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

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.

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 𝜕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, (1.1) where the matrices of coefficients 𝑎𝐴=¹¹𝑎₁₂𝑎₂₁𝑎₂₂𝑏,𝐵=¹¹𝑏₁₂𝑏₂₁𝑏₂₂ (1.2) 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 𝑠𝑆=¹¹𝑠₁₂𝑠₂₁𝑠₂₂, (1.3) simultaneously reducing both matrices 𝐴 and 𝐵 into diagonal forms 𝜆Λ=¹00𝜆₂𝜎,Σ=¹00𝜎₂, (1.4) that is, 𝑆⁻¹𝐴𝑆=Λ,𝑆⁻¹𝐵𝑆=Σ, (1.5) where 𝑆⁻¹=1Δ𝑠₂₂−𝑠₁₂−𝑠₂₁𝑠₁₁,Δ=𝑠₁₁𝑠₂₂−𝑠₁₂𝑠₂₂. (1.6) 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^∗=𝒜^𝑇, that is, the real matrix is normal if 𝒜^𝑇𝒜=𝒜𝒜^𝑇. 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.

Let 𝑙 be a positive constant, and let 𝜇_𝑖∶[)𝜃−𝜏,∞→ℝ,𝑖=1,2,_𝑖∶[)[]×[][]×[]−𝜏,∞⟶ℝ,𝑖=1,2,𝜑∶0,𝑙−𝜏,0→ℝ,𝜓∶0,𝑙−𝜏,0→ℝ (1.7) be continuously differentiable functions such that 𝜇₁(𝑡)()[],𝜇=𝜑0,𝑡,𝑡∈−𝜏,0₂(𝑡)()[],𝜃=𝜑𝑙,𝑡,𝑡∈−𝜏,0₁(𝑡)()[],𝜃=𝜓0,𝑡,𝑡∈−𝜏,0₂(𝑡)=𝜓(𝑙,𝑡),𝑡∈[−𝜏,0]. (1.8) Together with system (1.1), we consider the first boundary-value problem, that is, the boundary conditions 𝑢(0,𝑡)=𝜇₁(𝑡)[),𝑢(),𝑡∈−𝜏,∞𝑙,𝑡=𝜇₂(𝑡)[),𝑣(),𝑡∈−𝜏,∞0,𝑡=𝜃₁(𝑡)[),𝑣(),𝑡∈−𝜏,∞𝑙,𝑡=𝜃₂(𝑡),𝑡∈[−𝜏,∞) (1.9)(1.10)(1.11)(1.12) and the initial conditions 𝑢()(),()∈[]×[],𝑣()(),()∈[]×[].𝑥,𝑡=𝜑𝑥,𝑡𝑥,𝑡0,𝑙−𝜏,0𝑥,𝑡=𝜓𝑥,𝑡𝑥,𝑡0,𝑙−𝜏,0 (1.13)(1.14) A solution to the first boundary-value problem (1.1), (1.9)–(1.14) is defined as a pair of functions 𝑢,𝑣∶[0,𝑙]×[−𝜏,∞)⟶ℝ, (1.15) 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 of 𝑛 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. Let 𝑏∈ℝ. The delayed exponential function exp_𝜏{𝑏,𝑡}∶ℝ→ℝ is a function continuous on ℝ⧵{−𝜏} defined as exp_𝜏{}=𝑡𝑏,𝑡0if−∞<𝑡<−𝜏,1if−𝜏≤𝑡<0,1+𝑏⋯𝑡1!if0≤𝑡<𝜏,1+𝑏1!+𝑏²(𝑡−𝜏)²2!+⋯+𝑏^𝑘(𝑡−(𝑘−1)𝜏)^𝑘()⋯𝑘!if𝑘−1𝜏≤𝑡<𝑘𝜏, (2.1) where 𝑘=0,1,2,….

Lemma 2.2. For the differentiation of a delayed exponential function, the formula 𝑑𝑑𝑡exp_𝜏{𝑏,𝑡}=𝑏exp_𝜏{𝑏,𝑡−𝜏} (2.2) 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 exp_𝜏{}𝑡𝑏,𝑡=1+𝑏1!+𝑏²(𝑡−𝜏)²2!+𝑏³(𝑡−2𝜏)³3!+⋯+𝑏^𝑘(𝑡−(𝑘−1)𝜏)^𝑘.𝑘! (2.3) Differentiating this expression, we obtain 𝑑𝑑𝑡exp_𝜏{𝑏,𝑡}=𝑏+𝑏²𝑡−𝜏1!+𝑏³(𝑡−2𝜏)²2!+⋯+𝑏^𝑘(𝑡−(𝑘−1)𝜏)^𝑘−1(𝑘−1)!=𝑏1+𝑏𝑡−𝜏1!+𝑏²(𝑡−2𝜏)²2!+⋯+𝑏^𝑘−1[𝑡−(𝑘−1)𝜏]^𝑘−1(𝑘−1)!=𝑏exp_𝜏{𝑏,𝑡−𝜏}. (2.4)

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

Let us consider a linear homogeneous equation with a single delay 𝑥̇(𝑡)=𝑏𝑥(𝑡−𝜏), (2.5) where 𝑏∈ℝ, together with the initial Cauchy condition 𝑥(𝑡)=𝛽(𝑡),𝑡∈[−𝜏,0]. (2.6)

From (2.2), it immediately follows that the delayed exponential exp_𝜏{𝑏,𝑡} is a solution of the initial Cauchy problems (2.5), (2.6) with 𝛽(𝑡)≡1, 𝑡∈[−𝜏,0].

Theorem 2.3. Let 𝛽∶[−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 𝑥(𝑡)=exp_𝜏{𝑏,𝑡}𝛽(−𝜏)+⁰_−𝜏exp_𝜏{𝑏,𝑡−𝜏−𝑠}𝛽^′(𝑠)𝑑𝑠, (2.7) where 𝑡∈[−𝜏,∞).

Proof. The representation (2.7) is a linear functional of the delayed exponential function exp_𝜏{𝑏,𝑡} and exp_𝜏{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: 𝛽(𝑡)≡exp_𝜏{𝑏,𝑡}𝛽(−𝜏)+⁰_−𝜏exp_𝜏{𝑏,𝑡−𝜏−𝑠}𝛽^′(𝑠)𝑑𝑠. (2.8) We rewrite (2.7) as 𝑥(𝑡)=exp_𝜏{𝑏,𝑡}𝛽(−𝜏)+^𝑡_−𝜏exp_𝜏{𝑏,𝑡−𝜏−𝑠}𝛽^′(𝑠)𝑑𝑠+⁰_𝑡exp_𝜏{𝑏,𝑡−𝜏−𝑠}𝛽^′(𝑠)𝑑𝑠. (2.9) From Definition 2.1, it follows: exp_𝜏{𝑏,𝑡}≡1if−𝜏≤𝑡≤0,exp_𝜏{𝑏,𝑡−𝜏−𝑠}≡1if−𝜏≤𝑠≤𝑡,exp_𝜏{𝑏,𝑡−𝜏−𝑠}≡0if𝑡<𝑠<0. (2.10)
Therefore, 𝑥(𝑡)=𝛽(−𝜏)+^𝑡_−𝜏𝛽^′(𝑠)𝑑𝑠=𝛽(−𝜏)+𝛽(𝑡)−𝛽(−𝜏)=𝛽(𝑡). (2.11)

Remark 2.4. Computing the integral in formula (2.7) by parts, we obtain for 𝑡≥𝜏: 𝑥(𝑡)=exp_𝜏{𝑏,𝑡−𝜏}𝛽(0)+𝑏⁰_−𝜏exp_𝜏{𝑏,𝑡−2𝜏−𝑠}𝛽(𝑠)𝑑𝑠. (2.12) 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 𝑥̇(𝑡)=𝑎𝑥(𝑡)+𝑏𝑥(𝑡−𝜏), (2.13) where 𝑎,𝑏∈ℝ, together with initial Cauchy condition (2.6).

