Representation of Classical Solutions to a Linear Wave Equation with Pure Delay

Representation of Classical Solutions to a Linear Wave Equation with Pure Delay

Denys Ya. Khusainov111Department of Cybernetics, Kyiv National Taras Shevchenko University, Ukraine, Michael Pokojovy222Department of Mathematics and Statistics, University of Konstanz, Germany, Elvin I. Azizbayov333Department of Mechanics and Mathematics, Baku State University, Azerbaijan
July 16, 2019
Abstract

For a wave equation with pure delay, we study an inhomogeneous initial-boundary value problem in a bounded 1D domain. Under smoothness assumptions, we prove unique existence of classical solutions for any given finite time horizon and give their explicit representation. Continuous dependence on the data in a weak extrapolated norm is also shown.

1 Introduction

The wave equation is a typical linear hyperbolic second-order partial differential equation which naturally arises when modeling phenomena of continuum mechanics such as sound, light, water or other kind of waves in acoustics, (electro)magnetics, elasticity and fluid dynamics, etc. (cf. [6, 13]). Providing a rather adequate description of physical processes, partial differential equations, or equations with distributed parameters in general, have found numerous applications in mechanics, medicine, ecology, etc. Introducing after-effects such as delay into these equations has gained a lot of attention over several past decades. See e.g., [2, 3, 7, 8]. Mathematical treatment of such systems requires additional carefulness since distributed systems with delay often turn out to be even ill-posed (cf. [4, 5, 12]).

In the present paper, we consider an initial-boundary value problem for a general linear wave equation with pure delay and constant coefficients in a bounded interval subject to non-homogeneous Dirichlet boundary conditions. To solve the equation, we employ Fourier’s separation method as well as the special functions referred to as delay sine and cosine functions which were introduced in [9, 10]. We prove the existence of a unique classical solution on any finite time interval, show its continuous dependence on the data in a very weak extrapolated norm, give its representation as a Fourier series and prove its absolute and uniform convergence under certain conditions on the data.

2 Equation with pure delay

For , , we consider the following linear wave equation in a bounded interval with a single delay being a second order partial difference-differential equation for an unknown function

 ∂ttη(t,x)=a2∂xxη(t−τ,x)+b∂xη(t−τ,x)+dη(t−τ,x)+g(t,x) for (t,x)∈(0,T)×(0,l) (2.1)

subject to non-homogeneous Dirichlet boundary conditions and initial conditions

 η(t,0)=θ1(t),η(t,l)=θ2(t) for t>−τ,η(t,x)=ψ(t,x) for (t,x)∈(−τ,0)×(0,l). (2.2)

Since we are interested in studying classical solutions, the following compatibility conditions are required to assure for smoothness of the solution on the boundary of time-space cylinder

 ψ(t,0)=θ1(t),ψ(t,l)=θ2(t) for t>−τ.
Definition 2.1.

Under a classical solution to the problem (2.1), (2.2) we understand a function which satisfies as well as and, being plugged into Equations (2.1), (2.2), turns them into identity.

Remark 2.2.

The previous definition does not impose any continuity of time derivatives in . If the continuity or even smoothness are desired, additional compatibility conditions on the data, including , are required.

Let , , denote the standard Sobolev norm (cf. [1]) and denote the norm of corresponding negative Sobolev space. We introduce the norm and define the Hilbert space as a completion of with respect to . Obviously, , i.e., can be continuously embedded into the space of distributions.

With this notation, we easily see that (with denoting the distributional derivative) is a bounded linear operator on since

 ∥A∥L(X)=sup∥u∥X=1∥Au∥X=sup∥u∥X=1 ⎷∞∑k=0∥a2∂2xu+b∂xu+du∥2−k,2≤sup∥u∥X=1∞∑k=0(a2∥u∥−k−2,2+b∥u∥−k−1,2+d∥u∥−k,2)≤sup∥u∥X=1(a2+b+c)∥u∥X=a2+b+d.
Theorem 2.3.

There exists a constant , dependent only on , such that the estimate

 maxt∈[0,T](∥η(⋅,t)∥2X+∥ηt(⋅,t)∥2X)≤C(∥ψ(0,⋅)∥2X+∥ψt(0,⋅)∥2X)+C∫0−τ(∥ψ(s,⋅)∥2X+∥ψt(s,⋅)∥2X)dt+=C∫T0(∥g(s,⋅)∥2X+|θ1(s)|2+|θ2(s)|2)ds

holds true for any classical solution of Equations (2.1), (2.2).

