s)f(s)ds,
that is, formula (2.23) is proved.Combining Theorems2.5, and 2.6 we get the following Corollary.Corollary 2.7.
Let the functionβ in (2.6) be continuously differentiable. Then the unique solution of the problems (2.21), (2.6) is given as
(2.34)x(t)=exp⁡τ{b1,t}ea(t+τ)β(-τ)+∫-τ0‍exp⁡τ{b1,t-τ-s}ea(t-s)[β′(s)-aβ(s)]ds+∫0t‍exp⁡τ{b1,t-τ-s}ea(t-s)f(s)ds,
where b1=be-aτ.


## 2.1.   First-Order Homogeneous Linear Differential Equations with a Single Delay

Let us consider a linear homogeneous equation with a single delay(2.5)x˙(t)=bx(t-τ),
where b∈ℝ, together with the initial Cauchy condition
(2.6)x(t)=β(t),t∈[-τ,0].From (2.2), it immediately follows that the delayed exponential expτ{b,t} is a solution of the initial Cauchy problems (2.5), (2.6) with β(t)≡1,  t∈[-τ,0].Theorem 2.3.
Letβ:[-τ,0]→ℝ be a continuously differentiable function. Then the unique solution of the initial Cauchy problems (2.5), (2.6) can be represented as
(2.7)x(t)=exp⁡τ{b,t}β(-τ)+∫-τ0exp⁡τ{b,t-τ-s}β′(s)ds,
where t∈[-τ,∞).Proof.
The representation (2.7) is a linear functional of the delayed exponential function expτ{b,t} and expτ{b,t-τ-s}. Because by Lemma 2.2 the delayed exponential function is the solution of (2.5), the functional on the right-hand side of (2.7) is a solution of the homogeneous equation (2.5) for arbitrary (differentiable) β(t).
We will show that initial condition (2.6) is satisfied as well, that is, we will verify that, for -τ≤t≤0, the next identity is correct:
(2.8)β(t)≡exp⁡τ{b,t}β(-τ)+∫-τ0‍exp⁡τ{b,t-τ-s}β′(s)ds.
We rewrite (2.7) as
(2.9)x(t)=exp⁡τ{b,t}β(-τ)+∫-τt‍exp⁡τ{b,t-τ-s}β′(s)ds+∫t0‍exp⁡τ{b,t-τ-s}β′(s)ds.
From Definition 2.1, it follows:
(2.10)expτ{b,t}≡1if-τ≤t≤0,exp⁡τ{b,t-τ-s}≡1if-τ≤s≤t,expτ{b,t-τ-s}≡0ift<s<0.
Therefore,(2.11)x(t)=β(-τ)+∫-τtβ′(s)ds=β(-τ)+β(t)-β(-τ)=β(t).Remark 2.4.
Computing the integral in formula (2.7) by parts, we obtain for t≥τ:
(2.12)x(t)=exp⁡τ{b,t-τ}β(0)+b∫-τ0exp⁡τ{b,t-2τ-s}β(s)ds.
We remark that it is possible to prove this formula assuming only continuity of the function β,that is, continuous differentiability of β is, in general, not necessary when we represent x by formula (2.12).Further we will consider the linear nonhomogeneous differential equation with a single delay(2.13)x˙(t)=ax(t)+bx(t-τ),
where a,b∈ℝ, together with initial Cauchy condition (2.6).Theorem 2.5.
Let the functionβ in (2.6) be continuously differentiable. Then the unique solution of the initial Cauchy problems (2.13), (2.6) can be represented as
(2.14)x(t)=exp⁡τ{b1,t}ea(t+τ)β(-τ)+∫-τ0exp⁡τ{b1,t-τ-s}ea(t-s)[β′(s)-aβ(s)]ds,
where b1=be-aτ and t∈[-τ,∞).Proof.
Transformingx by a substitution
(2.15)x(t)=eaty(t),
where y is a new unknown function, we obtain
(2.16)aeaty(t)+eaty˙(t)=aeaty(t)+bea(t-τ)y(t-τ)
or
(2.17)y˙(t)=b1y(t-τ).
Correspondingly, the initial condition for (2.17) is
(2.18)y(t)=e-atβ(t),t∈[-τ,0].
As follows, from formula (2.7), the solution of the corresponding initial Cauchy problems (2.17), (2.18) is
(2.19)y(t)=exp⁡τ{b1,t}eaτβ(-τ)+∫-τ0exp⁡τ{b1,t-τ-s}[e-asβ′(s)-ae-asβ(s)]ds.
Using substitution (2.15), we obtain
(2.20)x(t)=exp⁡τ{b1,t}ea(t+τ)β(-τ)+∫-τ0‍exp⁡τ{b1,t-τ-s}ea(t-s)[β′(s)-aβ(s)]ds,
which is formula (2.14).


## 2.2. First-Order Nonhomogeneous Linear Differential Equations with a Single Delay

Let a linear non-homogeneous delay equation with a single delay(2.21)x˙(t)=ax(t)+bx(t-τ)+f(t)
be given, where a,b∈ℝ and f:[0,∞)→ℝ. We consider the Cauchy problem with a zero initial condition
(2.22)x(t)=0,t∈[-τ,0],
that is, we put β≡0 in (2.6).Theorem 2.6.
The unique solution of the problem (2.21), (2.22) is given by the formula
(2.23)x(t)=∫0t‍exp⁡τ{b1,t-τ-s}ea(t-s)f(s)ds,
where b1=be-aτ.Proof.
We apply substitution (2.15). Then
(2.24)aeaty(t)+eaty˙(t)=aeaty(t)+bea(t-τ)y(t-τ)+f(t),
or,  equivalently,
(2.25)y˙(t)=b1y(t-τ)+e-atf(t).
We will show that the solution of the non-homogeneous equation (2.25) satisfying a zero initial condition (deduced from (2.15) and (2.22)) is
(2.26)y(t)=∫0t‍exp⁡τ{b1,t-τ-s}e-asf(s)ds.
Substituting (2.26) in (2.25), we obtain
(2.27)exp⁡τ{b1,t-τ-s}e-asf(s)|s=t+b1∫0t‍exp⁡τ{b1,t-2τ-s}e-asf(s)ds=b1∫0t-τ‍exp⁡τ{b1,t-2τ-s}e-asf(s)ds+e-atf(t).
Since
(2.28)exp⁡τ{b1,t-τ-s}e-asf(s)|s=t=exp⁡{b1,-τ}e-atf(t)=e-atf(t),
we obtain
(2.29)e-atf(t)+b1∫0t-τ‍exp⁡τ{b1,t-2τ-s}e-asf(s)ds+b1∫t-τt‍exp⁡τ{b1,t-2τ-s}e-asf(s)ds=b1∫0t-τ‍exp⁡τ{b1,t-2τ-s}e-asf(s)ds+e-atf(t).
Hence,
(2.30)∫t-τt‍exp⁡τ{b1,t-2τ-s}e-asf(s)ds=0.
This equality is true since
(2.31)t-2τ-s≤t-2τ-(t-τ)=-τ
and, by formula (2.1) in Definition 2.1,
(2.32)exp⁡τ{b1,t-2τ-s}≡0
if t-2τ-s<-τ. In accordance with (2.15), we get
(2.33)x(t)=eaty(t)=∫0t‍exp⁡τ{b1,t-τ-s}ea(t-s)f(s)ds,
that is, formula (2.23) is proved.Combining Theorems2.5, and 2.6 we get the following Corollary.Corollary 2.7.
Let the functionβ in (2.6) be continuously differentiable. Then the unique solution of the problems (2.21), (2.6) is given as
(2.34)x(t)=exp⁡τ{b1,t}ea(t+τ)β(-τ)+∫-τ0‍exp⁡τ{b1,t-τ-s}ea(t-s)[β′(s)-aβ(s)]ds+∫0t‍exp⁡τ{b1,t-τ-s}ea(t-s)f(s)ds,
where b1=be-aτ.