Theorem 2.5. Let the function 𝛽 in (2.6) be continuously differentiable. Then the unique solution of the initial Cauchy problems (2.13), (2.6) can be represented as 𝑥(𝑡)=exp_𝜏𝑏₁,𝑡𝑒^{𝑎(𝑡+𝜏)}𝛽(−𝜏)+⁰_−𝜏exp_𝜏𝑏₁,𝑡−𝜏−𝑠𝑒^{𝑎(𝑡−𝑠)}𝛽^′(𝑠)−𝑎𝛽(𝑠)𝑑𝑠, (2.14) where 𝑏₁=𝑏𝑒^−𝑎𝜏 and 𝑡∈[−𝜏,∞).

Proof. Transforming 𝑥 by a substitution 𝑥(𝑡)=𝑒^𝑎𝑡𝑦(𝑡), (2.15) where 𝑦 is a new unknown function, we obtain 𝑎𝑒^𝑎𝑡𝑦(𝑡)+𝑒^𝑎𝑡𝑦̇(𝑡)=𝑎𝑒^𝑎𝑡𝑦(𝑡)+𝑏𝑒^{𝑎(𝑡−𝜏)}𝑦(𝑡−𝜏) (2.16) or 𝑦̇(𝑡)=𝑏₁𝑦(𝑡−𝜏). (2.17) Correspondingly, the initial condition for (2.17) is 𝑦(𝑡)=𝑒^−𝑎𝑡𝛽(𝑡),𝑡∈[−𝜏,0]. (2.18) As follows, from formula (2.7), the solution of the corresponding initial Cauchy problems (2.17), (2.18) is 𝑦(𝑡)=exp_𝜏𝑏₁,𝑡𝑒^𝑎𝜏𝛽(−𝜏)+⁰_−𝜏exp_𝜏𝑏₁,𝑡−𝜏−𝑠𝑒^−𝑎𝑠𝛽′(𝑠)−𝑎𝑒^−𝑎𝑠𝛽(𝑠)𝑑𝑠. (2.19) Using substitution (2.15), we obtain 𝑥(𝑡)=exp_𝜏𝑏₁,𝑡𝑒^{𝑎(𝑡+𝜏)}𝛽(−𝜏)+⁰_−𝜏exp_𝜏𝑏₁,𝑡−𝜏−𝑠𝑒^{𝑎(𝑡−𝑠)}𝛽^′(𝑠)−𝑎𝛽(𝑠)𝑑𝑠, (2.20) 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 𝑥̇(𝑡)=𝑎𝑥(𝑡)+𝑏𝑥(𝑡−𝜏)+𝑓(𝑡) (2.21) be given, where 𝑎,𝑏∈ℝ and 𝑓∶[0,∞)→ℝ. We consider the Cauchy problem with a zero initial condition 𝑥(𝑡)=0,𝑡∈[−𝜏,0], (2.22) 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 𝑥(𝑡)=^𝑡₀exp_𝜏𝑏₁,𝑡−𝜏−𝑠𝑒^{𝑎(𝑡−𝑠)}𝑓(𝑠)𝑑𝑠, (2.23) where 𝑏₁=𝑏𝑒^−𝑎𝜏.

Proof. We apply substitution (2.15). Then 𝑎𝑒^𝑎𝑡𝑦(𝑡)+𝑒^𝑎𝑡𝑦̇(𝑡)=𝑎𝑒^𝑎𝑡𝑦(𝑡)+𝑏𝑒^{𝑎(𝑡−𝜏)}𝑦(𝑡−𝜏)+𝑓(𝑡), (2.24) or, equivalently, 𝑦̇(𝑡)=𝑏₁𝑦(𝑡−𝜏)+𝑒^−𝑎𝑡𝑓(𝑡). (2.25) 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 𝑦(𝑡)=^𝑡₀exp_𝜏𝑏₁,𝑡−𝜏−𝑠𝑒^−𝑎𝑠𝑓(𝑠)𝑑𝑠. (2.26) Substituting (2.26) in (2.25), we obtain exp_𝜏𝑏₁,𝑡−𝜏−𝑠𝑒^−𝑎𝑠𝑓(𝑠)_𝑠=𝑡+𝑏₁^𝑡₀exp_𝜏𝑏₁,𝑡−2𝜏−𝑠𝑒^−𝑎𝑠𝑓(𝑠)𝑑𝑠=𝑏₁₀^𝑡−𝜏exp_𝜏𝑏₁,𝑡−2𝜏−𝑠𝑒^−𝑎𝑠𝑓(𝑠)𝑑𝑠+𝑒^−𝑎𝑡𝑓(𝑡). (2.27) Since exp_𝜏𝑏₁,𝑡−𝜏−𝑠𝑒^−𝑎𝑠𝑓(𝑠)_𝑠=𝑡=exp𝑏₁,−𝜏𝑒^−𝑎𝑡𝑓(𝑡)=𝑒^−𝑎𝑡𝑓(𝑡), (2.28) we obtain 𝑒^−𝑎𝑡𝑓(𝑡)+𝑏₁₀^𝑡−𝜏exp_𝜏𝑏₁,𝑡−2𝜏−𝑠𝑒^−𝑎𝑠𝑓(𝑠)𝑑𝑠+𝑏₁^𝑡_𝑡−𝜏exp_𝜏𝑏₁,𝑡−2𝜏−𝑠𝑒^−𝑎𝑠𝑓(𝑠)𝑑𝑠=𝑏₁₀^𝑡−𝜏exp_𝜏𝑏₁,𝑡−2𝜏−𝑠𝑒^−𝑎𝑠𝑓(𝑠)𝑑𝑠+𝑒^−𝑎𝑡𝑓(𝑡). (2.29) Hence, ^𝑡_𝑡−𝜏exp_𝜏𝑏₁,𝑡−2𝜏−𝑠𝑒^−𝑎𝑠𝑓(𝑠)𝑑𝑠=0. (2.30) This equality is true since 𝑡−2𝜏−𝑠≤𝑡−2𝜏−(𝑡−𝜏)=−𝜏 (2.31) and, by formula (2.1) in Definition 2.1, exp_𝜏𝑏₁,𝑡−2𝜏−𝑠≡0 (2.32) if 𝑡−2𝜏−𝑠<−𝜏. In accordance with (2.15), we get 𝑥(𝑡)=𝑒^𝑎𝑡𝑦(𝑡)=^𝑡₀exp_𝜏𝑏₁,𝑡−𝜏−𝑠𝑒^{𝑎(𝑡−𝑠)}𝑓(𝑠)𝑑𝑠, (2.33) that is, formula (2.23) is proved.

Combining Theorems 2.5, and 2.6 we get the following Corollary.

Corollary 2.7. Let the function 𝛽 in (2.6) be continuously differentiable. Then the unique solution of the problems (2.21), (2.6) is given as 𝑥(𝑡)=exp_𝜏𝑏₁,𝑡𝑒^{𝑎(𝑡+𝜏)}𝛽()+−𝜏_0−𝜏exp_𝜏𝑏₁,𝑡−𝜏−𝑠𝑒^{𝑎(𝑡−𝑠)}(𝑠)(𝑠+𝛽′−𝑎𝛽)𝑑𝑠_𝑡0exp_𝜏𝑏₁,𝑡−𝜏−𝑠𝑒^{𝑎(𝑡−𝑠)}𝑓(𝑠)𝑑𝑠, (2.34) 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 𝑢()𝑣()𝜉()𝜂()𝑥,𝑡𝑥,𝑡=𝑆𝑥,𝑡𝑥,𝑡, (3.1) the systems (1.1) can be reduced to a form 𝜕𝜉(𝑥,𝑡)()𝜕𝑡𝜕𝜂𝑥,𝑡𝜕𝜕𝑡=Λ²𝜉(𝑥,𝑡−𝜏)𝜕𝑥²𝜕²𝜂(𝑥,𝑡−𝜏)𝜕𝑥²𝜕+Σ²𝜉(𝑥,𝑡)𝜕𝑥²𝜕²𝜂(𝑥,𝑡)𝜕𝑥², (3.2) that is, into two independent scalar equations 𝜕𝜉(𝑥,𝑡)𝜕𝑡=𝜆₁𝜕²𝜉(𝑥,𝑡−𝜏)𝜕𝑥²+𝜎₁𝜕²𝜉(𝑥,𝑡)𝜕𝑥²,𝜕𝜂(𝑥,𝑡)𝜕𝑡=𝜆₂𝜕²𝜂(𝑥,𝑡−𝜏)𝜕𝑥²+𝜎₂𝜕²𝜂(𝑥,𝑡)𝜕𝑥². (3.3)(3.4) Initial and boundary conditions reduce to 𝜉(0,𝑡)=𝜇^∗₁(𝑡)[),𝜉(),𝑡∈−𝜏,∞𝑙,𝑡=𝜇^∗₂(𝑡)[),𝜂(),𝑡∈−𝜏,∞0,𝑡=𝜃^∗₁(𝑡)[),𝜂(),𝑡∈−𝜏,∞𝑙,𝑡=𝜃^∗₂(𝑡),𝑡∈[−𝜏,∞), (3.5)(3.6) and to 𝜉(𝑥,𝑡)=𝜑^∗(),()∈[]×[],𝜂()𝑥,𝑡𝑥,𝑡0,𝑙−𝜏,0𝑥,𝑡=𝜓^∗(𝑥,𝑡),(𝑥,𝑡)∈[0,𝑙]×[−𝜏,0], (3.7)(3.8) where 𝜇^∗₁(𝑡)𝜃^∗₁(𝑡)=𝑆⁻¹𝜇₁(𝑡)𝜃₁(𝑡),𝜇^∗₂(𝑡)𝜃^∗₂(𝑡)=𝑆⁻¹𝜇₂(𝑡)𝜃₂(𝑡),𝜑^∗()𝜓𝑥,𝑡_∗(𝑥,𝑡)=𝑆⁻¹𝜑()𝜓().𝑥,𝑡𝑥,𝑡 (3.9)

