Citation: Davide Bellandi. On the initial value problem for a class of discrete velocity models[J]. Mathematical Biosciences and Engineering, 2017, 14(1): 31-43. doi: 10.3934/mbe.2017003
[1] | Marco Di Francesco, Simone Fagioli, Massimiliano D. Rosini . Many particle approximation of the Aw-Rascle-Zhang second order model for vehicular traffic. Mathematical Biosciences and Engineering, 2017, 14(1): 127-141. doi: 10.3934/mbe.2017009 |
[2] | Francesca Marcellini . The Riemann problem for a Two-Phase model for road traffic with fixed or moving constraints. Mathematical Biosciences and Engineering, 2020, 17(2): 1218-1232. doi: 10.3934/mbe.2020062 |
[3] | Léon Masurel, Carlo Bianca, Annie Lemarchand . Space-velocity thermostatted kinetic theory model of tumor growth. Mathematical Biosciences and Engineering, 2021, 18(5): 5525-5551. doi: 10.3934/mbe.2021279 |
[4] | Le Li, Lihong Huang, Jianhong Wu . Flocking and invariance of velocity angles. Mathematical Biosciences and Engineering, 2016, 13(2): 369-380. doi: 10.3934/mbe.2015007 |
[5] | Yoichi Enatsu, Yukihiko Nakata, Yoshiaki Muroya . Global stability for a class of discrete SIR epidemic models. Mathematical Biosciences and Engineering, 2010, 7(2): 347-361. doi: 10.3934/mbe.2010.7.347 |
[6] | John Cleveland . Basic stage structure measure valued evolutionary game model. Mathematical Biosciences and Engineering, 2015, 12(2): 291-310. doi: 10.3934/mbe.2015.12.291 |
[7] | Max-Olivier Hongler, Roger Filliger, Olivier Gallay . Local versus nonlocal barycentric interactions in 1D agent dynamics. Mathematical Biosciences and Engineering, 2014, 11(2): 303-315. doi: 10.3934/mbe.2014.11.303 |
[8] | Nawaz Ali Zardari, Razali Ngah, Omar Hayat, Ali Hassan Sodhro . Adaptive mobility-aware and reliable routing protocols for healthcare vehicular network. Mathematical Biosciences and Engineering, 2022, 19(7): 7156-7177. doi: 10.3934/mbe.2022338 |
[9] | Hans F. Weinberger, Xiao-Qiang Zhao . An extension of the formula for spreading speeds. Mathematical Biosciences and Engineering, 2010, 7(1): 187-194. doi: 10.3934/mbe.2010.7.187 |
[10] | Carole Guillevin, Rémy Guillevin, Alain Miranville, Angélique Perrillat-Mercerot . Analysis of a mathematical model for brain lactate kinetics. Mathematical Biosciences and Engineering, 2018, 15(5): 1225-1242. doi: 10.3934/mbe.2018056 |
In recent years the mathematical modeling based on suitable generalizations of kinetic theory has shown its ability to describe many of the features characterizing the phenomenon of complexity [14]. These features include, among others, the emergence of collective behaviors that are not immediately deducible from the mutual microscopic interactions among the individual constituents. The systems exhibiting aspects typical of complexity that have been successfully approached by kinetic modeling include social [1] and biological systems [6,12,15], as well as pedestrian crowds [7] and vehicular traffic flow [4,5,8]. The phenomenology of the latter reveals a number of features, such as stop-and-go waves and ghost queues [17], that makes vehicular traffic a prototype of complex systems.