Proof.

Let be the classical solution to Equations (2.1), (2.2). We define

 w(t,x):={η(t,x) for (t,x)∈[−τ,0]×[0,l],η(t,x)−θ1(t)−xl(θ2(t)−θ1(t)) for (t,x)∈(0,T]×[0,l].

Then satisfies homogeneous Dirichlet boundary conditions and solves the equation

 ∂ttw(t,⋅)=Aw(t−τ,⋅)+f(t,⋅) (2.3)

in the extrapolated space with

 f(t,⋅)=g(t,⋅)+b(θ2(t)−θ1(t))+θ1(t)+dxl(θ2(t)−θ1(t)).

We multiply the equation with in the scalar product of and use Young’s inequality to get the estimate

 ∂t∥wt(t,⋅)∥2X=⟨Aw(t−τ,⋅),wt(t,⋅)⟩X+⟨f(t,⋅),wt(t,⋅)⟩X≤∥w(t−τ,⋅)∥2X+(1+∥A∥2L(X))∥wt(t,⋅)∥2X+∥f(t,⋅)∥2X. (2.4)

As in [11], we introduce the history variable

 z(s,t,x):=w(t−s,x) for (s,t,x)∈[0,τ]×[0,T]×[0,l]

and obtain

 zt(s,t,x)+zs(s,t,x)=0 for (s,t,x)∈(0,τ)×(0,T)×(0,l).

Multiplying these identities with in and performing a partial integration, we find

 ∂t∫τ0∥z(s,t,⋅)∥2Xds=−∫τ0∂s∥z(s,t,⋅)∥2Xds=∥w(t,⋅)∥2X−∥w(t−τ,⋅)∥2X. (2.5)

Adding equations (2.4) and (2.5) to the trivial identity

 ∂t∥w(t,⋅)∥2X≤∥w(t,⋅)∥2X+∥wt(t,⋅)∥2X,

we obtain

 ∂t(∥w(t,⋅)∥2X+∥wt(t,⋅)∥2X+∫0−τ∥w(s,t,⋅)∥2Xds)≤(2+∥A∥L(X))∥w(t,⋅)∥2X+∥wt(t,⋅)∥2X+∥f(t,⋅)∥2X.

Thus, we have shown

 ∂tE(t)≤(2+∥A∥L(X))E(t)+∥f(t,⋅)∥2X, (2.6)

where

 E(t):=∥w(t,⋅)∥2X+∥wt(t,⋅)∥2X+∫0−τ∥z(s,t,⋅)∥Xds.

From Equation (2.6) we conclude

 E(t)≤E(0)+(2+∥A∥L(X))∫t0E(s)ds+∫t0∥f(t,⋅)∥2Xds.

Using now the integral form of Gronwall’s inequality, we obtain

 E(t)≤E(0)+∫t0∥f(t,⋅)∥2Xds+∫t0e(2+∥A∥L(X))(t−s)(E(0)+∫s0∥f(ξ,⋅)∥2Xdξ)ds≤~C(E(0)+∫T0∥f(s,⋅)∥2Xds) (2.7)

for certain . Taking into account

for some constants and exploiting the definition of , the proof is a direct consequence of Equation (2.7). ∎

Corollary 2.4.

Solutions of Equation (2.1), (2.2) are unique. The solution map

 (ψ,g,θ1,θ2)↦η

is well-defined, linear and continuous in the norms from Theorem 2.3.

Remark 2.5.

It was essential to consider the weak space . If the space corresponding to the usual wave equation is used, i.e., , there follows from [5] that Equation (2.1), (2.2) is an ill-posed problem due to the lack of continuous dependence on the data even in the homogeneous case.