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 𝜉(𝑥,𝑡)=𝜉₀(𝑥,𝑡)+𝜉₁(𝑥,𝑡)+𝜇^∗₁(𝑡)+𝑥𝑙𝜇^∗₂(𝑡)−𝜇^∗₁(𝑡), (3.10) where (𝑥,𝑡)∈[0,𝑙]×[−𝜏,∞), 𝜉₀(𝑥,𝑡) is a solution of (3.3) with zero boundary conditions 𝜉₀(0,𝑡)=0,𝜉₀(𝑙,𝑡)=0,𝑡∈[−𝜏,∞) (3.11) and with a nonzero initial condition 𝜉₀(𝑥,𝑡)=Φ(𝑥,𝑡)∶=𝜑^∗(𝑥,𝑡)−𝜇^∗₁(𝑡)−𝑥𝑙𝜇^∗₂(𝑡)−𝜇^∗₁(𝑡),(𝑥,𝑡)∈[0,𝑙]×[−𝜏,0], (3.12) and 𝜉₁(𝑥,𝑡) is a solution of a non-homogeneous equation 𝜕𝜉(𝑥,𝑡)𝜕𝑡=𝜆₁𝜕²𝜉(𝑥,𝑡−𝜏)𝜕𝑥²+𝜎₁𝜕²𝜉(𝑥,𝑡)𝜕𝑥²+𝐹(𝑥,𝑡), (3.13) where 𝐹(𝑥,𝑡)∶=−𝜇̇^∗₁(𝑡)−𝑥𝑙𝜇̇^∗₂(𝑡)−𝜇̇^∗₁(𝑡), (3.14) with zero boundary conditions 𝜉₁(0,𝑡)=0,𝜉₁(𝑙,𝑡)=0,𝑡∈[−𝜏,∞) (3.15) and a zero initial condition 𝜉₁(𝑥,𝑡)=0,(𝑥,𝑡)∈[0,𝑙]×[−𝜏,0]. (3.16)

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

For finding a solution 𝜉=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, 𝜉₀(𝑥,𝑡)=𝑋(𝑥)𝑇(𝑡). (3.17) Substituting (3.17) into (3.3), we obtain 𝑋(𝑥)𝑇^′(𝑡)=𝜆₁𝑋^′′(𝑥)𝑇(𝑡−𝜏)+𝜎₁𝑋^′′(𝑥)𝑇(𝑡). (3.18) Separating variables, we have 𝑇^′(𝑡)𝜆₁𝑇(𝑡−𝜏)+𝜎₁𝑇(𝑡)=𝑋^′′(𝑥)𝑋(𝑥)=−𝜅², (3.19) where 𝜅 is a constant. We consider two differential equations 𝑇^′(𝑡)+𝜎₁𝜅²𝑇(𝑡)+𝜆₁𝜅²𝑇()𝑋𝑡−𝜏=0,_′′(𝑥)+𝜅²𝑋(𝑥)=0. (3.20)(3.21) Nonzero solutions of (3.21) that satisfy zero boundary conditions 𝑋(0)=0,𝑋(𝑙)=0, (3.22) exist for the choice 𝜅²=𝜅²_𝑛=(𝜋𝑛/𝑙)², 𝑛=1,2,…, and are defined by the formulas 𝑋(𝑥)=𝑋_𝑛(𝑥)=𝐴_𝑛sin𝜋𝑛𝑙𝑥,𝑛=1,2,…, (3.23) where 𝐴_𝑛 are arbitrary constants. Now we consider (3.20) with 𝜅=𝜅_𝑛: 𝑇^′_𝑛(𝑡)=−𝜎₁𝜋𝑛𝑙²𝑇_𝑛(𝑡)−𝜆₁𝜋𝑛𝑙²𝑇_𝑛(𝑡−𝜏),𝑛=1,2,…. (3.24) 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 Φ(𝑥,𝑡)=^∞_𝑛=1Φ_𝑛(𝑡)sin𝜋𝑛𝑙𝑥,(𝑥,𝑡)∈[0,𝑙]×[−𝜏,0], (3.25) where Φ_𝑛(𝑡)=2𝑙^𝑙₀Φ(𝑠,𝑡)sin𝜋𝑛𝑙=2𝑠𝑑𝑠𝑙^𝑙₀𝜑^∗(𝑠,𝑡)−𝜇^∗₁(𝑡)−𝑠𝑙𝜇^∗₂(𝑡)−𝜇^∗₁(𝑡)sin𝜋𝑛𝑙=2𝑠𝑑𝑠𝑙^𝑙₀𝜑^∗(𝑠,𝑡)sin𝜋𝑛𝑙−𝑠𝑑𝑠2𝜇^∗₁(𝑡)𝑙^𝑙₀sin𝜋𝑛𝑙2𝜇𝑠𝑑𝑠−^∗2(𝑡)−𝜇^∗₁(𝑡)𝑙^2𝑙₀𝑠⋅sin𝜋𝑛𝑙=2𝑠𝑑𝑠𝑙^𝑙₀𝜑^∗(𝑠,𝑡)sin𝜋𝑛𝑙2𝑠𝑑𝑠+)𝜋𝑛(−1^𝑛𝜇^∗₂(𝑡)−𝜇^∗₁(𝑡),𝑡∈[−𝜏,0]. (3.26) 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 𝑇^′_𝑛(𝑡)=−𝜎₁𝜋𝑛𝑙²𝑇_𝑛(𝑡)−𝜆₁𝜋𝑛𝑙²𝑇_𝑛(),𝑇𝑡−𝜏_𝑛(𝑡)=Φ_𝑛(𝑡),𝑡∈[−𝜏,0], (3.27) 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 𝑇_𝑛(𝑡)=exp_𝜏𝑟_1𝑛,𝑡𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡+𝜏)}Φ_𝑛()+−𝜏_0−𝜏exp_𝜏𝑟_1𝑛,𝑡−𝜏−𝑠𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡−𝑠)}Φ^′_𝑛(𝑠)+𝜎₁𝜋𝑛𝑙²Φ_𝑛(𝑠)𝑑𝑠, (3.28) where 𝑟_1𝑛=−𝜆₁𝜋𝑛𝑙²𝑒^{𝜎1(𝜋𝑛/𝑙)2𝜏},𝑛=1,2,…. (3.29) 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 𝐴_𝑛=1, 𝑛=1,2,… in (3.23)) is 𝜉₀(𝑥,𝑡)=^∞_𝑛=1exp_𝜏𝑟_1𝑛,𝑡𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡+𝜏)}Φ_𝑛()+−𝜏_0−𝜏exp_𝜏𝑟_1𝑛,𝑡−𝜏−𝑠𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡−𝑠)}Φ^′_𝑛(𝑠)+𝜎₁𝜋𝑛𝑙²Φ_𝑛(𝑠)𝑑𝑠sin𝜋𝑛𝑙𝑥, (3.30) where Φ_𝑛 is defined by (3.26), 𝑟_1𝑛 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 𝜉₁(𝑥,𝑡)=^∞_𝑛=1𝑇⁰_𝑛(𝑡)sin𝜋𝑛𝑙𝑥, (3.31) where (𝑥,𝑡)∈[0,𝑙]×[−𝜏,∞) and 𝑇⁰_𝑛∶[−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: 𝑇⁰_𝑛^′(𝑡)=−𝜎₁𝜋𝑛𝑙²𝑇⁰_𝑛(𝑡)−𝜆₁𝜋𝑛𝑙²𝑇⁰_𝑛(𝑡−𝜏)+𝑓_𝑛(𝑡),𝑡∈[0,∞),𝑛=1,2,…, (3.32) where 𝑓_𝑛∶[−𝜏,∞)→ℝ are Fourier coefficients of the function 𝐹(𝑥,𝑡), that is, 𝑓_𝑛(𝑡)=2𝑙^𝑙₀𝐹(𝑠,𝑡)sin𝜋𝑛𝑙2𝑠𝑑𝑠=−𝑙^𝑙₀𝜇̇^∗₁(𝑡)+𝑠𝑙𝜇̇^∗₂(𝑡)−𝜇̇^∗₁(𝑡)sin𝜋𝑛𝑙2𝑠𝑑𝑠=−𝑙𝜇̇^∗₁(𝑡)^𝑙₀sin𝜋𝑛𝑙2𝑠𝑑𝑠−𝑙²𝜇̇^∗₂(𝑡)−𝜇̇^∗₁(𝑡)^𝑙₀𝑠⋅sin𝜋𝑛𝑙2𝑠𝑑𝑠=−()𝜋𝑛−1^𝑛+1𝜇̇^∗₂(𝑡)+𝜇̇^∗₁(𝑡). (3.33) In accordance with (3.15), we assume zero initial conditions 𝑇⁰_𝑛(𝑡)=0,𝑡∈[−𝜏,0],𝑛=1,2,… (3.34) 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 𝑇⁰_𝑛(𝑡)=^𝑡₀exp_𝜏𝑟_1𝑛,𝑡−𝜏−𝑠𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡−𝑠)}𝑓_𝑛(𝑠)𝑑𝑠,𝑡∈[−𝜏,∞),𝑛=1,2,…, (3.35) where 𝑟_1𝑛 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 𝜉₁(𝑥,𝑡)=^∞_𝑛=1^𝑡₀exp_𝜏𝑟_1𝑛,𝑡−𝜏−𝑠𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡−𝑠)}𝑓_𝑛(𝑠)𝑑𝑠sin𝜋𝑛𝑙𝑥, (3.36) where 𝑓_𝑛 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 solutions 𝜉₀(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 𝜇^∗₁(𝑡), 𝜇^∗₂(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: 𝜉(𝑥,𝑡)=^∞_𝑛=1exp_𝜏𝑟_1𝑛,𝑡𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡+𝜏)}Φ_𝑛()+−𝜏_0−𝜏exp_𝜏𝑟_1𝑛,𝑡−𝜏−𝑠𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡−𝑠)}Φ^′_𝑛(𝑠)+𝜎₁𝜋𝑛𝑙²Φ_𝑛(𝑠)𝑑𝑠+^𝑡₀exp_𝜏𝑟_1𝑛,𝑡−𝜏−𝑠𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡−𝑠)}𝑓_𝑛(𝑠)𝑑𝑠sin𝜋𝑛𝑙𝑥+𝜇^∗₁(𝑡)+𝑥𝑙𝜇^∗₂(𝑡)−𝜇^∗₁(𝑡), (3.37) where (𝑥,𝑡)∈[0,𝑙]×[−𝜏,∞), coefficients Φ_𝑛 are defined by formulas (3.26), coefficients 𝑓_𝑛 by formulas (3.33), and the numbers 𝑟_1𝑛 by formula (3.29).