## 3.   Partial Differential Systems with Delay

Now we consider second-order autonomous systems of linear partial homogeneous differential equations of parabolic type with a single delay (1.1) where 0≤x≤l and t≥-τ. The initial conditions (1.13), (1.14) are defined for (x,t)∈[0,l]×[-τ,0]. Boundary conditions (1.9)–(1.12) are defined for t≥-τ and compatibility conditions (1.8) are fulfilled on the interval -τ≤t≤0.By the transformation(3.1)(u(x,t)v(x,t))=S(ξ(x,t)η(x,t)),
the systems (1.1) can be reduced to a form
(3.2)(∂ξ(x,t)∂t∂η(x,t)∂t)=Λ(∂2ξ(x,t-τ)∂x2∂2η(x,t-τ)∂x2)+Σ(∂2ξ(x,t)∂x2∂2η(x,t)∂x2),
that is, into two independent scalar equations
(3.3)∂ξ(x,t)∂t=λ1∂2ξ(x,t-τ)∂x2+σ1∂2ξ(x,t)∂x2,(3.4)∂η(x,t)∂t=λ2∂2η(x,t-τ)∂x2+σ2∂2η(x,t)∂x2.
Initial and boundary conditions reduce to
(3.5)ξ(0,t)=μ1*(t),t∈[-τ,∞),ξ(l,t)=μ2*(t),t∈[-τ,∞),(3.6)η(0,t)=θ1*(t),t∈[-τ,∞),η(l,t)=θ2*(t),t∈[-τ,∞),
and to
(3.7)ξ(x,t)=φ*(x,t),(x,t)∈[0,l]×[-τ,0],(3.8)η(x,t)=ψ*(x,t),(x,t)∈[0,l]×[-τ,0],
where
(3.9)(μ1*(t)θ1*(t))=S-1(μ1(t)θ1(t)),(μ2*(t)θ2*(t))=S-1(μ2(t)θ2(t)),(φ*(x,t)ψ*(x,t))=S-1(φ(x,t)ψ(x,t)).


### 3.1. Constructing of a Solution of (3.3)

We will consider  (3.3) with the boundary conditions  (3.5), (3.6) and the initial condition (3.7). We will construct a solution in the form
(3.10)ξ(x,t)=ξ0(x,t)+ξ1(x,t)+μ1*(t)+xl[μ2*(t)-μ1*(t)],
where (x,t)∈[0,l]×[-τ,∞), ξ0(x,t) is a solution of  (3.3) with zero boundary conditions
(3.11)ξ0(0,t)=0,ξ0(l,t)=0,t∈[-τ,∞)
and with a nonzero initial condition
(3.12)ξ0(x,t)=Φ(x,t):=φ*(x,t)-μ1*(t)-xl[μ2*(t)-μ1*(t)],(x,t)∈[0,l]×[-τ,0],
and ξ1(x,t) is a solution of a non-homogeneous equation
(3.13)∂ξ(x,t)∂t=λ1∂2ξ(x,t-τ)∂x2+σ1∂2ξ(x,t)∂x2+F(x,t),
where
(3.14)F(x,t):=-μ˙1*(t)-xl[μ˙2*(t)-μ˙1*(t)],
with zero boundary conditions
(3.15)ξ1(0,t)=0,ξ1(l,t)=0,t∈[-τ,∞)
and a zero initial condition
(3.16)ξ1(x,t)=0,(x,t)∈[0,l]×[-τ,0].


#### 3.1.1. Equation (3.3)—Solution of  the Problems (3.11), (3.12)

For finding a solutionξ=ξ0(x,t) of  (3.3), we will use the method of separation of variables. The solution ξ0(x,t) is seen as the product of two unknown functions X(x) and T(t), that is,
(3.17)ξ0(x,t)=X(x)T(t).
Substituting (3.17) into (3.3), we obtain
(3.18)X(x)T′(t)=λ1X′′(x)T(t-τ)+σ1X′′(x)T(t).
Separating variables, we have
(3.19)T′(t)λ1T(t-τ)+σ1T(t)=X′′(x)X(x)=-κ2,
where κ is a constant. We consider two differential equations
(3.20)T′(t)+σ1κ2T(t)+λ1κ2T(t-τ)=0,(3.21)X′′(x)+κ2X(x)=0.
Nonzero solutions of  (3.21) that satisfy zero boundary conditions
(3.22)X(0)=0,X(l)=0,
exist for the choice κ2=κn2=(πn/l)2, n=1,2,…, and are defined by the formulas
(3.23)X(x)=Xn(x)=Ansin⁡πnlx,n=1,2,…,
where An are arbitrary constants. Now we consider (3.20) with κ=κn:
(3.24)Tn′(t)=-σ1(πnl)2Tn(t)-λ1(πnl)2Tn(t-τ),n=1,2,….
Each of (3.24) represents a linear first-order delay differential equation with constant coefficients. We will specify initial conditions for each of (3.23), (3.24). To obtain such initial conditions, we expand the corresponding initial condition Φ(x,t) (see (3.12)) into Fourier series
(3.25)Φ(x,t)=∑n=1∞Φn(t)sin⁡πnlx,(x,t)∈[0,l]×[-τ,0],
where
(3.26)Φn(t)=2l∫0l‍Φ(s,t)sin⁡πnlsds=2l∫0l‍[φ*(s,t)-μ1*(t)-sl(μ2*(t)-μ1*(t))]sin⁡πnlsds=2l∫0l‍φ*(s,t)sin⁡πnlsds-2μ1*(t)l∫0l‍sin⁡πnlsds-2(μ2*(t)-μ1*(t))l2∫0l‍s⋅sin⁡πnlsds=2l∫0l‍φ*(s,t)sin⁡πnlsds+2πn[(-1)nμ2*(t)-μ1*(t)],t∈[-τ,0].
We will find an analytical solution of the problem (3.24) with initial function (3.26), that is, we will find an analytical solution of the Cauchy initial problem
(3.27)Tn′(t)=-σ1(πnl)2Tn(t)-λ1(πnl)2Tn(t-τ),Tn(t)=Φn(t),t∈[-τ,0],
for every n=1,2,…. Using the results of Part 2, we will solve the problem (3.27). According to formula (2.14), we get
(3.28)Tn(t)=exp⁡τ{r1n,t}e-σ1(πn/l)2(t+τ)Φn(-τ)+∫-τ0‍exp⁡τ{r1n,t-τ-s}e-σ1(πn/l)2(t-s)[Φn′(s)+σ1(πnl)2Φn(s)]ds,
where
(3.29)r1n=-λ1(πnl)2eσ1(πn/l)2τ,n=1,2,….
Thus, the solution ξ0(x,t) of the homogeneous equation (3.3) that satisfies zero boundary conditions (3.11) and a nonzero initial condition (3.12) (to satisfy (3.12) we set An=1, n=1,2,… in (3.23)) is
(3.30)ξ0(x,t)=∑n=1∞[∫-τ0exp⁡τ{r1n,t-τ-s}e-σ1(πnl)2(t-s)exp⁡τ{r1n,t}e-σ1(πn/l)2(t+τ)Φn(-τ)+∫-τ0exp⁡τ{r1n,t-τ-s}e-σ1(πn/l)2(t-s)[Φn′(s)+σ1(πnl)2Φn(s)]dse-σ1(πnl)2(t+τ)]sin⁡πnlx,
where Φn is defined by (3.26), r1n by (3.29), and (x,t)∈[0,l]×[-τ,∞).