Models referring to generalized kinetic theory describe the system under investigation using a distribution function
This paper deals with the existence and uniqueness of solutions to a large class of discrete velocity kinetic models related the mathematical description of vehicular traffic flow. The question of well-posedness is of fundamental relevance both for a proper selection of mathematically meaningful class of models and for the robustness of their numerical implementation and simulation. A basic requirement for any admissible mathematical model of vehicular flow is that it should posses global (i.e., for large times) existence and uniqueness of solutions in any traffic condition, provided the interaction terms are "sufficiently reasonable". It is worth observing that the mathematical problem related to the well-posedness for large times of discrete velocities traffic models is a difficult task due to the nonlinearity of interactions and to their hyperbolic multi-characteristics aspect. This issue has been addressed in [2], while the recent, more comprehensive, results are given in [3], which is the starting point of the present work. In the latter the Authors obtain global in time existence of solutions under general hypotheses. Their approach, however, requires a key tool the introduction of an a priori cut off on the interaction terms that freezes the interactions when the traffic density reaches a critical value. As the Authors of [3] themselves emphasize in their concluding remarks, such an assumption, though mimicking the phenomenological observations that show the existence of different phases in the traffic flow corresponding to different densities [17], should possibly appear as a byproduct of the model rather than being introduced into it from the very beginning in order to achieve the well-posedness.
In this paper we prove global existence and uniqueness under hypotheses similar to that in [3], but under weaker assumptions on the structure of the interaction terms. Specifically, we prove global in time existence and uniqueness of solution without assuming that the interaction terms as the density of vehicles becomes large. In Sect. 2 the mathematical problem is formulated and the main hypotheses on the interaction terms are introduced. The main results of this paper are given in Sect. 3 where existence and uniqueness of solutions for any time is proven in the case of non-local interactions. Finally, Sect. 4 reports a toy model that explicitly shows how the global existence results of Sects. 3 are in fact more general than those available in the existing literature, and ends with some concluding remarks and perspectives.
In this paper we study the initial-value problem for a class of discrete velocities models of the form:
{∂tfi(t,x)+vi∂xfi(t,x)=Ji(f)(t,x),fi(0,x)=¯fi(x),i=1,…,n. | (1) |
Such a class of models, here applied to the study of vehicles traffic flow, find applications al in the context of mathematical biology.
Here the vector distribution function
fi=fi(t,x):R+×R→R+, i=1…,n, |
and the velocity can assume only a finite set
ρ(t,x):=n∑i=1fi(t,x), | (2) |
such that
Ji(f)=Γi(f)−fiΛi(f), | (3) |
where:
Γi(f)(t,x):=n∑h,k=1fh(t,x)∫Dxη(ρ(t,y))Bihk(ρ(t,y))fk(t,y)dy, | (4) |
is a gain term that describes the inflow of the
Λi(f)(t,x):=n∑h=1∫Dxη(ρ(t,y))fh(t,y)dy. | (5) |
is the corresponding loss term, i.e., the flow of cars leaving the
Dx=[x−Δ−,x+Δ+], |
where
•
•
From the definition of
Bihk(ρ)≥0,n∑i=1Bihk(ρ)=1,∀ρ≥0,∀h,k=1,…,n. | (6) |
hold true. Thanks to (6) we have that:
n∑i=1Ji(f)=0. | (7) |
This relation reflects the absence in the system of proliferative and destructive effects, and will lead to the conservation of the total number of vehicles, as we shall see later on.
Now we state the basic assumptions we need in order to get local and global existence of solutions for (1).
• The encounter rate
|η(ρ1)−η(ρ2)|≤Lη|ρ1−ρ2|, ∀ρ1,ρ2∈R,|η(ρ)|≤Cη,∀ρ∈R. | (8) |
• Transition probability density
|Bihk(ρ1)−Bihk(ρ2)|≤CBihk,r|ρ1−ρ2|, ∀ρ1,ρ2∈R with |ρ1|<r,|ρ2|<r, | (9) |
and verifies (6).
These hypotheses are crucial in order to have global solutions to system (1) in a suitable space.