Next, we want to establish conditions on the data allowing for the existence of a classical solution. Performing the substitution

 ξ(t,x):=e−b2a2xη(t,x) for (t,x)∈[−τ,T]×[0,l] (2.8)

with a new unknown function (cp. [11]), the initial boundary value problem (2.1), (2.2) can be written in the following simplified form with a self-adjoint operator on the right-hand side

 ∂ttξ(t,x)=a2∂xxξ(t−τ,x)+cξ(t−τ,x)+f(t,x) for (t,x)∈(0,T)×(0,l) (2.9)

with complemented by the following boundary and initial conditions

 ξ(t,0) =μ1(t),ξ(t,l)=μ2(t) for t>−τ with μ1(t):=θ1(t),μ2(t):=eb2a2lθ2, (2.10) ξ(t,x) =φ(t,x) for (t,x)∈(−τ,0)×(0,l) with % φ(t,x):=eb2a2xψ(t,x) (2.11)

and

 f(t,x):=eb2a2xg(t,x) for (t,x)∈[0,T]×[0,l].

The solution will be determined in the form

 ξ(t,x)=ξ0(t,x)+ξ1(t,x)+G(t,x).

Here, is an arbitrary function with satisfying the boundary conditions

 G(t,0)=μ1(t),G(t,l)=μ2(t).

Assuming , we let

 G(t,x):=μ1(t)+xl(μ2(t)−μ1(t)) for (t,x)∈[−τ,T]×[0,l]. (2.12)
• solves the homogeneous equation

 ∂ttξ0(t,x)=a2∂xxξ0(t−τ,x)+cξ0(t−τ,x) (2.13)

subject to homogeneous boundary and non-homogeneous initial conditions

 ξ0(t,0)≡0,ξ0(t,l)=0 in (−τ,T),ξ0(t,x)=Φ(t,x) for (t,x)∈(−τ,0)×(0,l) with Φ(t,x):=φ(t,x)−G(t,x). (2.14)

In particular, with the function selected as in Equation (2.12), we obtain

 Φ(t,x)=φ(t,x)−μ1(t)−xl(μ2(t)−μ1(t)) for (t,x)∈[−τ,0]×[0,l]. (2.15)
• solves the non-homogeneous equation

 ∂ttξ1(t,x)=a2∂xxξ1(t−τ,x)+cξ1(t−τ,x)+F(t,x) for (t,x)∈(0,T)×(0,l) (2.16)

with

 F(t,x):=a2∂xxG(t−τ,x)+cG(t−τ,x)−∂ttG(t,x) % for (t,x)∈[0,T]×[0,l] (2.17)

subject to homogeneous boundary and initial conditions. For from Equation (2.12), we have

 F(t,x)=f(t,x)+c(μ1(t−τ)+xl(μ2(t−τ)−μ1(t−τ)))−(¨μ1(t)+xl(¨μ2(t)−¨μ1(t))). (2.18)

3 Homogeneous equation

In this section, we obtain a formal solution to the initial-boundary value problem (2.13) with initial and boundary conditions given in Equations (2.10), (2.11). We exploit Fourier’s separation method to determine in the product form . After plugging this ansatz into Equation (2.13), we find

 X(x)¨T(t)=a2X′′(x)T(t−τ)+cX(x)T(t−τ).

Hence,

 X(x)(¨T(t)−cT(t−τ))=a2X′′(x)T(t−τ).

By formally separating the variables, we deduce

 X′′(x)X(x)=¨T(t)−cT(t−τ)a2T(t−τ)=−λ2.

Thus, the equation can be decoupled as follows

 ¨T(t)+(a2λ2−c)T(t−τ)=0,X′′(x)+λ2X(x)=0. (3.1)

These are linear second order ordinary (delay) differential equations with constant coefficients.

Due to the zero boundary conditions for , the boundary conditions for the second equation in (3.1) will also be homogeneous, i.e.,

 X(0)=0,X(l)=0.

Therefore, we obtain a Sturm & Liouville problem admitting non-trivial solutions only for the eigennumbers

 λ2=λ2n=(πnl)n for n∈N

and the corresponding eigenfunctions

 Xn(x)=sin(πnlx) for n∈N.

Assuming

 (πal)2−c>0,

we denote

 ωn=√(πna)2−c for n∈N

and consider the first equation in (3.1), i.e.,

 ¨T(t)+ω2nT(t−τ)=0 for n∈N. (3.2)

The initial conditions for each of the equations in (3.2) can be obtained by expanding the initial data into a Fourier series with respect to the eigenfunction basis of the second equation in (3.1)

 (3.3)

for . Let us further determine the solution of the Cauchy problem associated with each of the equations in (3.2) subject to the initial conditions from Equation (3.3).