#### 3.1.2. Nonhomogeneous Equation (3.13)

Further, we will consider the non-homogeneous equation (3.13) with zero boundary conditions (3.15) and a zero initial condition (3.16). We will try to find the solution in the form of an expansion
(3.31)ξ1(x,t)=∑n=1∞Tn0(t)sin⁡πnlx,
where (x,t)∈[0,l]×[-τ,∞) and Tn0:[-τ,∞)→ℝ are unknown functions. Substituting (3.31) into (3.13) and equating the coefficients of the same functional terms, we will obtain a system of equations:
(3.32)(Tn0)′(t)=-σ1(πnl)2Tn0(t)-λ1(πnl)2Tn0(t-τ)+fn(t),t∈[0,∞),n=1,2,…,
where fn:[-τ,∞)→ℝ are Fourier coefficients of the function F(x,t), that is,
(3.33)fn(t)=2l∫0l‍F(s,t)sin⁡πnlsds=-2l∫0l‍(μ˙1*(t)+sl[μ˙2*(t)-μ˙1*(t)])sin⁡πnlsds=-2lμ˙1*(t)∫0l‍sin⁡πnlsds-2l2(μ˙2*(t)-μ˙1*(t))∫0l‍s⋅sin⁡πnlsds=-2πn((-1)n+1μ˙2*(t)+μ˙1*(t)).
In accordance with (3.15), we assume zero initial conditions
(3.34)Tn0(t)=0,t∈[-τ,0],n=1,2,…
for every equation (3.32). Then, by formula (2.23) in Theorem 2.6, a solution of each of the problems (3.32), (3.34) can be written as
(3.35)Tn0(t)=∫0t‍exp⁡τ{r1n,t-τ-s}e-σ1(πn/l)2(t-s)fn(s)ds,t∈[-τ,∞),n=1,2,…,
where  r1n is defined by formula (3.29).Hence, the solution of the non-homogeneous equation (3.13) with zero boundary conditions and a zero initial condition is
(3.36)ξ1(x,t)=∑n=1∞‍[∫0t‍exp⁡τ{r1n,t-τ-s}e-σ1(πn/l)2(t-s)fn(s)ds]sin⁡πnlx,
where fn is given by formula (3.33).


### 3.2. Formal Solution of the Boundary Value Problem

Now we complete the particular results giving a solution of the boundary value problem of the initial system (1.1) satisfying conditions (1.9)–(1.14) in the form of a formal series. Conditions of their convergence will be discussed in the following Part 4.Since the solutionsξ0(x,t) and ξ1(x,t) of auxiliary problems are formally differentiable once with respect to t and twice with respect to x, and functions μ1*(t), μ2*(t) are once differentiable, we conclude that a formal solution of the first boundary value problem (3.5), (3.6) and (3.7) for (3.3) can be expressed by the formula:
(3.37)ξ(x,t)=∑n=1∞[exp⁡τ{r1n,t}e-σ1(πn/l)2(t+τ)Φn(-τ)+∫-τ0exp⁡τ{r1n,t-τ-s}e-σ1(πn/l)2(t-s)[Φn′(s)+σ1(πnl)2Φn(s)]ds+∫0texp⁡τ{r1n,t-τ-s}e-σ1(πn/l)2(t-s)fn(s)ds]sin⁡πnlx+μ1*(t)+xl[μ2*(t)-μ1*(t)],
where (x,t)∈[0,l]×[-τ,∞), coefficients Φn are defined by formulas (3.26), coefficients fn by formulas (3.33), and the numbers r1n by formula (3.29).Similarly, a formal solution of the first boundary value problem for (3.4) is given by the formula:
(3.38)η(x,t)=∑n=1∞[exp⁡τ{r2n,t}e-σ2(πn/l)2(t+τ)Ψn(-τ)+∫-τ0‍exp⁡τ{r2n,t-τ-s}e-σ2(πn/l)2(t-s)[Ψn′(s)+σ2(πnl)2Ψn(s)]ds+∫0t‍exp⁡τ{r2n,t-τ-s}e-σ2(πn/l)2(t-s)gn(s)ds]sin⁡πnlx+θ1*(t)+xl[θ2*(t)-θ1*(t)],
where (x,t)∈[0,l]×[-τ,∞) and (by analogy with (3.26), (3.33) and (3.29))
(3.39)Ψn(t)=2l∫0l‍ψ*(s,t)sinπnlsds+2πn[(-1)nθ2*(t)-θ1*(t)],t∈[-τ,0],gn(t)=-2πn((-1)n+1θ˙2*(t)+θ˙1*(t)),t∈[-τ,∞),r2n=-λ2(πnl)2eσ2(πn/l)2τ.Then, a formal solution of the boundary value problem of the initial system (1.1) satisfying conditions (1.9)–(1.14) is given by the formulas
(3.40)u(x,t)=s11ξ(x,t)+s12η(x,t),v(x,t)=s21ξ(x,t)+s22η(x,t),
where ξ(x,t), η(x,t) are defined by (3.37) and (3.38).


## 3.1. Constructing of a Solution of (3.3)

We will consider  (3.3) with the boundary conditions  (3.5), (3.6) and the initial condition (3.7). We will construct a solution in the form
(3.10)ξ(x,t)=ξ0(x,t)+ξ1(x,t)+μ1*(t)+xl[μ2*(t)-μ1*(t)],
where (x,t)∈[0,l]×[-τ,∞), ξ0(x,t) is a solution of  (3.3) with zero boundary conditions
(3.11)ξ0(0,t)=0,ξ0(l,t)=0,t∈[-τ,∞)
and with a nonzero initial condition
(3.12)ξ0(x,t)=Φ(x,t):=φ*(x,t)-μ1*(t)-xl[μ2*(t)-μ1*(t)],(x,t)∈[0,l]×[-τ,0],
and ξ1(x,t) is a solution of a non-homogeneous equation
(3.13)∂ξ(x,t)∂t=λ1∂2ξ(x,t-τ)∂x2+σ1∂2ξ(x,t)∂x2+F(x,t),
where
(3.14)F(x,t):=-μ˙1*(t)-xl[μ˙2*(t)-μ˙1*(t)],
with zero boundary conditions
(3.15)ξ1(0,t)=0,ξ1(l,t)=0,t∈[-τ,∞)
and a zero initial condition
(3.16)ξ1(x,t)=0,(x,t)∈[0,l]×[-τ,0].