The well-posedness of problem (1) in the spatially homogeneous case is well known. We refer to the paper [13], in which the result is achieved under similar hypotheses. Here we study the non-homogeneous problem, pointing out that (1) is a system of
ξ=γi(τ,t,x)=x+vi(τ−t). |
Finding no problem for existence of characteristics we can introduce the idea of mild solution, presented in [16] for kinetic equations, that is related integrability properties of
Definition 3.1. Let
fi(t,x)=¯fi(γi(0,t,x))+∫t0Ji(f(τ,γi(τ,t,x)))dτ, | (10) |
for all
We put:
X=(L1(R)∩L∞(R))n, |
and define:
‖u‖X:=‖u‖1+‖u‖∞,u∈X, |
where:
‖u‖∞:=maxi‖ui‖L∞, and ‖u‖1:=n∑i=1‖ui‖L1. |
It is easy to verify that
Lemma 3.2. Let
Proof. We have to prove that if
‖J(f)‖X=‖J(f)‖1+‖J(f)‖∞<+∞. |
We start analyzing the
∫R|Ji(f(x))|dx≤∫R|Γi(f(x))|dx+∫R|fi(x)Λi(f(x))|dx. |
Evaluating the gain term, we find
∫R|Γi(f(x))|dx≤n∑h,k=1∫R|fh(x)|(∫Dx|η(ρ(y))Bihk(ρ(y))fk(y)|dy)dx≤n∑h,k=1‖fh‖L1∫R|η(ρ(y))Bihk(ρ(y))fk(y)|dy, |
and using the boundedness of hich gives the boundedness of
∫R|Γi(f(x))|dx≤n∑h,k=1Cη‖fh‖L1∫R|fk(y)|dy=n∑h,k=1Cη‖fh‖L1‖fk‖L1<+∞. |
Similarly, we have:
∫R|fi(x)Λi(f(x))|dx≤‖fi‖L1n∑h=1∫R|η(ρ(y))fh(y)|dy≤Cη‖fi‖L1n∑h=1∫R|fh(y)|dy≤Cη‖fi‖n∑h=1‖fh‖<+∞. |
Coming to analyze the
|Γi(f(x))|≤n∑h,k=1‖fh‖L∞∫Dx|η(ρ(y))Bihk(ρ(y))fk(y)|dy≤Cηn∑h,k=1‖fh‖L∞‖fk‖L1<+∞, |
which gives the boundedness of
|fi(x)Λi(f(x))|≤Cη‖fi‖L∞n∑h,k=1‖fh‖L1<+∞, |
which ends the proof.
Lemma 3.3. Let
Proof. Let
|Ji(f(x))−Ji(g(x))|≤|Γi(f(x))−Γi(g(x))|+|fi(x)Λi(f(x))−gi(x)Λi(g(x))|. |
Now:
|Γi(f(x))−Γi(g(x))|≤n∑h,k=1|fh(x)∫Dxη(ρf(y))Bihk(ρf(y))fk(y)dy−gh(x)∫Dxη(ρg(y))Bihk(ρg(y))gk(y)dy|≤n∑h,k=1|fh(x)−gh(x)|∫R|η(ρf(y))Bihk(ρf(y))fk(y)|dy+n∑h,k=1|gh(x)|∫R|η(ρf(y))Bihk(ρf(y))fk(y)−η(ρg(y))Bihk(ρg(y))gk(y)|dy=I1(x)+I2(x). |
Reminding that, by 9, the quantities
I1(x)≤Cη|fh(x)−fk(x)|∫Rfk(y)dy≤Cηn∑h,k=1‖fk‖L1|fh(x)−gh(x)|≤Cηnrn∑h=1|fh(x)−gh(x)|, |
from which we have
‖I1‖L1≤Cηnr‖f−g‖1, | (11) |
‖I1‖L∞≤Cηn2r‖f−g‖∞. | (12) |
For
I2(x)≤n∑h,k=1|gh(x)|(∫R|η(ρf(y))Bihk(ρf(y))−η(ρg(y))Bihk(ρg(y))||fk(y)|dy+∫R|Bihk(ρg(y))η(ρg(y))(fk(y)−gk(y))|dy)≤n∑h,k=1|gh(x)|(‖fk‖L∞∫R(Lη+LBCη)|ρf(y)−ρg(y)|dy+Cη∫R|fk(y)−gk(y)|dy)≤rn(Lη+(1+Lβr)Cη)‖f−g‖1n∑h=1|gh(x)|, |
with
‖I2‖L1≤r2n(Lη+(1+Lβr)Cη)‖f−g‖1, | (13) |
‖I2‖L∞≤r2n2(Lη+(1+Lβr)Cη)‖f−g‖1. | (14) |
In a similar way we have
|fi(x)Λi(f)(x)−gi(x)Λi(g)(x)|≤n∑h=1|fi(x)−gi(x)|∫R|η(ρf(y))fh(y)|dy+n∑h=1|gi(x)|∫R|η(ρf(y))fh(y)−η(ρg(y))gh(y)|dy, |
which leads to the following inequalities
‖fiΛi(f)−giΛi(g)‖L∞≤Cηr‖f−g‖∞+(Cηr+Lηr2)‖f−g‖1, | (15) |
and
‖fiΛi(f)−giΛi(g)‖L1≤(2Cηr+Lηr2)‖f−g‖1. | (16) |
Combining the previous inequalities we conclude the proof.
