# Analytic Solutions for a Functional Differential Equation Related to a Traffic Flow Model

Authors: Houyu Zhao
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We study the existence of analytic solutions of a functional differential equation(z(s)+α)2z'(s)=β(z(s+z(s))-z(s)) which comes from traffic flow model. By reducing the equation with the Schröder transformation to an auxiliary equation, the author discusses not only that the constant λ at resonance, that is, at a root of the unity, but also those λ near resonance under the Brjuno condition.


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## Body

## 1. Introduction

Traffic flow models have found much attention [1–13] in the last few years. They mostly fall into two types: one is “macroscopic” which was introduced by Aw and Rascle [14], and Zhang [13], we also refer the reader to [2–5, 9, 12], and the other is called “microscopic” which has been discussed in [8, 10, 11, 15]. In particular, Illner et al. [4, 7] investigated kinetic models which can be seen as a bridge between macroscopic and microscopic models.Recently,   Illner and McGregor [6] studied
(1.1)(z(s)+α)2z′(s)=β(z(s+z(s))-z(s)),
where α, β are positive parameters arising from a nonlocal traffic flow model in a travelling wave approximation. Analytical and numerical studies of (1.1) exist, in particular on the existence and properties of nonconstant travelling wave solutions.In this paper, we prove the existence of analytic solutions for (1.1) by locally reducing the equation to another functional differential equation, which we called auxiliary equation. In fact, if we let g(s)=s+z(s), then z(s)=g(s)-s, and (1.1) can be written in
(1.2)(g(s)-s+α)2(g′(s)-1)=β(g(g(s))-2g(s)+s).
As in [16, 17], we reduce (1.2) with g(s)=h(λh-1(s)) to the auxiliary equation
(1.3)(h(λs)-h(s)+α)2(h′(λs)-h′(s))=βh′(s)(h(λ2s)-2h(λs)+h(s)),s∈ℂ.
If we can prove the existence of analytic solutions for (1.3), then the analytic solutions of (1.1) can be obtained.Throughout this paper, we will assume thatα,β>0, and λ in (1.3) satisfies one of the following conditions:(C1)
0<|λ|<1;(C2)
λ=e2πiθ, θ∈ℝ∖ℚ, and θ is a Brjuno number [18, 19]: B(θ)=∑n=0∞‍(logqn+1/qn)<∞, where {pn/qn} denotes the sequence of partial fraction of the continued fraction expansion of θ;(C3)
λ=e2πiq/p for some integer p∈ℕ with p≥2 and q∈ℤ∖{0}, and λ≠e2πiξ/v for all 1≤v≤p-1 and ξ∈ℤ∖{0}.We observe thatλ is inside the unit circle S1 in case  (C1) but on S1 in the rest of cases. More difficulties are encountered for λ on S1 since the small divisor λn-1 is involved in the latter (2.24). Under Diophantine condition, “λ=e2πiθ, where θ∈ℝ∖ℚ and there exist constants ζ>0 and δ>0 such that|λn-1|≥ζ-1n-δ for all n≥1,” the number λ∈S1 is “far” from all roots of the unity. Since then, we have been striving to give a result of analytic solutions for those λ “near” a root of the unity, that is, neither being roots of the unity nor satisfying the Diophantine condition. The Brjuno condition in  (C2) provides such a chance for us. Moreover, we also discuss the so-called resonance case, that is, the case of  (C3).