Similarly, a formal solution of the first boundary value problem for (3.4) is given by the formula: 𝜂(𝑥,𝑡)=^∞_𝑛=1exp_𝜏𝑟_2𝑛,𝑡𝑒^{−𝜎2(𝜋𝑛/𝑙)2(𝑡+𝜏)}Ψ_𝑛()+−𝜏_0−𝜏exp_𝜏𝑟_2𝑛,𝑡−𝜏−𝑠𝑒^{−𝜎2(𝜋𝑛/𝑙)2(𝑡−𝑠)}Ψ^′_𝑛(𝑠)+𝜎₂𝜋𝑛𝑙²Ψ_𝑛(𝑠)𝑑𝑠+^𝑡₀exp_𝜏𝑟_2𝑛,𝑡−𝜏−𝑠𝑒^{−𝜎2(𝜋𝑛/𝑙)2(𝑡−𝑠)}𝑔_𝑛(𝑠)𝑑𝑠sin𝜋𝑛𝑙𝑥+𝜃^∗₁(𝑡)+𝑥𝑙𝜃^∗₂(𝑡)−𝜃^∗₁(𝑡), (3.38) where (𝑥,𝑡)∈[0,𝑙]×[−𝜏,∞) and (by analogy with (3.26), (3.33) and (3.29)) Ψ_𝑛(𝑡)=2𝑙^𝑙₀𝜓^∗(𝑠,𝑡)sin𝜋𝑛𝑙2𝑠𝑑𝑠+)𝜋𝑛(−1^𝑛𝜃^∗₂(𝑡)−𝜃^∗₁(𝑡[],𝑔),𝑡∈−𝜏,0_𝑛(𝑡)2=−()𝜋𝑛−1^𝑛+1𝜃̇^∗₂(𝑡)+𝜃̇^∗₁(𝑡)[),𝑟,𝑡∈−𝜏,∞_2𝑛=−𝜆₂𝜋𝑛𝑙²𝑒^{𝜎2(𝜋𝑛/𝑙)2𝜏}. (3.39)