### 3.1.1. Equation (3.3)—Solution of  the Problems (3.11), (3.12)

For finding a solutionξ=ξ0(x,t) of  (3.3), we will use the method of separation of variables. The solution ξ0(x,t) is seen as the product of two unknown functions X(x) and T(t), that is,
(3.17)ξ0(x,t)=X(x)T(t).
Substituting (3.17) into (3.3), we obtain
(3.18)X(x)T′(t)=λ1X′′(x)T(t-τ)+σ1X′′(x)T(t).
Separating variables, we have
(3.19)T′(t)λ1T(t-τ)+σ1T(t)=X′′(x)X(x)=-κ2,
where κ is a constant. We consider two differential equations
(3.20)T′(t)+σ1κ2T(t)+λ1κ2T(t-τ)=0,(3.21)X′′(x)+κ2X(x)=0.
Nonzero solutions of  (3.21) that satisfy zero boundary conditions
(3.22)X(0)=0,X(l)=0,
exist for the choice κ2=κn2=(πn/l)2, n=1,2,…, and are defined by the formulas
(3.23)X(x)=Xn(x)=Ansin⁡πnlx,n=1,2,…,
where An are arbitrary constants. Now we consider (3.20) with κ=κn:
(3.24)Tn′(t)=-σ1(πnl)2Tn(t)-λ1(πnl)2Tn(t-τ),n=1,2,….
Each of (3.24) represents a linear first-order delay differential equation with constant coefficients. We will specify initial conditions for each of (3.23), (3.24). To obtain such initial conditions, we expand the corresponding initial condition Φ(x,t) (see (3.12)) into Fourier series
(3.25)Φ(x,t)=∑n=1∞Φn(t)sin⁡πnlx,(x,t)∈[0,l]×[-τ,0],
where
(3.26)Φn(t)=2l∫0l‍Φ(s,t)sin⁡πnlsds=2l∫0l‍[φ*(s,t)-μ1*(t)-sl(μ2*(t)-μ1*(t))]sin⁡πnlsds=2l∫0l‍φ*(s,t)sin⁡πnlsds-2μ1*(t)l∫0l‍sin⁡πnlsds-2(μ2*(t)-μ1*(t))l2∫0l‍s⋅sin⁡πnlsds=2l∫0l‍φ*(s,t)sin⁡πnlsds+2πn[(-1)nμ2*(t)-μ1*(t)],t∈[-τ,0].
We will find an analytical solution of the problem (3.24) with initial function (3.26), that is, we will find an analytical solution of the Cauchy initial problem
(3.27)Tn′(t)=-σ1(πnl)2Tn(t)-λ1(πnl)2Tn(t-τ),Tn(t)=Φn(t),t∈[-τ,0],
for every n=1,2,…. Using the results of Part 2, we will solve the problem (3.27). According to formula (2.14), we get
(3.28)Tn(t)=exp⁡τ{r1n,t}e-σ1(πn/l)2(t+τ)Φn(-τ)+∫-τ0‍exp⁡τ{r1n,t-τ-s}e-σ1(πn/l)2(t-s)[Φn′(s)+σ1(πnl)2Φn(s)]ds,
where
(3.29)r1n=-λ1(πnl)2eσ1(πn/l)2τ,n=1,2,….
Thus, the solution ξ0(x,t) of the homogeneous equation (3.3) that satisfies zero boundary conditions (3.11) and a nonzero initial condition (3.12) (to satisfy (3.12) we set An=1, n=1,2,… in (3.23)) is
(3.30)ξ0(x,t)=∑n=1∞[∫-τ0exp⁡τ{r1n,t-τ-s}e-σ1(πnl)2(t-s)exp⁡τ{r1n,t}e-σ1(πn/l)2(t+τ)Φn(-τ)+∫-τ0exp⁡τ{r1n,t-τ-s}e-σ1(πn/l)2(t-s)[Φn′(s)+σ1(πnl)2Φn(s)]dse-σ1(πnl)2(t+τ)]sin⁡πnlx,
where Φn is defined by (3.26), r1n by (3.29), and (x,t)∈[0,l]×[-τ,∞).


### 3.1.2. Nonhomogeneous Equation (3.13)

Further, we will consider the non-homogeneous equation (3.13) with zero boundary conditions (3.15) and a zero initial condition (3.16). We will try to find the solution in the form of an expansion
(3.31)ξ1(x,t)=∑n=1∞Tn0(t)sin⁡πnlx,
where (x,t)∈[0,l]×[-τ,∞) and Tn0:[-τ,∞)→ℝ are unknown functions. Substituting (3.31) into (3.13) and equating the coefficients of the same functional terms, we will obtain a system of equations:
(3.32)(Tn0)′(t)=-σ1(πnl)2Tn0(t)-λ1(πnl)2Tn0(t-τ)+fn(t),t∈[0,∞),n=1,2,…,
where fn:[-τ,∞)→ℝ are Fourier coefficients of the function F(x,t), that is,
(3.33)fn(t)=2l∫0l‍F(s,t)sin⁡πnlsds=-2l∫0l‍(μ˙1*(t)+sl[μ˙2*(t)-μ˙1*(t)])sin⁡πnlsds=-2lμ˙1*(t)∫0l‍sin⁡πnlsds-2l2(μ˙2*(t)-μ˙1*(t))∫0l‍s⋅sin⁡πnlsds=-2πn((-1)n+1μ˙2*(t)+μ˙1*(t)).
In accordance with (3.15), we assume zero initial conditions
(3.34)Tn0(t)=0,t∈[-τ,0],n=1,2,…
for every equation (3.32). Then, by formula (2.23) in Theorem 2.6, a solution of each of the problems (3.32), (3.34) can be written as
(3.35)Tn0(t)=∫0t‍exp⁡τ{r1n,t-τ-s}e-σ1(πn/l)2(t-s)fn(s)ds,t∈[-τ,∞),n=1,2,…,
where  r1n is defined by formula (3.29).Hence, the solution of the non-homogeneous equation (3.13) with zero boundary conditions and a zero initial condition is
(3.36)ξ1(x,t)=∑n=1∞‍[∫0t‍exp⁡τ{r1n,t-τ-s}e-σ1(πn/l)2(t-s)fn(s)ds]sin⁡πnlx,
where fn is given by formula (3.33).


## 3.1.1. Equation (3.3)—Solution of  the Problems (3.11), (3.12)

For finding a solutionξ=ξ0(x,t) of  (3.3), we will use the method of separation of variables. The solution ξ0(x,t) is seen as the product of two unknown functions X(x) and T(t), that is,
(3.17)ξ0(x,t)=X(x)T(t).
Substituting (3.17) into (3.3), we obtain
(3.18)X(x)T′(t)=λ1X′′(x)T(t-τ)+σ1X′′(x)T(t).
Separating variables, we have
(3.19)T′(t)λ1T(t-τ)+σ1T(t)=X′′(x)X(x)=-κ2,
where κ is a constant. We consider two differential equations
(3.20)T′(t)+σ1κ2T(t)+λ1κ2T(t-τ)=0,(3.21)X′′(x)+κ2X(x)=0.
Nonzero solutions of  (3.21) that satisfy zero boundary conditions
(3.22)X(0)=0,X(l)=0,
exist for the choice κ2=κn2=(πn/l)2, n=1,2,…, and are defined by the formulas
(3.23)X(x)=Xn(x)=Ansin⁡πnlx,n=1,2,…,
where An are arbitrary constants. Now we consider (3.20) with κ=κn:
(3.24)Tn′(t)=-σ1(πnl)2Tn(t)-λ1(πnl)2Tn(t-τ),n=1,2,….
Each of (3.24) represents a linear first-order delay differential equation with constant coefficients. We will specify initial conditions for each of (3.23), (3.24). To obtain such initial conditions, we expand the corresponding initial condition Φ(x,t) (see (3.12)) into Fourier series
(3.25)Φ(x,t)=∑n=1∞Φn(t)sin⁡πnlx,(x,t)∈[0,l]×[-τ,0],
where
(3.26)Φn(t)=2l∫0l‍Φ(s,t)sin⁡πnlsds=2l∫0l‍[φ*(s,t)-μ1*(t)-sl(μ2*(t)-μ1*(t))]sin⁡πnlsds=2l∫0l‍φ*(s,t)sin⁡πnlsds-2μ1*(t)l∫0l‍sin⁡πnlsds-2(μ2*(t)-μ1*(t))l2∫0l‍s⋅sin⁡πnlsds=2l∫0l‍φ*(s,t)sin⁡πnlsds+2πn[(-1)nμ2*(t)-μ1*(t)],t∈[-τ,0].
We will find an analytical solution of the problem (3.24) with initial function (3.26), that is, we will find an analytical solution of the Cauchy initial problem
(3.27)Tn′(t)=-σ1(πnl)2Tn(t)-λ1(πnl)2Tn(t-τ),Tn(t)=Φn(t),t∈[-τ,0],
for every n=1,2,…. Using the results of Part 2, we will solve the problem (3.27). According to formula (2.14), we get
(3.28)Tn(t)=exp⁡τ{r1n,t}e-σ1(πn/l)2(t+τ)Φn(-τ)+∫-τ0‍exp⁡τ{r1n,t-τ-s}e-σ1(πn/l)2(t-s)[Φn′(s)+σ1(πnl)2Φn(s)]ds,
where
(3.29)r1n=-λ1(πnl)2eσ1(πn/l)2τ,n=1,2,….
Thus, the solution ξ0(x,t) of the homogeneous equation (3.3) that satisfies zero boundary conditions (3.11) and a nonzero initial condition (3.12) (to satisfy (3.12) we set An=1, n=1,2,… in (3.23)) is
(3.30)ξ0(x,t)=∑n=1∞[∫-τ0exp⁡τ{r1n,t-τ-s}e-σ1(πnl)2(t-s)exp⁡τ{r1n,t}e-σ1(πn/l)2(t+τ)Φn(-τ)+∫-τ0exp⁡τ{r1n,t-τ-s}e-σ1(πn/l)2(t-s)[Φn′(s)+σ1(πnl)2Φn(s)]dse-σ1(πnl)2(t+τ)]sin⁡πnlx,
where Φn is defined by (3.26), r1n by (3.29), and (x,t)∈[0,l]×[-τ,∞).