## 2. Analytic Solutions of the Auxiliary Equation

In this section, we discuss local invertible analytic solutions of (1.3) with the initial condition
(2.1)h(s)=∑n=1∞‍ansn,h(0)=0,h′(0)=η≠0,η∈ℂ.Lemma 2.1.
Equation (1.3) has a formal solution of the form
(2.2)h(s)=ηs+∑n=2∞‍ansn,
where η is  as in (2.1).Proof.
If we leth(s)=∑n=1∞‍ansn and substituting into (1.3), we have
(2.3)α2∑n=0∞‍(n+1)(λn-1)an+1sn+2α∑n=1∞‍(∑i=0n-1‍(i+1)(λn-i-1)(λi-1)an-iai+1)sn+∑n=2∞‍(∑i=0n-2‍∑k=1n-i-1‍(i+1)(λn-k-i-1)(λk-1)(λi-1)an-k-iakai+1)sn=β∑n=1∞‍(∑i=0n-1‍(i+1)(λn-i-1)2an-iai+1)sn.
Comparing coefficients we obtain
(2.4)α2(λ0-1)a1=0,(2.5)α2(n+1)(λn-1)an+1=β∑i=0n-1‍(i+1)(λn-i-1)2an-iai+1-2α∑i=0n-1‍(i+1)(λn-i-1)(λi-1)an-iai+1-∑i=0n-2‍∑k=1n-i-1‍(i+1)(λn-k-i-1)(λk-1)(λi-1)an-k-iakai+1,n≥1.
Then for arbitrarily chosen a1=η≠0, the sequence {an}n=2∞ is successively determined by (2.5) in a unique manner.This shows that (1.3) has a formal power series solution of the form (2.2).Theorem 2.2.
Suppose  that (C1) holds, then (1.3) in a neighborhood of the origin has an analytic solution of the form (2.2).Proof.
From  (C1), we have(2.6)limn→∞1|λn-1|=1.
There exists L>0 such that 1/|λn-1|≤L, for all n≥1. It follows from (2.5) that
(2.7)|an+1|≤8Lα2[(α+β)∑i=0n-1‍|an-i||ai+1|+∑i=0n-2‍∑k=1n-i-1‍|an-k-i||ak||ai+1|]
for n≥1.
We consider the implicit function equation:(2.8)B(s)=|η|s+8Lα2[(α+β)B2(s)+B3(s)].
Define the function
(2.9)Θ(s,ω;L,η,α,β)=|η|s-ω+8Lα2[(α+β)ω2+ω3]
for (s,ω) from a neighborhood of (0,0), then the function B(s) satisfies
(2.10)Θ(s,B(s);L,η,α,β)=0.
In view of Θ(0,0;L,η,α,β)=0,
(2.11)Θω′(0,0;L,η,α,β)=-1≠0,
and the implicit function theorem, there exists a unique function Φ(s), analytic in a neighborhood of zero, such that
(2.12)Φ(0)=0,Φ′(0)=-Θs′(0,0;L,η,α,β)Θω′(0,0;L,η,α,β)=|η|,
and Θ(s,Φ(s);L,η,α,β)=0. According to (2.10), we have B(s)=Φ(s).
If we assume that the power series expansion ofB(s) is as follows:
(2.13)B(s)=∑n=1∞‍Bnsn,B1=|η|,
substituting the series in (2.10) and comparing coefficients, we obtain B1=|η| and
(2.14)Bn+1=8Lα2[(α+β)∑i=0n-1‍Bn-iBi+1+∑i=0n-2‍∑k=1n-i-1‍Bn-k-iBkBi+1],n≥2.
From (2.7) we obtain immediately that |an|≤Bn for all n by induction. This implies that (2.2) converges in a neighborhood of the origin. This completes the proof.Next we devote to the existence of analytic solutions of (1.3) under the Brjuno condition. First, we recall briefly the definition of Brjuno numbers and some basic facts. As stated in [20], for a real number θ, we let [θ] denote its integer part and {θ}=θ-[θ] its fractional part. Then every irrational number θ has a unique expression of the Gauss' continued fraction:
(2.15)θ=d0+θ0=d0+1d1+θ1=⋯,
denoted simply by θ=[d0,d1,…,dn,…], where dj's and θj's are calculated by the algorithm: (a) d0=[θ], θ0={θ}, and (b) dn=[1/θn-1],  θn={1/θn-1} for all n≥1. Define the sequences (pn)n∈ℕ and (qn)n∈ℕ as follows:
(2.16)q-2=1,q-1=0,qn=dnqn-1+qn-2,p-2=0,p-1=1,pn=dnpn-1+pn-2.