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 𝑢(𝑥,𝑡)=𝑠₁₁𝜉(𝑥,𝑡)+𝑠₁₂𝜂(),𝑣()𝑥,𝑡𝑥,𝑡=𝑠₂₁𝜉(𝑥,𝑡)+𝑠₂₂𝜂(𝑥,𝑡), (3.40) 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 Φ_𝑛∶[−𝜏,0]⟶ℝ,𝑓_𝑛∶[−𝜏,∞)⟶ℝ,𝑛=1,2,… (4.1) 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 𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡∗−(𝑘−1)𝜏)}𝑛^2𝑘max_{−𝜏≤𝑡≤𝑘𝜏}𝑓_𝑛(𝑡≤𝑀)𝑛^1+𝛼,𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡∗−(𝑘−1)𝜏)}𝑛^2𝑘max_{−𝜏≤𝑡≤0}Φ_𝑛(𝑡≤𝑀)𝑛^3+𝛼. (4.2) 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 𝜉(𝑥,𝑡)=𝑆₁(𝑥,𝑡)+𝑆₂(𝑥,𝑡)+𝑆₃(𝑥,𝑡)+𝜇^∗₁(𝑡)+𝑥𝑙𝜇^∗₂(𝑡)−𝜇^∗₁(𝑡), (4.3) where 𝑆₁(𝑥,𝑡)=^∞_𝑛=1𝐴_𝑛(𝑡)sin𝜋𝑛𝑙𝐴𝑥,_𝑛(𝑡)∶=exp_𝜏𝑟_1𝑛,𝑡𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡+𝜏)}Φ_𝑛(),𝑆−𝜏₂(𝑥,𝑡)=^∞_𝑛=1𝐵_𝑛(𝑡)sin𝜋𝑛𝑙𝐵𝑥,_𝑛(𝑡)∶=⁰_−𝜏exp_𝜏𝑟_1𝑛,𝑡−𝜏−𝑠𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡−𝑠)}Φ^′_𝑛(𝑠)+𝜎₁𝜋𝑛𝑙²Φ_𝑛(𝑠𝑆)𝑑𝑠,₃(𝑥,𝑡)=^∞_𝑛=1𝐶_𝑛(𝑡)sin𝜋𝑛𝑙𝐶𝑥,_𝑛(𝑡)∶=^𝑡₀exp_𝜏𝑟_1𝑛,𝑡−𝜏−𝑠𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡−𝑠)}𝑓_𝑛(𝑠)𝑑𝑠. (4.4) In the following parts, we will prove the uniform convergence of each of the series 𝑆_𝑖(𝑥,𝑡), 𝑖=1,2,3 separately.
Throughout the proof, we use the delayed exponential function defined by Definition 2.1, 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 Series 𝑆₁(𝑥,𝑡)
We consider the coefficients 𝐴_𝑛(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: 𝐴_𝑛𝑡^∗=exp_𝜏𝑟_1𝑛,𝑡^∗𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡∗+𝜏)}Φ_𝑛(−𝜏)=𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡∗+𝜏)}Φ_𝑛(−𝜏)1+𝑟_1𝑛𝑡^∗1!+𝑟²_1𝑛𝑡^∗−𝜏²2!+⋯+𝑟^𝑘_1𝑛𝑡^∗−(𝑘−1)𝜏^𝑘𝑘!=𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡∗+𝜏)}Φ_𝑛(−𝜏)1−𝜆₁𝜋𝑛𝑙²𝑒^{𝜎1(𝜋𝑛/𝑙)2𝜏}𝑡^∗1!+𝜆²₁𝜋𝑛𝑙⁴𝑒^{2𝜎1(𝜋𝑛/𝑙)2𝜏}𝑡^∗−𝜏²2!+⋯+−𝜆₁^𝑘𝜋𝑛𝑙^2𝑘𝑒^{𝑘𝜎1(𝜋𝑛/𝑙)2𝜏}𝑡^∗−(𝑘−1)𝜏^𝑘.𝑘! (4.5) Therefore, 𝑆₁𝑥,𝑡^∗=^∞_𝑛=1𝐴_𝑛𝑡^∗sin𝜋𝑛𝑙𝑥=^∞_𝑛=1exp_𝜏𝑟_1𝑛,𝑡^∗𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡∗+𝜏)}Φ_𝑛(−𝜏)sin𝜋𝑛𝑙𝑥=^∞_𝑛=1𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡∗+𝜏)}Φ_𝑛(−𝜏)sin𝜋𝑛𝑙𝑥−𝜆₁𝑡^∗1!^∞_𝑛=1𝑒^{−𝜎1(𝜋𝑛/𝑙)2𝑡∗}𝜋𝑛𝑙²Φ_𝑛(−𝜏)sin𝜋𝑛𝑙𝑥+𝜆²₁𝑡^∗−𝜏²2!^∞_𝑛=1𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡∗−𝜏)}𝜋𝑛𝑙⁴Φ_𝑛(−𝜏)sin𝜋𝑛𝑙+()𝑥+⋯−1^𝑘𝜆^𝑘₁𝑡^∗−(𝑘−1)𝜏^𝑘𝑘!^∞_𝑛=1𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡∗−(𝑘−1)𝜏)}𝜋𝑛𝑙^2𝑘Φ_𝑛(−𝜏)sin𝜋𝑛𝑙𝑥. (4.6) Due to condition (4.2), we conclude that 𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡∗−(𝑘−1)𝜏)}𝑛^2𝑘Φ_𝑛(≤𝑀−𝜏)𝑛^3+𝛼. (4.7) 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 max_{−𝜏≤𝑡≤𝑘∗𝜏}𝑓_𝑛(𝑡≤𝑀)𝑛^2𝑘+1+𝛼≤𝑀𝑛^{2𝑘∗+1+𝛼},𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡∗−(𝑘∗−1)𝜏)}𝑛^2𝑘max_{−𝜏≤𝑡≤0}Φ_𝑛(𝑡≤𝑀)𝑛^3+𝛼. (4.8) 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 Series 𝑆₂(𝑥,𝑡)
We consider the coefficients 𝐵_𝑛(𝑡), 𝑛=1,2,… of the second series 𝑆₂(𝑥,𝑡). In the representation 𝐵_𝑛(𝑡)=𝜎₁𝜋𝑛𝑙²⁰_−𝜏exp_𝜏𝑟_1𝑛,𝑡−𝜏−𝑠𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡−𝑠)}Φ_𝑛(𝑠)+𝑑𝑠_0−𝜏exp_𝜏𝑟_1𝑛,𝑡−𝜏−𝑠𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡−𝑠)}Φ^′_𝑛(𝑠)𝑑𝑠, (4.9) we calculate the second integral by parts and use formula (2.2) in Lemma 2.2: 𝐵_𝑛(𝑡)=𝜎₁𝜋𝑛𝑙²⁰_−𝜏exp_𝜏𝑟_1𝑛,𝑡−𝜏−𝑠𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡−𝑠)}Φ_𝑛(𝑠)𝑑𝑠+exp_𝜏𝑟_1𝑛,𝑡−𝜏𝑒^{−𝜎1(𝜋𝑛/𝑙)2𝑡}Φ_𝑛(0)−exp_𝜏𝑟_1𝑛,𝑡𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡+𝜏)}Φ_𝑛()+−𝜏_0−𝜏𝜆₁𝜋𝑛𝑙²exp_𝜏𝑟_1𝑛,𝑡−2𝜏−𝑠𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡−𝑠−𝜏)}Φ_𝑛(𝑠)𝑑𝑠−𝜎₁𝜋𝑛𝑙²⁰_−𝜏exp_𝜏𝑟_1𝑛,𝑡−𝜏−𝑠𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡−𝑠)}Φ_𝑛(𝑠)𝑑𝑠=exp_𝜏𝑟_1𝑛,𝑡−𝜏𝑒^{−𝜎1(𝜋𝑛/𝑙)2𝑡}Φ_𝑛(0)−exp_𝜏𝑟_1𝑛,𝑡𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡+𝜏)}Φ_𝑛()+−𝜏_0−𝜏𝜆₁𝜋𝑛𝑙²exp_𝜏𝑟_1𝑛,𝑡−2𝜏−𝑠𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡−𝑠−𝜏)}Φ_𝑛(𝑠)𝑑𝑠=𝐵_𝑛1(𝑡)−𝐵_𝑛2(𝑡)+𝐵_𝑛3(𝑡), (4.10) where 𝐵_𝑛1(𝑡)=exp_𝜏𝑟_1𝑛,𝑡−𝜏𝑒^{−𝜎1(𝜋𝑛/𝑙)2𝑡}Φ_𝑛(0),𝐵_𝑛2(𝑡)=exp_𝜏𝑟_1𝑛,𝑡𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡+𝜏)}Φ_𝑛(),𝐵−𝜏_𝑛3(𝑡)=⁰_−𝜏𝜆₁𝜋𝑛𝑙²exp_𝜏𝑟_1𝑛,𝑡−2𝜏−𝑠𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡−𝑠−𝜏)}Φ_𝑛(𝑠)𝑑𝑠. (4.11) As follows from the Definition 2.1 of the delayed exponential function, for 𝑡^∗∈[(𝑘−1)𝜏,𝑘𝜏], the following equality holds: 𝐵_𝑛1𝑡^∗=exp_𝜏𝑟_1𝑛,𝑡^∗−𝜏𝑒^{−𝜎1(𝜋𝑛/𝑙)2𝑡∗}Φ_𝑛(0)=𝑒^{−𝜎1(𝜋𝑛/𝑙)2𝑡∗}Φ_𝑛(0)×1−𝜆₁𝜋𝑛𝑙²𝑒^{𝜎1(𝜋𝑛/𝑙)2𝜏}𝑡^∗−𝜏1!