First, we briefly present some useful results from the theory of second order delay differential equations with pure delay obtained in [9]. The authors considered a linear homogeneous second order delay differential equation

 ¨x(t)+ω2x(t−τ)=0 for t∈(0,∞),x(t)=β(t) for t∈[−τ,0]. (3.4)

They introduced two special functions referred to as delay cosine and sine functions. Exploiting these functions, a unique solution to the initial value problem (3.4) was obtained.

Definition 3.1.

Delay cosine is the function given as

 (3.5)

with -order polynomials on each of the intervals continuously adjusted at the nodes , .

Definition 3.2.

Delay sine is the function given as

 cosτ(ω,t)=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩0,−∞

with -order polynomials on each of the intervals continuously adjusted at the nodes , .

There has further been proved that delay cosine uniquely solves the linear homogeneous second order ordinary delay differential equation with pure delay subject to the unit initial conditions in and the delay sine in its turn solves Equation (3.4) subject to the initial conditions for .

Using the fact above, the solution of the Cauchy problem was represented in the integral form. In particular, the solution to the homogeneous delay differential equation (3.4) with the initial conditions in for an arbitrary was shown to be given as

 x(t)=β(−τ)cosτ(ω,t)+1ω˙β(−τ)sinτ(ω,t)+1ω∫0−τsinτ(ω,t−τ−s)¨β(s)ds. (3.7)

Turning back to the delay differential equation (3.2) with the initial conditions (2.4), we obtain their unique solution in the form

 Tn(t)=Φn(−τ)cosτ(ωn,t)+1ωn˙Φn(−τ)sinτ(ωn,t)+1ωn∫0−τsinτ(ωn,t−τ−s)¨Φ(s)ds. (3.8)

Thus, assuming sufficient smoothness of the data to be specified later, the solution to the homogeneous equation (2.13) satisfying homogeneous boundary and non-homogeneous initial conditions in reads as

 ξ0(t,x)=∞∑n=1(Φn(−τ)cosτ(ωn,t)+1ωn˙Φn(−τ)sinτ(ωn,t)+1ωn∫0−τsinτ(ωn,t−τ−s)¨Φn(s)ds)sin(πnlx),Φn(t)=2l∫l0(φ(t,s)−G(t,s))sin(πnls)ds for n∈N. (3.9)

4 Non-homogeneous equation

Next, we consider the non-homogeneous equation (2.16) with the right-hand side from Equation (2.18) subject to homogeneous initial and boundary conditions

 ∂ttξ1(t,x)=a2∂xxξ1(t−τ,x)+cξ1(t−τ,x)+F(t,x) for (t,x)∈(0,T)×(0,l),

where

 F(t,x)=f(t,x)+c(μ1(t−τ)+xl(μ2(t−τ)−μ1(t−τ)))−(¨μ1(t)−xl(¨μ2(t)−¨μ1(t))).

The solution will be constructed as as Fourier series with respect to the eigenfunctions of the Sturm & Liouville problem from the previous section, i.e.,

 ξ1(t,x)=∞∑n=1Tn(t)sin(πnlx). (4.1)

Plugging the ansatz from (4.1) into Equation (2.6) and comparing the time-dependent Fourier coefficients, we obtain a system of countably many second order delay differential equations

 ¨Tn(t)+ωnTn(t−τ)=Fn(t) for t∈(0,T) % with Fn(t)=2l∫l0F(t,s)sin(πnlx)ds. (4.2)

In [9], the initial value problem for the non-homogeneous delay differential equation

 ¨x(t)+ω2x(t−τ)=f(t) for t≥0

with homogeneous initial conditions in was shown to be uniquely solved by

 x(t)=∫t0sinτ(ω,t−τ−s)f(s)ds. (4.3)

Exploiting Equation (4.3), the equations in (4.2) subject to zero initial conditions are uniquely solved by

 Tn(t)=∫t0sinτ(ωn,t−τ−s)Fn(s)ds. (4.4)

Therefore, the non-homogeneous partial delay differential equation with homogeneous initial and boundary condition formally reads as

 ξ1(t,x)=∞∑n=1(∫t0sinτ(ωn,t−τ−s)Fn(s)ds)sin(πnlx) for % (t,x)∈[0,T]×[0,l]. (4.5)

5 General case solution

The solution in the general case can thus formally be represented as the following series

 (5.1)

5.1 Convergence of the Fourier series

Theorem 5.1.