It is easy to show that pn/qn=[d0,d1,…,dn]. Thus, for every θ∈ℝ∖ℚ we associate, using its convergence, an arithmetical function B(θ)=∑n≥0‍(logqn+1/qn). We say that θ is a Brjuno number or that it satisfies Brjuno condition if B(θ)<+∞. The Brjuno condition is weaker than the Diophantine condition. For example, if dn+1≤cedn for all n≥0, where c>0 is a constant, then θ=[d0,d1,…,dn,…] is a Brjuno number but is not a Diophantine number. So the case  (C2) contains both Diophantine condition and a part of λ “near” resonance. Let θ∈ℝ∖ℚ and (qn)n∈ℕ be the sequence of partial denominators of the Gauss's continued fraction for θ. As in [20], let
(2.17)Ak={n≥0∣∥nθ∥≤18qk},Ek=max(qk,qk+14),ηk=qkEk.
Let Ak* be the set of integers j≥0 such that either j∈Ak or for some j1 and j2 in Ak, with j2-j1<Ek, one has j1<j<j2 and qk divides j-j1. For any integer n≥0, define
(2.18)lk(n)=max((1+ηk)nqk-2,(mnηk+n)1qk-1),
where mn=max{j∣0≤j≤n,j∈Ak*}. We then define function hk:ℕ→ℝ+ as follows:
(2.19)hk(n)={mn+ηknqk-1,ifmn+qk∈Ak*,lk(n),ifmn+qk∉Ak*.
Let gk(n):=max(hk(n),[n/qk]), and define k(n) by the condition qk(n)≤n≤qk(n)+1. Clearly, k(n) is nondecreasing. Then we are able to state the following result.Lemma 2.3 (Davie's lemma [21]).
LetK(n)=nlog2+∑k=0k(n)‍gk(n)log(2qk+1). Then(a)
there is a universal constantϱ>0 (independent of n and θ) such that
(2.20)K(n)≤n(∑k=0k(n)‍logqk+1qk+ϱ),(b)
K(n1)+K(n2)≤K(n1+n2) for all n1 and n2, and(c)
-log|λn-1|≤K(n)-K(n-1).Theorem 2.4.
Suppose  that (C2) holds, then (1.3) has an analytic solution of the form (2.2) in a neighborhood of the origin.Proof.
As in the Theorem2.2, we seek a power series solution of the form (2.2). First, we have
(2.21)|an+1|≤1|λn-1|8α2[(α+β)∑i=0n-1‍|an-i||ai+1|+∑i=0n-2‍∑k=1n-i-1‍|an-k-i||ak||ai+1|]
for n≥2.
To construct a majorant series, we consider the implicit functional equation:(2.22)Θ(s,ψ;L,η,α,β)=0,
where Θ is defined in (2.9) and L=1. Similarly to the proof of Theorem 2.2, using the implicit function theorem we can prove that (2.22) has a unique analytic solution ψ(s) in a neighborhood of the origin such that ψ(0)=0,  ψ′(0)=|η| and Θ(s,ψ(s);L,η,α,β)=0. Thus ψ(s) in (2.22) can be expanded into a convergent series:
(2.23)ψ(s)=∑n=1∞‍Bnsn,
in a neighborhood of the origin. Replacing (2.23) into (2.22) and comparing coefficients, we obtain that B1=|η| and
(2.24)Bn=8α2[(α+β)∑i=0n-1‍Bn-iBi+1+∑i=0n-2‍∑k=1n-i-1‍Bn-k-iBkBi+1],n≥2.
Note that the series (2.23) converges in a neighborhood of the origin. Now, we can deduce, by induction, that |an|≤BneK(n-1) for n≥1, where K:ℕ→ℝ is defined in Lemma 2.3.
In fact,|a1|=|η|=B1. For inductive proof we assume that |aj|≤BjeK(j-1), for j=1,2,…,n. From (2.21) we know
(2.25)|an+1|≤1|λn-1|8α2[(α+β)∑i=0n-1‍|an-i||ai+1|+∑i=0n-2‍∑k=1n-i-1‍|an-k-i||ak||ai+1|]≤1|λn-1|8α2[(α+β)∑i=0n-1‍Bn-iBi+1eK(n-i-1)+K(i)1|λn-1|8α2+∑i=0n-2‍∑k=1n-i-1‍Bn-k-iBkBi+1eK(n-k-i-1)+K(k-1)+K(i)].
Note that(2.26)K(n-i-1)+K(i)≤K(n-1),K(n-k-i-1)+K(k-1)+K(i)≤K(n-1).
Then from Lemma 2.3, we have
(2.27)|an+1|≤eK(n-1)|λn-1|8α2[(α+β)∑i=0n-1‍Bn-iBi+1+∑i=0n-2‍∑k=1n-i-1‍Bn-k-iBkBi+1],n≥1.