## 3.1.2. Nonhomogeneous Equation (3.13)

Further, we will consider the non-homogeneous equation (3.13) with zero boundary conditions (3.15) and a zero initial condition (3.16). We will try to find the solution in the form of an expansion
(3.31)ξ1(x,t)=∑n=1∞Tn0(t)sin⁡πnlx,
where (x,t)∈[0,l]×[-τ,∞) and Tn0:[-τ,∞)→ℝ are unknown functions. Substituting (3.31) into (3.13) and equating the coefficients of the same functional terms, we will obtain a system of equations:
(3.32)(Tn0)′(t)=-σ1(πnl)2Tn0(t)-λ1(πnl)2Tn0(t-τ)+fn(t),t∈[0,∞),n=1,2,…,
where fn:[-τ,∞)→ℝ are Fourier coefficients of the function F(x,t), that is,
(3.33)fn(t)=2l∫0l‍F(s,t)sin⁡πnlsds=-2l∫0l‍(μ˙1*(t)+sl[μ˙2*(t)-μ˙1*(t)])sin⁡πnlsds=-2lμ˙1*(t)∫0l‍sin⁡πnlsds-2l2(μ˙2*(t)-μ˙1*(t))∫0l‍s⋅sin⁡πnlsds=-2πn((-1)n+1μ˙2*(t)+μ˙1*(t)).
In accordance with (3.15), we assume zero initial conditions
(3.34)Tn0(t)=0,t∈[-τ,0],n=1,2,…
for every equation (3.32). Then, by formula (2.23) in Theorem 2.6, a solution of each of the problems (3.32), (3.34) can be written as
(3.35)Tn0(t)=∫0t‍exp⁡τ{r1n,t-τ-s}e-σ1(πn/l)2(t-s)fn(s)ds,t∈[-τ,∞),n=1,2,…,
where  r1n is defined by formula (3.29).Hence, the solution of the non-homogeneous equation (3.13) with zero boundary conditions and a zero initial condition is
(3.36)ξ1(x,t)=∑n=1∞‍[∫0t‍exp⁡τ{r1n,t-τ-s}e-σ1(πn/l)2(t-s)fn(s)ds]sin⁡πnlx,
where fn is given by formula (3.33).


## 3.2. Formal Solution of the Boundary Value Problem

Now we complete the particular results giving a solution of the boundary value problem of the initial system (1.1) satisfying conditions (1.9)–(1.14) in the form of a formal series. Conditions of their convergence will be discussed in the following Part 4.Since the solutionsξ0(x,t) and ξ1(x,t) of auxiliary problems are formally differentiable once with respect to t and twice with respect to x, and functions μ1*(t), μ2*(t) are once differentiable, we conclude that a formal solution of the first boundary value problem (3.5), (3.6) and (3.7) for (3.3) can be expressed by the formula:
(3.37)ξ(x,t)=∑n=1∞[exp⁡τ{r1n,t}e-σ1(πn/l)2(t+τ)Φn(-τ)+∫-τ0exp⁡τ{r1n,t-τ-s}e-σ1(πn/l)2(t-s)[Φn′(s)+σ1(πnl)2Φn(s)]ds+∫0texp⁡τ{r1n,t-τ-s}e-σ1(πn/l)2(t-s)fn(s)ds]sin⁡πnlx+μ1*(t)+xl[μ2*(t)-μ1*(t)],
where (x,t)∈[0,l]×[-τ,∞), coefficients Φn are defined by formulas (3.26), coefficients fn by formulas (3.33), and the numbers r1n by formula (3.29).Similarly, a formal solution of the first boundary value problem for (3.4) is given by the formula:
(3.38)η(x,t)=∑n=1∞[exp⁡τ{r2n,t}e-σ2(πn/l)2(t+τ)Ψn(-τ)+∫-τ0‍exp⁡τ{r2n,t-τ-s}e-σ2(πn/l)2(t-s)[Ψn′(s)+σ2(πnl)2Ψn(s)]ds+∫0t‍exp⁡τ{r2n,t-τ-s}e-σ2(πn/l)2(t-s)gn(s)ds]sin⁡πnlx+θ1*(t)+xl[θ2*(t)-θ1*(t)],
where (x,t)∈[0,l]×[-τ,∞) and (by analogy with (3.26), (3.33) and (3.29))
(3.39)Ψn(t)=2l∫0l‍ψ*(s,t)sinπnlsds+2πn[(-1)nθ2*(t)-θ1*(t)],t∈[-τ,0],gn(t)=-2πn((-1)n+1θ˙2*(t)+θ˙1*(t)),t∈[-τ,∞),r2n=-λ2(πnl)2eσ2(πn/l)2τ.Then, a formal solution of the boundary value problem of the initial system (1.1) satisfying conditions (1.9)–(1.14) is given by the formulas
(3.40)u(x,t)=s11ξ(x,t)+s12η(x,t),v(x,t)=s21ξ(x,t)+s22η(x,t),
where ξ(x,t), η(x,t) are defined by (3.37) and (3.38).