+𝜆²₁𝜋𝑛𝑙⁴𝑒^{2𝜎1(𝜋𝑛/𝑙)2𝜏}𝑡^∗−2𝜏²()2!+⋯+−1^𝑘−1𝜆₁^𝑘−1𝜋𝑛𝑙^2(𝑘−1)𝑒^{(𝑘−1)𝜎1(𝜋𝑛/𝑙)2𝜏}𝑡^∗−(𝑘−1)𝜏^𝑘−1()!.𝑘−1 (4.12) Therefore, for 𝑆₂₁𝑥,𝑡^∗∶=^∞_𝑛=1𝐵_𝑛1𝑡^∗sin𝜋𝑛𝑙𝑥, (4.13) we get 𝑆₂₁𝑥,𝑡^∗=^∞_𝑛=1𝐵_𝑛1𝑡^∗sin𝜋𝑛𝑙𝑥=^∞_𝑛=1exp_𝜏𝑟_1𝑛,𝑡^∗−𝜏𝑒^{−𝜎1(𝜋𝑛/𝑙)2𝑡∗}Φ_𝑛(0)sin𝜋𝑛𝑙𝑥=^∞_𝑛=1𝑒^{−𝜎1(𝜋𝑛/𝑙)2𝑡∗}Φ_𝑛(0)sin𝜋𝑛𝑙𝑥−𝜆₁𝑡^∗−𝜏1!^∞_𝑛=1𝜋𝑛𝑙²𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡∗−𝜏)}Φ_𝑛(0)sin𝜋𝑛𝑙𝑥+𝜆²₁𝑡^∗−2𝜏²2!^∞_𝑛=1𝜋𝑛𝑙⁴𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡∗−2𝜏)}Φ_𝑛(0)sin𝜋𝑛𝑙+()𝑥+⋯−1^𝑘−1𝜆₁^𝑘−1𝑡^∗−(𝑘−1)𝜏^𝑘−1(𝑘−1)!^∞_𝑛=1𝜋𝑛𝑙^2(𝑘−1)𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡∗−(𝑘−1)𝜏)}Φ_𝑛(0)sin𝜋𝑛𝑙𝑥. (4.14) Due to condition (4.2), we conclude that 𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡∗−(𝑘−1)𝜏)}𝑛^2𝑘Φ_𝑛(0≤𝑀)𝑛^3+𝛼. (4.15) 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 coefficients 𝐵_𝑛2(𝑡), the following holds: 𝐵_𝑛2𝑡^∗=exp_𝜏𝑟_1𝑛,𝑡^∗𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡∗+𝜏)}Φ_𝑛(−𝜏)=𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡∗+𝜏)}Φ_𝑛(−𝜏)×1−𝜆₁𝜋𝑛𝑙²𝑒^{𝜎1(𝜋𝑛/𝑙)2𝜏}𝑡^∗1!+𝜆²₁𝜋𝑛𝑙⁴𝑒^{2𝜎1(𝜋𝑛/𝑙)2𝜏}𝑡^∗−𝜏²()2!+⋯+−1^𝑘𝜆^𝑘₁𝜋𝑛𝑙^2𝑘𝑒^{𝑘𝜎1(𝜋𝑛/𝑙)2𝜏}𝑡^∗−(𝑘−1)𝜏^𝑘.𝑘! (4.16) Therefore, for 𝑆₂₂𝑥,𝑡^∗∶=^∞_𝑛=1𝐵_𝑛2𝑡^∗sin𝜋𝑛𝑙𝑥, (4.17) we get 𝑆₂₂𝑥,𝑡^∗=^∞_𝑛=1𝐵_𝑛2𝑡^∗sin𝜋𝑛𝑙𝑥=^∞_𝑛=1exp_𝜏𝑟_1𝑛,𝑡^∗𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡∗+𝜏)}Φ_𝑛(−𝜏)sin𝜋𝑛𝑙𝑥=^∞_𝑛=1𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡∗+𝜏)}Φ_𝑛(−𝜏)sin𝜋𝑛𝑙𝑥−𝜆₁𝑡^∗1!^∞_𝑛=1𝜋𝑛𝑙²𝑒^{−𝜎1(𝜋𝑛/𝑙)2𝑡∗}Φ_𝑛(−𝜏)sin𝜋𝑛𝑙𝑥+𝜆²₁𝑡^∗−𝜏²2!^∞_𝑛=1𝜋𝑛𝑙⁴𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡∗−𝜏)}Φ_𝑛(−𝜏)sin𝜋𝑛𝑙𝑥+⋯+𝜆^𝑘₁𝑡^∗−(𝑘−1)𝜏^𝑘𝑘!^∞_𝑛=1𝜋𝑛𝑙^2𝑘𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡∗−(𝑘−1)𝜏)}Φ_𝑛(−𝜏)sin𝜋𝑛𝑙𝑥. (4.18) 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 coefficients 𝐵_𝑛3(𝑡) at 𝑡=𝑡^∗∈[(𝑘−1)𝜏,𝑘𝜏]. Substituting 𝑡^∗−2𝜏−𝑠=𝜔, we obtain the following: 𝐵_𝑛3𝑡^∗=⁰_−𝜏𝜆₁𝜋𝑛𝑙²𝑒^{𝜎1(𝜋𝑛/𝑙)2𝜏}exp_𝜏𝑟_1𝑛,𝑡^∗−2𝜏−𝑠𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡∗−𝑠)}Φ_𝑛(𝑠)𝑑𝑠=𝜆₁𝜋𝑛𝑙²𝑒^{𝜎1(𝜋𝑛/𝑙)2𝜏𝑡∗}_𝑡^−𝜏_∗−2𝜏exp_𝜏𝑟_1𝑛,𝜔𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝜔+2𝜏)}Φ_𝑛𝑡^∗−2𝜏−𝜔𝑑𝜔. (4.19) We will split this integral in two: 𝐵_𝑛3𝑡^∗=𝜆₁𝜋𝑛𝑙²𝑒^{𝜎1(𝜋𝑛/𝑙)2𝜏}_𝑡^{(𝑘−2)𝜏}_∗−2𝜏exp_𝜏𝑟_1𝑛,𝜔𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝜔+2𝜏)}Φ_𝑛𝑡^∗−2𝜏−𝜔𝑑𝜔+𝜆₁𝜋𝑛𝑙²𝑒^{𝜎1(𝜋𝑛/𝑙)2𝜏𝑡∗}_()𝜏^−𝜏_𝑘−2exp_𝜏𝑟_1𝑛,𝜔𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝜔+2𝜏)}Φ_𝑛𝑡^∗−2𝜏−𝜔𝑑𝜔. (4.20) Therefore, owing to the mean value theorem, there are values 𝜔₁ and 𝜔₂ such that 𝑡^∗−2𝜏≤𝜔₁≤()()𝑘−2𝜏,𝑘−2𝜏≤𝜔₂≤𝑡^∗−𝜏, (4.21) and (using the Definition 2.1 of the delayed exponential function) we have 𝐵_𝑛3𝑡^∗=𝜆₁𝜋𝑛𝑙²𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝜔1+𝜏)}Φ_𝑛𝑡^∗−2𝜏−𝜔₁𝑘𝜏−𝑡^∗exp_𝜏𝑟_1𝑛,𝜔₁+𝜆₁𝜋𝑛𝑙²𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝜔2+𝜏)}Φ_𝑛𝑡^∗−2𝜏−𝜔₂𝑡^∗−(𝑘−1)𝜏exp_𝜏𝑟_1𝑛,𝜔₂=𝜆₁𝜋𝑛𝑙²𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝜔1+𝜏)}Φ_𝑛𝑡^∗−2𝜏−𝜔₁𝑘𝜏−𝑡^∗×1−𝜆₁𝜋𝑛𝑙²𝑒^{𝜎1(𝜋𝑛/𝑙)2𝜏}𝜔₁1!+𝜆²₁𝜋𝑛𝑙⁴𝑒^{2𝜎1(𝜋𝑛/𝑙)2𝜏}𝜔₁−𝜏²()2!+⋯+−1^𝑘−2𝜆₁^𝑘−2𝜋𝑛𝑙^2(𝑘−2)𝑒^{(𝑘−2)𝜎1(𝜋𝑛/𝑙)2𝜏}𝜔₁−(𝑘−3)𝜏^𝑘−2(𝑘−2)!+𝜆₁𝜋𝑛𝑙²𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝜔2+𝜏)}Φ_𝑛𝑡^∗−2𝜏−𝜔₂𝑡^∗−()𝜏×𝑘−11−𝜆₁𝜋𝑛𝑙²𝑒^{𝜎1(𝜋𝑛/𝑙)2𝜏}𝜔₂1!+𝜆²₁𝜋𝑛𝑙⁴𝑒^{2𝜎1(𝜋𝑛/𝑙)2𝜏}𝜔₂−𝜏²()2!+⋯+−1^𝑘−1𝜆₁^𝑘−1𝜋𝑛𝑙^2(𝑘−1)𝑒^{(𝑘−1)𝜎1(𝜋𝑛/𝑙)2𝜏}𝜔₂−(𝑘−2)𝜏^𝑘−1()!.𝑘−1 (4.22) Hence, for 𝑆₂₃𝑥,𝑡^∗∶=^∞_𝑛=1𝐵_𝑛3𝑡^∗sin𝜋𝑛𝑙𝑥, (4.23) we get 𝑆₂₃𝑥,𝑡^∗=^∞_𝑛=1𝐵_𝑛3𝑡^∗sin𝜋𝑛𝑙𝑥=^∞_𝑛=1𝜆₁𝜋𝑛𝑙²𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝜔1+𝜏)}Φ_𝑛×𝑡^∗−2𝜏−𝜔₁𝑘𝜏−𝑡^∗1−𝜆₁𝜋𝑛𝑙²𝑒^{𝜎1(𝜋𝑛/𝑙)2𝜏}𝜔₁1!+𝜆²₁𝜋𝑛𝑙⁴𝑒^{2𝜎1(𝜋𝑛/𝑙)2𝜏}×𝜔₁−𝜏²()2!+⋯+−1^𝑘−2𝜆₁^𝑘−2𝜋𝑛𝑙^2(𝑘−2)×𝑒^{(𝑘−2)𝜎1(𝜋𝑛/𝑙)2𝜏}𝜔₁−(𝑘−3)𝜏^𝑘−2(𝑘−2)!+𝜆₁𝜋𝑛𝑙²𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝜔2+𝜏)}Φ_𝑛𝑡^∗−2𝜏−𝜔₂𝑡^∗−()𝜏×𝑘−11−𝜆₁𝜋𝑛𝑙²𝑒^{𝜎1(𝜋𝑛/𝑙)2𝜏}𝜔₂1!+𝜆²₁𝜋𝑛𝑙⁴𝑒^{2𝜎1(𝜋𝑛/𝑙)2𝜏}𝜔₂−𝜏²+()2!+⋯−1^𝑘−1𝜆₁^𝑘−1𝜋𝑛𝑙^2(𝑘−1)𝑒^{(𝑘−1)𝜎1(𝜋n/𝑙)2𝜏}𝜔₂−(𝑘−2)𝜏^𝑘−1(𝑘−1)!sin𝜋𝑛𝑙𝑥. (4.24) After some rearranging, we get 𝑆₂₃𝑥,𝑡^∗=𝜆₁^∞_𝑛=1𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝜔1+𝜏)}Φ_𝑛𝑡^∗−2𝜏−𝜔₁𝑘𝜏−𝑡^∗+𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝜔2+𝜏)}Φ_𝑛𝑡^∗−2𝜏−𝜔₂𝑡^∗−(𝑘−1)𝜏𝜋𝑛𝑙²sin𝜋𝑛𝑙𝑥−𝜆²₁^∞_𝑛=1𝑒^{−𝜎1(𝜋𝑛/𝑙)2𝜔1}Φ_𝑛𝑡^∗−2𝜏−𝜔₁𝑘𝜏−𝑡^∗×𝜔₁1!