Since ∑n=1∞‍Bnsn is convergent in a neighborhood of the origin, there exists a constant Λ>0 such that
(2.28)Bn<Λn,n≥1.
Moreover, from Lemma 2.3, we know that K(n)≤n(B(θ)+ϱ) for some universal constant ϱ>0. Then
(2.29)|an|≤BneK(n-1)≤Λne(n-1)(B(θ)+ϱ),
that is,
(2.30)limn→∞sup(|an|)1/n≤limn→∞sup(Λne(n-1)(B(θ)+ϱ))1/n=ΛeB(θ)+ϱ.
This implies that the convergence radius of (2.2) is at least (ΛeB(θ)+ϱ)-1. This completes the proof.In the case  (C3) both the Diophantine condition and Brjuno condition are not satisfied. We need to define a sequence{Cn}n=1∞ by C1=|η| and
(2.31)Cn+1=8Γα2[(α+β)∑i=0n-1‍Cn-iCi+1+∑i=0n-2‍∑k=1n-i-1‍Cn-k-iCkCi+1],n≥1,
where Γ:=max{1,1/|1-λ|,1/|1-λ2|,…,1/|1-λ(p-1)|}, and p is defined in  (C3).Theorem 2.5.
Assume that  (C3) holds. Let{an}n=0∞ be determined by a1=η and
(2.32)α2(n+1)(λn-1)an+1=Ξ(n,λ),n≥1,
where
(2.33)Ξ(n,λ)=β∑i=0n-1‍(i+1)(λn-i-1)2an-iai+1-2α∑i=0n-1‍(i+1)(λn-i-1)(λi-1)an-iai+1-∑i=0n-2‍∑k=1n-i-1‍(i+1)(λn-k-i-1)(λk-1)(λi-1)an-k-iakai+1.
If Ξ(lp,λ)=0 for all l=1,2,…, then (1.3) has an analytic solution of the form
(2.34)h(s)=ηs+∑n=lp,l∈ℕ‍μlp+1slp+1+∑n≠lp,l∈ℕ‍an+1sn+1,ℕ={1,2,3,…}
in a neighborhood of the origin, where all μlp+1,s are arbitrary constants satisfying the inequality |μlp+1|≤Clp+1 and the sequence {Cn}n=1∞ is defined in  (2.31). Otherwise, if Ξ(lp,α)≠0 for some l=1,2,…, then (1.3) has no analytic solutions in any neighborhood of the origin.Proof.
Analogously to the proof of Lemma2.1, let  (2.2) be the expansion of a formal solution h(s) of  (1.3); we also have  (2.5) or  (2.32). If Ξ(lp,λ)≠0 for some natural number l, then the equality in  (2.32) does not hold for n=lp since λlp-1=0. In such a circumstance (1.3) has no formal solutions.
IfΞ(lp,λ)=0 for all natural numbers l, then there are infinitely many choices of corresponding alp+1 in  (2.32) and the formal solutions  (2.2) form a family of functions of infinitely many parameters. We can arbitrarily choose alp+1=μlp+1 such that |μlp+1|≤Clp+1,  l=1,2,…. In what follows we prove that the formal solution  (2.2) converges in a neighborhood of the origin. First of all, note that |λn-1|-1≤Γ, for n≠lp. It follows from  (2.32) that
(2.35)|an+1|≤8Γα2[(α+β)∑i=0n-1‍|an-i||ai+1|+∑i=0n-2‍∑k=1n-i-1‍|an-k-i||ak||ai+1|],
for all n≠lp,  l=1,2,…. Further, we can prove that
(2.36)|an|≤Cn,n=1,2,….
In fact, for inductive proof we assume that |ar|≤Cr for all 1≤r≤n. When n=lp, we have |an+1|=|μn+1|≤Cn+1. On the other hand, when n≠lp, from  (2.36) we get
(2.37)|an+1|≤8Γα2[(α+β)∑i=0n-1‍Cn-iCi+1+∑i=0n-2‍∑k=1n-i-1‍Cn-k-iCkCi+1]=Cn+1
as desired. Set
(2.38)F(s)=∑n=1∞‍Cnsn,C1=|η|.
It is easy to check that (2.38) satisfies
(2.39)Θ(x,F;Γ,η,α,β)=0,
where the function Θ is defined in (2.9). Moreover, similarly to the proof of Theorem 2.2, we can prove that (2.39) has a unique analytic solution F(s) in a neighborhood of the origin such that F(0)=0 and F′(0)=|η|≠0. Thus, (2.38) converges in a neighborhood of the origin. By the convergence of (2.38) and inequality (2.36), the series (2.2) converges in a neighborhood of the origin. This completes the proof.