## 4. Convergence of Formal Series

A solution of the first boundary value problem for (3.3), (3.4) is presented in the form of formal series (3.37), (3.38). We will show that, when certain conditions are satisfied, the series (together with its relevant partial derivatives) converges for (x,t)∈[0,l]×[-τ,t*] where t*>0 is arbitrarily large and, consequently, is a solution of partial delay differential equations (3.3), (3.4).Theorem 4.1.
Let, for the functions(4.1)Φn:[-τ,0]⟶R,fn:[-τ,∞)⟶R,n=1,2,…
defined by (3.26) and (3.33), for an integer k≥1 and arbitrary t*∈[(k-1)τ,kτ], there exist constants M≥0, α>0 such that
(4.2)e-σ1(πn/l)2(t*-(k-1)τ)n2kmax⁡-τ≤t≤kτ|fn(t)|≤Mn1+α,e-σ1(πn/l)2(t*-(k-1)τ)n2kmax⁡-τ≤t≤0|Φn(t)|≤Mn3+α.
Then, for (x,t)∈[0,l]×[0,kτ], the formal series on the right-hand side of expression (3.37) as well as its first derivative with respect to t and its second derivative with respect to x converge uniformly. Moreover, equality (3.37) holds, and the function ξ(x,t) is a solution of (3.3) for (x,t)∈[0,l]×[-τ,kτ].Proof.
First we prove that the right-hand side of expression (3.37) uniformly converges. Decompose the function ξ(x,t) as
(4.3)ξ(x,t)=S1(x,t)+S2(x,t)+S3(x,t)+μ1*(t)+xl[μ2*(t)-μ1*(t)],
where
(4.4)S1(x,t)=∑n=1∞‍An(t)sinπnlx,An(t):=exp⁡τ{r1n,t}e-σ1(πn/l)2(t+τ)Φn(-τ),S2(x,t)=∑n=1∞‍Bn(t)sinπnlx,Bn(t):=∫-τ0‍expτ{r1n,t-τ-s}e-σ1(πn/l)2(t-s)[Φn′(s)+σ1(πnl)2Φn(s)]ds,S3(x,t)=∑n=1∞‍Cn(t)sinπnlx,Cn(t):=∫0t‍expτ{r1n,t-τ-s}e-σ1(πn/l)2(t-s)fn(s)ds.
In the following parts, we will prove the uniform convergence of each of the series Si(x,t), i=1,2,3 separately.
Throughout the proof, we use the delayed exponential function defined by Definition2.1, formula (2.1). Note that this function is continuous on ℝ∖{-τ} and at knots t=kτ where k=0,1,… two lines in formula (2.1) can be applied. We use this property in the proof without any special comment.Uniform Convergence of the SeriesS1(x,t)
We consider the coefficientsAn(t), n=1,2,… of the first series S1(x,t). As follows from Definition 2.1 of the delayed exponential function, the following equality holds:
(4.5)An(t*)=exp⁡τ⁡{r1n,t*}e-σ1(πn/l)2(t*+τ)Φn(-τ)=e-σ1(πn/l)2(t*+τ)Φn(-τ)[1+r1nt*1!+r1n2(t*-τ)22!+⋯+r1nk(t*-(k-1)τ)kk!]=e-σ1(πn/l)2(t*+τ)Φn(-τ)[1-λ1(πnl)2eσ1(πn/l)2τt*1!+λ12(πnl)4e2σ1(πn/l)2τ(t*-τ)22!+⋯+(-λ1)k(πnl)2kekσ1(πn/l)2τ(t*-(k-1)τ)kk!].
Therefore,
(4.6)S1(x,t*)=∑n=1∞An(t*)sin⁡πnlx=∑n=1∞exp⁡τ{r1n,t*}e-σ1(πn/l)2(t*+τ)Φn(-τ)sin⁡πnlx=∑n=1∞e-σ1(πn/l)2(t*+τ)Φn(-τ)sin⁡πnlx-λ1t*1!∑n=1∞e-σ1(πn/l)2t*(πnl)2Φn(-τ)sin⁡πnlx+λ12(t*-τ)22!∑n=1∞e-σ1(πn/l)2(t*-τ)(πnl)4Φn(-τ)sin⁡πnlx+⋯+(-1)kλ1k(t*-(k-1)τ)kk!∑n=1∞e-σ1(πn/l)2(t*-(k-1)τ)(πnl)2kΦn(-τ)sin⁡πnlx.
Due to condition (4.2), we conclude that
(4.7)e-σ1(πn/l)2(t*-(k-1)τ)n2k|Φn(-τ)|≤Mn3+α.
Therefore, the series S1(x,t*) converges uniformly with respect to x∈[0,l] and t*∈[(k-1)τ,kτ]. If t*∈[(k*-1)τ,k*τ] where k*∈{1,2,…,k-1}, then the estimations remain valid. Note that inequalities (4.2) are also valid for k=k* because
(4.8)max-τ≤t≤k*τ|fn(t)|≤Mn2k+1+α≤Mn2k*+1+α,e-σ1(πn/l)2(t*-(k*-1)τ)n2kmax⁡-τ≤t≤0|Φn(t)|≤Mn3+α.
Consequently, it is easy to see that the series S1(x,t) converges uniformly for x∈[0,l] and t∈[0,kτ].Uniform Convergence of the SeriesS2(x,t)
We consider the coefficientsBn(t), n=1,2,… of the second series S2(x,t). In the representation
(4.9)Bn(t)=σ1(πnl)2∫-τ0‍exp⁡τ{r1n,t-τ-s}e-σ1(πn/l)2(t-s)Φn(s)ds+∫-τ0‍exp⁡τ{r1n,t-τ-s}e-σ1(πn/l)2(t-s)Φn′(s)ds,
we calculate the second integral by parts and use formula (2.2) in Lemma 2.2:
(4.10)Bn(t)=σ1(πnl)2∫-τ0‍exp⁡τ{r1n,t-τ-s}e-σ1(πn/l)2(t-s)Φn(s)ds+exp⁡τ{r1n,t-τ}e-σ1(πn/l)2tΦn(0)-exp⁡τ{r1n,t}e-σ1(πn/l)2(t+τ)Φn(-τ)+∫-τ0‍λ1(πnl)2exp⁡τ{r1n,t-2τ-s}e-σ1(πn/l)2(t-s-τ)Φn(s)ds-σ1(πnl)2∫-τ0‍exp⁡τ{r1n,t-τ-s}e-σ1(πn/l)2(t-s)Φn(s)ds=exp⁡τ{r1n,t-τ}e-σ1(πn/l)2tΦn(0)-exp⁡τ{r1n,t}e-σ1(πn/l)2(t+τ)Φn(-τ)+∫-τ0‍λ1(πnl)2exp⁡τ{r1n,t-2τ-s}e-σ1(πn/l)2(t-s-τ)Φn(s)ds=Bn1(t)-Bn2(t)+Bn3(t),
where
(4.11)Bn1(t)=expτ{r1n,t-τ}e-σ1(πn/l)2tΦn(0),Bn2(t)=exp⁡τ{r1n,t}e-σ1(πn/l)2(t+τ)Φn(-τ),Bn3(t)=∫-τ0λ1(πnl)2expτ{r1n,t-2τ-s}e-σ1(πn/l)2(t-s-τ)Φn(s)ds.
As follows from the Definition 2.1 of the delayed exponential function, for t*∈[(k-1)τ,kτ], the following equality holds:
(4.12)Bn1(t*)=exp⁡τ{r1n,t*-τ}e-σ1(πn/l)2t*Φn(0)=e-σ1(πn/l)2t*Φn(0)×[1-λ1(πnl)2eσ1(πn/l)2τt*-τ1!+λ12(πnl)4e2σ1(πn/l)2τ(t*-2τ)22!+⋯+(-1)k-1λ1k-1(πnl)2(k-1)e(k-1)σ1(πn/l)2τ(t*-(k-1)τ)k-1(k-1)!(πnl)2].