+𝑒^{−𝜎1(𝜋𝑛/𝑙)2𝜔2}Φ_𝑛𝑡^∗−2𝜏−𝜔₂𝑡^∗−()𝜏𝜔𝑘−1²1!𝜋𝑛𝑙⁴sin𝜋𝑛𝑙𝑥+𝜆³₁^∞_𝑛=1𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝜔1−𝜏)}Φ_𝑛𝑡^∗−2𝜏−𝜔₁𝑘𝜏−𝑡^∗𝜔₁−𝜏²2!+𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝜔2−𝜏)}Φ_𝑛𝑡^∗−2𝜏−𝜔₂𝑡^∗−()𝜏𝜔𝑘−1²−𝜏²2!𝜋𝑛𝑙⁶sin𝜋𝑛𝑙𝑥+()+⋯−1^𝑘−2𝜆₁^∞𝑘−1_𝑛=1𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝜔1−(𝑘−3)𝜏)}Φ_𝑛𝑡^∗−2𝜏−𝜔₁𝑘𝜏−𝑡^∗×𝜔₁−(𝑘−3)𝜏^𝑘−2(𝑘−2)!+𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝜔2−(𝑘−3)𝜏)}Φ_𝑛𝑡^∗−2𝜏−𝜔₂×𝑡^∗−()𝜏𝜔𝑘−1²−(𝑘−3)𝜏^𝑘−2(𝑘−2)!𝜋𝑛𝑙^2𝑘−2sin𝜋𝑛𝑙𝑥+(−1)^𝑘−1𝜆^𝑘₁𝑡^∗−()𝜏𝜔𝑘−1²−(𝑘−2)𝜏^𝑘−1()!×𝑘−1_∞𝑛=1𝜋𝑛𝑙^2𝑘𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝜔2−(𝑘−2)𝜏)}Φ_𝑛𝑡^∗−2𝜏−𝜔₂sin𝜋𝑛𝑙𝑥. (4.25) Due to condition (4.2), we conclude that 𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝜔2−(𝑘−2)𝜏)}𝑛^2𝑘max_{−𝜏≤𝑡≤0}Φ_𝑛(𝑡≤𝑀)𝑛^3+𝛼. (4.26) 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 Series 𝑆₃(𝑥,𝑡)
We will consider the coefficients 𝐶_𝑛(𝑡), 𝑛=1,2,… of the series 𝑆₃(𝑥,𝑡) at 𝑡=𝑡^∗∈[(𝑘−1)𝜏,𝑘𝜏]. Substituting 𝑡^∗−𝜏−𝑠=𝜔, we obtain 𝐶_𝑛𝑡^∗=^𝑡∗₀exp_𝜏𝑟_1𝑛,𝑡^∗−𝜏−𝑠𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡∗−𝑠)}𝑓_𝑛(𝑠)=𝑑𝑠_{𝑡∗−𝜏−𝜏}exp_𝜏𝑟_1𝑛,𝜔𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝜔+𝜏)}𝑓_𝑛𝑡^∗=−𝜏−𝜔𝑑𝜔_0−𝜏exp_𝜏𝑟_1𝑛,𝜔𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝜔+𝜏)}𝑓_𝑛𝑡^∗+−𝜏−𝜔𝑑𝜔_𝜏0exp_𝜏𝑟_1𝑛,𝜔𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝜔+𝜏)}𝑓_𝑛𝑡^∗+−𝜏−𝜔𝑑𝜔_𝜏2𝜏exp_𝜏𝑟_1𝑛,𝜔𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝜔+𝜏)}𝑓_𝑛𝑡^∗−𝜏−𝜔𝑑𝜔+⋯+^𝑡∗_()𝜏^−𝜏_𝑘−2exp_𝜏𝑟_1𝑛,𝜔𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝜔+𝜏)}𝑓_𝑛𝑡^∗−𝜏−𝜔𝑑𝜔. (4.27) Owing to the mean value theorem, there are values 𝜔_𝑖, 𝑖=1,2,…,𝑘 such that −𝜏≤𝜔₁≤0,0≤𝜔₂≤𝜏,…,(𝑘−2)𝜏≤𝜔_𝑘≤𝑡^∗𝐶−𝜏_𝑛𝑡^∗=𝜏𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝜔1+𝜏)}𝑓_𝑛𝑡^∗−𝜏−𝜔₁+𝜏1−𝜆₁𝜋𝑛𝑙²𝑒^{𝜎1(𝜋𝑛/𝑙)2𝜏}𝜔₂𝑒1!^{−𝜎1(𝜋𝑛/𝑙)2(𝜔2+𝜏)}𝑓_𝑛𝑡^∗−𝜏−𝜔₂+𝜏1−𝜆₁𝜋𝑛𝑙²𝑒^{𝜎1(𝜋𝑛/𝑙)2𝜏}𝜔₃1!+𝜆²₁𝜋𝑛𝑙⁴𝑒^{2𝜎1(𝜋𝑛/𝑙)2𝜏}𝜔₃−𝜏²𝑒2!^{−𝜎1(𝜋𝑛/𝑙)2(𝜔3+𝜏)}×𝑓_𝑛𝑡^∗−𝜏−𝜔₃+⋯+𝜏1−𝜆₁𝜋𝑛𝑙²𝑒^{𝜎1(𝜋𝑛/𝑙)2𝜏}𝜔_𝑘−1+()1!+…−1^𝑘−2𝜆₁^𝑘−2𝜋𝑛𝑙^2(𝑘−2)𝑒^{(𝑘−2)𝜎1(𝜋𝑛/𝑙)2𝜏}×𝜔_𝑘−1−(𝑘−3)𝜏^𝑘−2()!𝑒𝑘−2^{−𝜎1(𝜋𝑛/𝑙)2(𝜔𝑘−1+𝜏)}×𝑓_𝑛𝑡^∗−𝜏−𝜔_𝑘−1+𝑡^∗−(𝑘−1)𝜏1−𝜆₁𝜋𝑛𝑙²𝑒^{𝜎1(𝜋𝑛/𝑙)2𝜏}𝜔_𝑘()1!+…+−1^𝑘−1×𝜆₁^𝑘−1𝜋𝑛𝑙^2(𝑘−1)𝑒^{(𝑘−1)𝜎1(𝜋𝑛/𝑙)2𝜏}𝜔_𝑘−(𝑘−2)𝜏^𝑘−1(𝑘−1)!×𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝜔𝑘+𝜏)}𝑓_𝑛𝑡^∗−𝜏−𝜔_𝑘. (4.28) Hence, 𝑆₃𝑥,𝑡^∗=^∞_𝑛=1𝐶_𝑛𝑡^∗sin𝜋𝑛𝑙𝑥=^∞_𝑛=1𝜏^𝑘−1_𝑖=1𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝜔𝑖+𝜏)}𝑓_𝑛𝑡^∗−𝜏−𝜔_𝑖+𝑡^∗−(𝑘−1)𝜏×𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝜔𝑘+𝜏)}𝑓_𝑛𝑡^∗−𝜏−𝜔_𝑘×sin𝜋𝑛𝑙𝑥−𝜆₁^∞_𝑛=1𝜏^𝑘−1_𝑖=2𝜔_𝑖𝑒1!^{−𝜎1(𝜋𝑛/𝑙)2𝜔𝑖}𝑓_𝑛𝑡^∗−𝜏−𝜔_𝑖+𝑡^∗−()𝜏×𝜔𝑘−1_𝑘𝑒1!^{−𝜎1(𝜋𝑛/𝑙)2𝜔𝑘}𝑓_𝑛𝑡^∗−𝜏−𝜔_𝑘×𝜋𝑛𝑙²sin𝜋𝑛𝑙𝑥+𝜆²₁^∞_𝑛=1𝜏^𝑘−1_𝑖=3𝜔_𝑖−𝜏²𝑒2!^{−𝜎1(𝜋𝑛/𝑙)2(𝜔𝑖−𝜏)}𝑓_𝑛𝑡^∗−𝜏−𝜔_𝑖+𝑡^∗−()𝜏𝜔𝑘−1^𝑘−𝜏²𝑒2!^{−𝜎1(𝜋𝑛/𝑙)2(𝜔𝑘−𝜏)}𝑓_𝑛𝑡^∗−𝜏−𝜔_𝑘×𝜋𝑛𝑙⁴sin𝜋𝑛𝑙𝑥+…+(−1)^𝑘−2𝜆₁^𝑘−2×^∞_𝑛=1𝜏𝜔_𝑘−1−(𝑘−3)𝜏^𝑘−2()!𝑒𝑘−2^{−𝜎1(𝜋𝑛/𝑙)2(𝜔𝑘−1−(𝑘−3)𝜏)}𝑓_𝑛𝑡^∗−𝜏−𝜔_𝑘−1+𝑡^∗−()𝜏𝜔𝑘−1^𝑘−(𝑘−3)𝜏^𝑘−2()!𝑒𝑘−2^{−𝜎1(𝜋𝑛/𝑙)2(𝜔𝑘−(𝑘−3)𝜏)}𝑓_𝑛𝑡^∗−𝜏−𝜔_𝑘×𝜋𝑛𝑙^2(𝑘−2)sin𝜋𝑛𝑙𝑥+(−1)^𝑘−1𝜆₁^𝑘−1𝑡^∗−()𝜏𝜔𝑘−1^𝑘−(𝑘−2)𝜏^𝑘−1()!×𝑘−1_∞𝑛=1𝜋𝑛𝑙^2(𝑘−1)𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝜔𝑘−(𝑘−2)𝜏)}𝑓_𝑛𝑡^∗−𝜏−𝜔_𝑘sin𝜋𝑛𝑙𝑥. (4.29)
Due to condition (4.2), we conclude that 𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝜔𝑘−(𝑘−2)𝜏)}𝑛^2(𝑘−1)max_{−𝜏≤𝑡≤𝑘𝜏}𝑓_𝑛(𝑡≤𝑀)𝑛^3+𝛼. (4.30) 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 for 𝜉(𝑥,𝑡)
Above, the absolute and uniform convergence of the series 𝑆₁(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 for 𝜉^′_𝑡(𝑥,𝑡) and 𝜉^′′_𝑥𝑥(𝑥,𝑡)
To prove the uniform convergence of the series for 𝜉^′_𝑡(𝑥,𝑡) and 𝜉^′′_𝑥𝑥(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 𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡∗−(𝑘−1)𝜏)}𝑛^2𝑘+2Φ_𝑛(≤𝑀−𝜏)𝑛^1+𝛼, (4.31) inequality (4.15) by 𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝑡∗−(𝑘−1)𝜏)}𝑛^2𝑘+2Φ_𝑛(0≤𝑀)𝑛^1+𝛼, (4.32) inequality (4.26) by 𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝜔2−(𝑘−2)𝜏)}𝑛^2𝑘+2max_{−𝜏≤𝑡≤0}Φ_𝑛(𝑡≤𝑀)𝑛^1+𝛼, (4.33) and inequality (4.30) by 𝑒^{−𝜎1(𝜋𝑛/𝑙)2(𝜔𝑘−(𝑘−2)𝜏)}𝑛^2𝑘max_{−𝜏≤𝑡≤𝑘𝜏}𝑓_𝑛(𝑡≤𝑀)𝑛^1+𝛼. (4.34)