## 3. Analytic Solutions of (1.2)

Theorem 3.1.
Suppose that conditions of Theorems2.2, 2.4, or 2.5 are fulfilled. Then (1.2) has an invertible analytic solution of the form
(3.1)g(s)=h(λh-1(s))
in a neighborhood of the origin, where h(s) is an analytic solutions of  (1.3) satisfying the initial conditions  (2.1).Proof.
In a view of Theorems2.2–2.5, we may find an analytic solution h(s) of the auxiliary equation (1.3) in the form of (2.2) such that h(0)=0 and h′(0)=η≠0. Clearly the inverse h-1(s) exists and is analytic in a neighborhood of the h(0)=0. Define
(3.2)g(s)∶=h(λh-1(s)).
Then g(s) is invertible analytic in a neighborhood of s=0. From (3.2) it is easy to see
(3.3)g(0)=h(λh-1(0))=h(0)=0,g′(0)=λh′(λh-1(0))(h-1)′(0)=λh′(λh-1(0))h′(h-1(0))=λh′(0)h′(0)=λ≠0.
From (1.3), we have
(3.4)(g(s)-s+α)2(g′(s)-1)=(h(λh-1(s))-h(h-1(s))+α)2(h′(λh-1(s))h′(h-1(s))-1)=β(h(λ2h-1(s))-2h(λh-1(s))+h(h-1(s)))=β(g(g(s))-2g(s)+s)
as required. This completes the proof.By Theorem3.1, we have shown that under the conditions of Theorems 2.2, 2.4, or 2.5, (1.2) has an analytic solution g(s)=h(λh-1(s)) in a neighborhood of the number 0, where h(s) is an analytic solution of (1.3). Since the function h(s) in (2.2) can be determined by (2.5), it is possible to calculate, at least in theory, the explicit form of h(s), an analytic solution of (1.3), in a neighborhood of the fixed point 0 of h(s). However, knowing that an analytic solution of (1.3) exists, we can take an alternative route as follows.Example 3.2.
Consider(3.5)(z(s)+1)2z′(s)=z(s+z(s))-z(s),
where
(3.6)α=β=1.
If taking λ=1/2, then by (1.2) and (1.3), we have
(3.7)(g(s)-s+1)2(g′(s)-1)=g(g(s))-2g(s)+s,
and the auxiliary equation is
(3.8)(h(12s)-h(s)+1)2(h′(12s)-h′(s))=h′(s)(h(14s)-2h(12s)+h(s)),s∈ℂ.From Lemma2.1, a1=η≠0 is given arbitrarily, and an,  n≥2 can be determined by
(3.9)(n+1)(12n-1)an+1=∑i=0n-1‍(i+1)(12n-i-1)2an-iai+1-2∑i=0n-1‍(i+1)(12n-i-1)(12i-1)an-iai+1-∑i=0n-2‍∑k=1n-i-1‍(i+1)(12n-k-i-1)(12k-1)(12i-1)an-k-iakai+1,n≥1.Now, by (3.9),
(3.10)a2=h′′(0)2!=-14η2,a3=h′′′(0)3!=-136a1a2=1144η3,⋯.
Because h(0)=0, h′(0)=η≠0, and the inverse h-1(s) is analytic near the origin, we can calculate that
(3.11)(h-1)′(0)=1h′(h-1(0))=1η,(h-1)′′(0)=-h′′(h-1(0))(h-1)′(0)(h′(h-1(0)))2=12η,(h-1)′′′(0)=-[h′′′(h-1(0))((h-1)′(0))2+h′′(h-1(0))(h-1)′′(0)](h′(h-1(0)))2(h′(h-1(0)))4+h′′(h-1(0))(h-1)′(0)·2h′(h-1(0))h′′(h-1(0))(h-1)′(0)(h′(h-1(0)))4=(5/24)η3+(1/2)η3η4=1724·1η⋯.
Now, we have
(3.12)g(0)=h(12h-1(0))=h(0)=0,g′(0)=h′(12h-1(0))12(h-1)′(0)=12h′(0)(h-1)′(0)=12,g′′(0)=14h′′(12h-1(0))[(h-1)′(0)]2+12h′(12h-1(0))(h-1)′′(0)=18,g′′′(0)=18h′′′(12h-1(0))[(h-1)′(0)]3+12h′′(12h-1(0))(h-1)′(0)(h-1)′′(0)+14h′′(12h-1(0))(h-1)′(0)(h-1)′′(0)+12h′(12h-1(0))(h-1)′′′(0)=18h′′′(0)[(h-1)′(0)]3+34h′′(0)(h-1)′(0)(h-1)′′(0)+12h′(0)(h-1)′′′(0)=1164,⋯.Thus, near the origin, (3.7) has an analytic solution:
(3.13)g(s)=12s-116s2+11384s3+⋯,z(s)=g(s)-s=-12s-116s2+11384s3+⋯is the analytic solution of (3.5).Remark 3.3.
If we restrict our arguments to the real number field, then by Theorem3.1, (1.1) has an invertible analytic real solution. We can define a real sequence {bm}m=0∞ and obtain a solution h(s) of the form of (2.2) with real coefficients. Restricted on ℝ both the function h(s) and its inverse are valued in ℝ. Hence, the function g(s)=h(λh-1(s)) is also a real function and Theorem 3.1 implies its invertible analyticity.



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