Therefore, for
(4.13)S21(x,t*):=∑n=1∞Bn1(t*)sin⁡πnlx,
we get
(4.14)S21(x,t*)=∑n=1∞‍Bn1(t*)sin⁡πnlx=∑n=1∞‍exp⁡τ{r1n,t*-τ}e-σ1(πn/l)2t*Φn(0)sin⁡πnlx=∑n=1∞‍e-σ1(πn/l)2t*Φn(0)sin⁡πnlx-λ1t*-τ1!∑n=1∞‍(πnl)2e-σ1(πn/l)2(t*-τ)Φn(0)sin⁡πnlx+λ12(t*-2τ)22!∑n=1∞(πnl)4e-σ1(πn/l)2(t*-2τ)Φn(0)sin⁡πnlx+⋯+(-1)k-1λ1k-1(t*-(k-1)τ)k-1(k-1)!∑n=1∞(πnl)2(k-1)e-σ1(πn/l)2(t*-(k-1)τ)Φn(0)sin⁡πnlx.
Due to condition (4.2), we conclude that
(4.15)e-σ1(πn/l)2(t*-(k-1)τ)n2k|Φn(0)|≤Mn3+α.
The proof of the uniform convergence of the series S21(x,t) for x∈[0,l] and t∈[0,kτ] can now be performed in a way similar to the proof of the uniform convergence of the series S1(x,t) for x∈[0,l] and t∈[0,kτ].
For the coefficientsBn2(t), the following holds:
(4.16)Bn2(t*)=exp⁡τ{r1n,t*}e-σ1(πn/l)2(t*+τ)Φn(-τ)=e-σ1(πn/l)2(t*+τ)Φn(-τ)×[1-λ1(πnl)2eσ1(πn/l)2τt*1!+λ12(πnl)4e2σ1(πn/l)2τ(t*-τ)22!+⋯+(-1)kλ1k(πnl)2kekσ1(πn/l)2τ(t*-(k-1)τ)kk!].
Therefore, for
(4.17)S22(x,t*):=∑n=1∞‍Bn2(t*)sin⁡πnlx,
we get
(4.18)S22(x,t*)=∑n=1∞‍Bn2(t*)sin⁡πnlx=∑n=1∞‍exp⁡τ{r1n,t*}e-σ1(πn/l)2(t*+τ)Φn(-τ)sin⁡πnlx=∑n=1∞‍e-σ1(πn/l)2(t*+τ)Φn(-τ)sin⁡πnlx-λ1t*1!∑n=1∞‍(πnl)2e-σ1(πn/l)2t*Φn(-τ)sin⁡πnlx+λ12(t*-τ)22!∑n=1∞‍(πnl)4e-σ1(πn/l)2(t*-τ)Φn(-τ)sin⁡πnlx+⋯+λ1k(t*-(k-1)τ)kk!∑n=1∞‍(πnl)2ke-σ1(πn/l)2(t*-(k-1)τ)Φn(-τ)sin⁡πnlx.
Due to (4.7), the proof of the uniform convergence of the series S22(x,t) for x∈[0,l] and t∈[0,kτ] can now be performed in a way similar to the proof of the uniform convergence of the series S1(x,t) for x∈[0,l] and t∈[0,kτ].
Finally, we consider the coefficientsBn3(t) at t=t*∈[(k-1)τ,kτ]. Substituting t*-2τ-s=ω, we obtain the following:
(4.19)Bn3(t*)=∫-τ0‍(λ1(πnl)2eσ1(πn/l)2τexp⁡τ{r1n,t*-2τ-s})e-σ1(πn/l)2(t*-s)Φn(s)ds=λ1(πnl)2eσ1(πn/l)2τ∫t*-2τt*-τ‍exp⁡τ{r1n,ω}e-σ1(πn/l)2(ω+2τ)Φn(t*-2τ-ω)dω.
We will split this integral in two:
(4.20)Bn3(t*)=λ1(πnl)2eσ1(πn/l)2τ∫t*-2τ(k-2)τexp⁡τ{r1n,ω}e-σ1(πn/l)2(ω+2τ)Φn(t*-2τ-ω)dω+λ1(πnl)2eσ1(πn/l)2τ∫(k-2)τt*-τ‍exp⁡τ{r1n,ω}e-σ1(πn/l)2(ω+2τ)Φn(t*-2τ-ω)dω.
Therefore, owing to the mean value theorem, there are values ω1 and ω2 such that
(4.21)t*-2τ≤ω1≤(k-2)τ,(k-2)τ≤ω2≤t*-τ,
and (using the Definition 2.1 of the delayed exponential function) we have
(4.22)Bn3(t*)=λ1(πnl)2e-σ1(πn/l)2(ω1+τ)Φn(t*-2τ-ω1)(kτ-t*)exp⁡τ{r1n,ω1}+λ1(πnl)2e-σ1(πn/l)2(ω2+τ)Φn(t*-2τ-ω2)(t*-(k-1)τ)exp⁡τ{r1n,ω2}=λ1(πnl)2e-σ1(πn/l)2(ω1+τ)Φn(t*-2τ-ω1)(kτ-t*)×[1-λ1(πnl)2eσ1(πn/l)2τω11!+λ12(πnl)4e2σ1(πn/l)2τ(ω1-τ)22!+⋯+(-1)k-2λ1k-2(πnl)2(k-2)e(k-2)σ1(πn/l)2τ(ω1-(k-3)τ)k-2(k-2)!]+λ1(πnl)2e-σ1(πn/l)2(ω2+τ)Φn(t*-2τ-ω2)(t*-(k-1)τ)×[1-λ1(πnl)2eσ1(πn/l)2τω21!+λ12(πnl)4e2σ1(πn/l)2τ(ω2-τ)22!+⋯+(-1)k-1λ1k-1(πnl)2(k-1)e(k-1)σ1(πn/l)2τ(ω2-(k-2)τ)k-1(k-1)!].
Hence, for
(4.23)S23(x,t*):=∑n=1∞Bn3(t*)sin⁡πnlx,
we get
(4.24)S23(x,t*)=∑n=1∞‍Bn3(t*)sin⁡πnlx=∑n=1∞‍{(ω2-(k-2)τ)k-1(k-1)!λ1(πnl)2e-σ1(πn/l)2(ω1+τ)Φn×(t*-2τ-ω1)(kτ-t*)[e(k-2)σ1(πnl)2τ(ω1-(k-3)τ)k-2(k-2)!1-λ1(πnl)2eσ1(πn/l)2τω11!+λ12(πnl)4e2σ1(πn/l)2τ×(ω1-τ)22!+⋯+(-1)k-2λ1k-2(πnl)2(k-2)×e(k-2)σ1(πn/l)2τ(ω1-(k-3)τ)k-2(k-2)!]+λ1(πnl)2e-σ1(πn/l)2(ω2+τ)Φn(t*-2τ-ω2)(t*-(k-1)τ)×[1-λ1(πnl)2eσ1(πn/l)2τω21!+λ12(πnl)4e2σ1(πn/l)2τ(ω2-τ)22!+⋯+(-1)k-1λ1k-1(πnl)2(k-1)e(k-1)σ1(πn/l)2τ(ω2-(k-2)τ)k-1(k-1)!]}sin⁡πnlx.
After some rearranging, we get
(4.25)S23(x,t*)=λ1∑n=1∞‍[e-σ1(πn/l)2(ω1+τ)Φn(t*-2τ-ω1)(kτ-t*)+e-σ1(πn/l)2(ω2+τ)Φn(t*-2τ-ω2)(t*-(k-1)τ)](πnl)2sin⁡πnlx-λ12∑n=1∞‍[e-σ1(πn/l)2ω1Φn(t*-2τ-ω1)(kτ-t*)×ω11!+e-σ1(πn/l)2ω2Φn(t*-2τ-ω2)(t*-(k-1)τ)ω21!](πnl)4sin⁡πnlx+λ13∑n=1∞‍[e-σ1(πn/l)2(ω1-τ)Φn(t*-2τ-ω1)(kτ-t*)(ω1-τ)22!+e-σ1(πn/l)2(ω2-τ)Φn(t*-2τ-ω2)(t*-(k-1)τ)(ω2-τ)22!](πnl)6sin⁡πnlx+⋯+(-1)k-2λ1k-1∑n=1∞‍[(ω2-(k-3)τ)k-2(k-2)!e-σ1(πn/l)2(ω1-(k-3)τ)Φn(t*-2τ-ω1)(kτ-t*)×(ω1-(k-3)τ)k-2(k-2)!+e-σ1(πn/l)2(ω2-(k-3)τ)Φn(t*-2τ-ω2)×(t*-(k-1)τ)(ω2-(k-3)τ)k-2(k-2)!e-σ1(πnl)2(ω1-(k-3)τ)](πnl)2k-2sin⁡πnlx+(-1)k-1λ1k(t*-(k-1)τ)(ω2-(k-2)τ)k-1(k-1)!×∑n=1∞‍(πnl)2ke-σ1(πn/l)2(ω2-(k-2)τ)Φn(t*-2τ-ω2)sin⁡πnlx.
Due to condition (4.2), we conclude that
(4.26)e-σ1(πn/l)2(ω2-(k-2)τ)n2kmax⁡-τ≤t≤0|Φn(t)|≤Mn3+α.
The proof of the uniform convergence of the series S23(x,t) for x∈[0,l] and t∈[0,kτ] can now be performed in a way similar to the proof of uniform convergence of the series S1(x,t) for x∈[0,l] and t∈[0,kτ].Uniform Convergence of the SeriesS3(x,t)
We will consider the coefficientsCn(t), n=1,2,… of the series S3(x,t) at t=t*∈[(k-1)τ,kτ]. Substituting t*-τ-s=ω, we obtain