The proof of the uniform convergence of the series, which represents the solution 𝜂(𝑥,𝑡) by formula (3.38), and the proof of the uniform convergence of the series for 𝜂^′_𝑥(𝑥,𝑡) and 𝜂^′′_𝑥𝑥(𝑥,𝑡) are much the same.

Corollary 4.2. Functions 𝑢(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 Theorem 4.1, we see that inequalities (4.2) for functions Φ_𝑛∶[−𝜏,0]→ℝ, 𝑛=1,2,…, and 𝑓_𝑛∶[−𝜏,∞)→ℝ, 𝑛=1,2,… are too restrictive if 𝑡^∗∈[(𝑘^∗−1)𝜏,(𝑘^∗−1)𝜏+𝜀], 𝑘^∗∈{1,2,…,𝑘}, where 𝜀 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+𝜀], 𝑘^∗∈{1,2,…,𝑘} remains open if, for example, Φ_𝑛∶[−𝜏,0]→ℝ, 𝑛=1,2,…, and 𝑓_𝑛∶[−𝜏,∞)→ℝ, 𝑛=1,2,… satisfy only the inequalities max_{−𝜏≤𝑡≤𝑘𝜏}𝑓_𝑛(𝑡≤𝑀)^∗𝑛,max_{−𝜏≤𝑡≤0}Φ_𝑛(𝑡≤𝑀)^∗𝑛 (4.35) 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+𝜀], 𝑘^∗∈{1,2,…,𝑘} and uniformly for 𝑥∈[0,𝑙]. In other words, in such a case, the series converge uniformly for (𝑥,𝑡)∈[0,𝑙]×[0,𝑘𝜏]⧵^𝑘_𝑘∗=1𝑘^∗−1𝜏,𝑘^∗−1𝜏+𝜀, (4.36) (where 𝜀>0 is fixed but arbitrarily small).

Acknowledgments

The paper was supported by the Grants P201/10/1032 and P201/11/0768 of the Czech Grant Agency (Prague), by the Council of Czech Government Grant MSM 00216 30519, and by the Grant FEKT-S-11-2-921 of Faculty of Electrical Engineering and Communication, Brno, University of Technology.