Let , and . Further, let the data functions , , and be such that their Fourier coefficients and given in Equations (3.3) and (4.5) satisfy the conditions

 limn→∞(|Φn(−τ)|+|˙Φn(−τ)|)n2m+3+α=0,limn→∞maxs∈[−τ,0]|¨Φn|n2m+3+α=0,limn→∞maxk=1,…,mmaxt∈[(k−1)τ,max{kτ,T}]|Fn(t)|n2k+3+α=0 (5.2)

for an arbitrary, but fixed . Let

 (πla)2>c.

Then the classical solution to problem (2.9)–(2.11) can be represented as an absolutely and uniformly convergent Fourier series given in Equation (5.1). The latter series is a twice continuously differentiable function with respect to both variables. Its derivatives of order less or equal two with respect to and can be obtained by a term-wise differentiation of the series and the resulting series are also absolutely and uniformly convergent in .

Proof.

We regroup the series from Equation (5.1) into the following sum

 ξ(t,x)=S1(t,x)+S2(t,x)+S3(t,x)+G(t,x),

where

 S1(t,x)=∞∑n=1An(t)sin(πnlx),S2(t,x)=∞∑n=1Bn(t)sin(πnlx),S3(t,x)=∞∑n=1Cn(t)sin(πnlx),An(t)=Φn(−τ)cosτ(ωn,t)+1ωn˙Φn(−τ)sinτ(ωn,t),Bn(t)=1ωn∫0−τsinτ(ωn,t−τ−s)¨Φnds,Cn(t)=1ωn∫0−τsinτ(ωn,t−τ−s)Fn(s)ds

and

 ωn=√(πnla)2−c for n∈N.
1. First, we consider the coefficient functions . For an arbitrary with , we find

 An(t)=Φn(−τ)cosτ(ωn,t)+1ωn˙Φn(−τ)sinτ(ωn,t)=(1−(πnla)2t32!+⋯+(−1)k(πnla)2k(t−(k−1)τ)2k(2k)!)Φn(−τ)+=;((1+τ)−(πnla)2t23!+⋯+(−1)k(πnla)2k(t−(k−1)τ)2k+1(2k+1)!)Φn(−τ).

If and , , are such that the condition

 limn→∞(|Φn(−τ)|+|˙Φn(−τ)|)n2k+3+α=0

holds true, the series as well as its derivatives of order less or equal 2 converge absolutely and uniformly. Note that a single differentiation with respect to corresponds, roughly speaking, to a multiplication with .

2. Next, we consider the coefficients . For an arbitrary with , we perform the substitution and exploit the mean value theorem to estimate

Applying the theorem on differentiation under the integral sign to and taking into account that is twice weakly differentiable in , namely: , its derivatives are polynomials of order lower than those of and their convolution with is continuous, analogous estimates can be obtained for and which, in their turn, also follow to be continuous functions.

Now, if the condition

 limn→∞maxs∈[−τ,0]|¨Φn|n2m+3+α=0

is satisfied, the series as well as its derivatives of order less or equal 2 converge absolutely and uniformly.

3. Finally, we look at the Fourier coefficients . Again, for an arbitrary period of time with , , we substitute . Once again, using the mean value theorem, we estimate

 |Cn(t)|=∣∣∣1ωn∫tt−τsinτ(πnla,ξ)Fn(t−τ−ξ)dξ∣∣∣≤τmaxt−τ≤s≤t|¨Φn(s)|maxj=k−1,kmaxt−τ≤s≤t∣∣∣(s−τ)−(πnla)2s33!+…+(−1)j(πnla)2j(s−(j−1)τ)2j+1(2j+1)!∣∣∣.

As before, can be shown to be twice continuously differentiable. If now

 limn→∞maxk=1,…,mmaxt∈[(k−1)τ,max{kτ,T}]|Fn(t)|n2k+3+α=0 (5.3)

is satisfied, then both and its derivatives of order less or equal 2 converge absolutely and uniformly.

Since all three conditions are guaranteed by the assumptions of the Theorem due to the fact , the proof is finished. ∎

Remark 5.2.

From the practical point of view, the rapid decay condition on the Fourier coefficients of the data given in Equation (5.2) mean a sufficiently high Sobolev regularity of the data and corresponding higher order compatibility conditions at the boundary of (cf. [11]).

References

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