(4.27)Cn(t*)=∫0t*‍exp⁡τ{r1n,t*-τ-s}e-σ1(πn/l)2(t*-s)fn(s)ds=∫-τt*-τ‍exp⁡τ{r1n,ω}e-σ1(πn/l)2(ω+τ)fn(t*-τ-ω)dω=∫-τ0‍exp⁡τ{r1n,ω}e-σ1(πn/l)2(ω+τ)fn(t*-τ-ω)dω+∫0τ‍exp⁡τ{r1n,ω}e-σ1(πn/l)2(ω+τ)fn(t*-τ-ω)dω+∫τ2τ‍exp⁡τ{r1n,ω}e-σ1(πn/l)2(ω+τ)fn(t*-τ-ω)dω+⋯+∫(k-2)τt*-τ‍exp⁡τ{r1n,ω}e-σ1(πn/l)2(ω+τ)fn(t*-τ-ω)dω.
Owing to the mean value theorem, there are values ωi, i=1,2,…,k such that
(4.28)-τ≤ω1≤0,0≤ω2≤τ,…,(k-2)τ≤ωk≤t*-τCn(t*)=τe-σ1(πn/l)2(ω1+τ)fn(t*-τ-ω1)+τ[1-λ1(πnl)2eσ1(πn/l)2τω21!]e-σ1(πn/l)2(ω2+τ)fn(t*-τ-ω2)+τ[1-λ1(πnl)2eσ1(πn/l)2τω31!+λ12(πnl)4e2σ1(πn/l)2τ(ω3-τ)22!]e-σ1(πn/l)2(ω3+τ)×fn(t*-τ-ω3)+⋯+τ[[ωk-1-(k-3)τ]k-2(k-2)!+(-1)k-2λ1k-2(πnl)2(k-2)e(k-2)σ1(πn/l)2τ[ωk-1-(k-3)τ]k-2(k-2)![ωk-1-(k-3)τ]k-2(k-2)!1-λ1(πnl)2eσ1(πn/l)2τωk-11!+…+(-1)k-2λ1k-2(πnl)2(k-2)[ωk-1-(k-3)τ]k-2(k-2)!e(k-2)σ1(πn/l)2τ×[ωk-1-(k-3)τ]k-2(k-2)!]e-σ1(πn/l)2(ωk-1+τ)×fn(t*-τ-ωk-1)+[t*-(k-1)τ][[ωk-(k-2)τ]k-1(k-1)!1-λ1(πnl)2eσ1(πn/l)2τωk1!+…+(-1)k-1×λ1k-1(πnl)2(k-1)λ1(πnl)2e(k-1)σ1(πn/l)2τ[ωk-(k-2)τ]k-1(k-1)!]×e-σ1(πn/l)2(ωk+τ)fn(t*-τ-ωk).
Hence,
(4.29)S3(x,t*)=∑n=1∞Cn(t*)sin⁡πnlx=∑n=1∞[τ∑i=1k-1e-σ1(πn/l)2(ωi+τ)fn(t*-τ-ωi)+(t*-(k-1)τ)×e-σ1(πn/l)2(ωk+τ)fn(t*-τ-ωk)τ∑i=1k-1e-σ1(πnl)2(ωi+τ)fn(t*-τ-ωi)+(t*-(k-1)τ)]×sin⁡πnlx-λ1∑n=1∞[τ∑i=2k-1ωi1!e-σ1(πn/l)2ωifn(t*-τ-ωi)+(t*-(k-1)τ)×ωk1!e-σ1(πn/l)2ωkfn(t*-τ-ωk)τ∑i=2k-1ωi1!e-σ1(πnl)2ωifn(t*-τ-ωi)+(t*-(k-1)τ)]×(πnl)2sin⁡πnlx+λ12∑n=1∞[τ∑i=3k-1(ωi-τ)22!e-σ1(πn/l)2(ωi-τ)fn(t*-τ-ωi)+(t*-(k-1)τ)(ωk-τ)22!e-σ1(πn/l)2(ωk-τ)fn(t*-τ-ωk)τ∑i=3k-1(ωi-τ)22!e-σ1(πnl)2(ωi-τ)fn(t*-τ-ωi)]×(πnl)4sin⁡πnlx+…+(-1)k-2λ1k-2×∑n=1∞[τ(ωk-1-(k-3)τ)k-2(k-2)!e-σ1(πn/l)2(ωk-1-(k-3)τ)fn(t*-τ-ωk-1)+[t*-(k-1)τ](ωk-(k-3)τ)k-2(k-2)!e-σ1(πn/l)2(ωk-(k-3)τ)fn(t*-τ-ωk)(ωk-1-(k-3)τ)k-2(k-2)!]×(πnl)2(k-2)sin⁡πnlx+(-1)k-1λ1k-1[t*-(k-1)τ](ωk-(k-2)τ)k-1(k-1)!×∑n=1∞(πnl)2(k-1)e-σ1(πn/l)2(ωk-(k-2)τ)fn(t*-τ-ωk)sin⁡πnlx.
Due to condition (4.2), we conclude that
(4.30)e-σ1(πn/l)2(ωk-(k-2)τ)n2(k-1)max⁡-τ≤t≤kτ|fn(t)|≤Mn3+α.
The proof of the uniform convergence of the series S3(x,t) for x∈[0,l] and t∈[0,kτ] can now be performed in a way similar to the proof of uniform convergence of the series S1(x,t) for x∈[0,l]  and  t∈[0,kτ].Uniform Convergence of the Formal Series forξ(x,t)
Above, the absolute and uniform convergence of the seriesS1(x,t), S2(x,t), S3(x,t) was proved. Therefore, the series for ξ(x,t) converges absolutely and uniformly as well.Uniform Convergence of the Formal Series forξt′(x,t) and ξxx′′(x,t)
To prove the uniform convergence of the series forξt′(x,t) and ξxx′′(x,t), we can proceed as in the above proof of the uniform convergence of ξ(x,t). The above scheme can be repeated, and the final inequalities (4.7), (4.15), (4.26), and (4.30) are replaced as follows: inequality (4.7) by
(4.31)e-σ1(πn/l)2(t*-(k-1)τ)n2k+2|Φn(-τ)|≤Mn1+α,
inequality (4.15) by
(4.32)e-σ1(πn/l)2(t*-(k-1)τ)n2k+2|Φn(0)|≤Mn1+α,
inequality (4.26) by
(4.33)e-σ1(πn/l)2(ω2-(k-2)τ)n2k+2max⁡-τ≤t≤0|Φn(t)|≤Mn1+α,
and inequality (4.30) by
(4.34)e-σ1(πn/l)2(ωk-(k-2)τ)n2kmax⁡-τ≤t≤kτ|fn(t)|≤Mn1+α.The proof of the uniform convergence of the series, which represents the solutionη(x,t) by formula (3.38), and the proof of the uniform convergence of the series for ηx′(x,t) and  ηxx′′(x,t) are much the same.Corollary 4.2.
Functionsu(x,t), v(x,t) are linear combinations of ξ(x,t) and η(x,t). Therefore, representations (3.40) are the solutions of the system system (1.1) satisfying all boundary and initial conditions (1.9)–(1.14).Remark 4.3.
Tracing the proof of Theorem4.1, we see that inequalities (4.2) for functions Φn:[-τ,0]→ℝ, n=1,2,…, and fn:[-τ,∞)→ℝ, n=1,2,… are too restrictive if t*∈[(k*-1)τ,(k*-1)τ+ɛ], k*∈{1,2,…,k}, where ɛ is an arbitrarily small positive number. The question whether the series are uniformly convergent for (x,t)∈[0,l]×[(k*-1)τ,(k*-1)τ+ɛ], k*∈{1,2,…,k} remains open if, for example, Φn:[-τ,0]→ℝ, n=1,2,…, and fn:[-τ,∞)→ℝ, n=1,2,… satisfy only the inequalities
(4.35)max⁡-τ≤t≤kτ|fn(t)|≤M*n,max⁡-τ≤t≤0|Φn(t)|≤M*n
for a positive constant M* because inequalities (4.2) cannot be valid. Nevertheless, in such a case, the series converge at least point-wise for t∈[(k*-1)τ,(k*-1)τ+ɛ], k*∈{1,2,…,k} and uniformly for x∈[0,l]. In other words, in such a case, the series converge uniformly for
(4.36)(x,t)∈[0,l]×([0,kτ]∖⋃k*=1k‍[(k*-1)τ,(k*-1)τ+ɛ]),
(where  ɛ>0 is fixed but arbitrarily small).



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*Source: 219040-2012-07-31.xml*