The previous lemma is crucial in order to use a fixed point argument in the following theorem.
Theorem 3.4 (Local existence and uniqueness). Let
Proof. Let
Dr(a):={u∈C([0,a],X)|‖u‖∗:=supt∈[0,a]‖u(t)‖X≤r}. |
Let
Φ(u)i(t,x):=¯fi(γi(0,t,x))+∫t0Ji(u(τ,γi(τ,t,x)))(τ,γi(τ,t,x))dt. | (17) |
Clearly
‖Φ(u)(t)‖X≤δ+∫t0(‖J(u(τ))−J(0)‖X+‖J(0)‖X)ds, |
and thus, reminding that
‖Φ(u)(t)‖X≤δ+aLr, |
with
‖Φ(u)(t)−Φ(v)(t)‖X≤∫t0‖J(u(τ))−J(v(τ))‖Xdτ≤aLr‖u−v‖∗, |
and thus:
‖Φ(u)−Φ(v)‖∗≤aLr‖u−v‖∗. | (18) |
Then, for every
Theorem 3.4 states that if the initial datum
N(t):=∫Rρ(t,x)dx=∫R(n∑i=1fi(t,x))dx, | (19) |
Integrating on
n∑i=1∫R(∂tfi(t,x)+vi∂xfi(t,x))dx=0, | (20) |
from which:
n∑i=1(∫R∂tfi(t,x)dx+vi[fi(t,x))]+∞−∞)=n∑i=1∫R∂tfi(t,x)dx=0, | (21) |
and finally:
dNdt(t)=0, | (22) |
that is, the total vehicles number is conserved and is equal to its initial value
The following proposition shows that if the system has non-negative initial data then the solution remains non-negative.
Lemma 3.5. Let
Proof.If
Ji(¯f(x))=Γi(¯f(x))≥0, |
and thus along characteristics they are non-decreasing functions, while remaining components remains non-negative for a certain time interval, always along characteristics. If
The previous proposition, together with the conservation of the total number of vehicles, ensures that:
‖f(t)‖1=n∑i=1∫R|fi(t,x)|dx=n∑i=1∫Rfi(t,x)dx=N0. | (23) |
In Theorem 3.4 we established existence and uniqueness of local in-time solutions to (1). Putting:
T¯f:=sup(T>0|∃u∈C([0,T],X) solution of (1)), |
then, stitching together the solutions, we obtain the existence of a unique maximal mild solution
Lemma 3.6. Let
limt→T¯f‖f(t)‖X=+∞ |
Proof. First of all we observe that the length
Thanks to the previous Lemma we have only to verify that:
limt→T¯f‖u(t)‖X<+∞. |
Moreover, since (23) says that the norm
|fi(t,x)|=fi(t,x)=¯fi(γi(0,t,x))+∫t0Ji(f(τ,γi(τ,t,x)))dτ≤¯fi(γi(0,t,x))+∫t0Γi(f(τ,γi(τ,t,x)))dτ≤‖¯fi‖L∞+n∑h,k=1Cη∫t0fh(τ,γi(τ,t,x))∫Dγi(τ,t,x)fk(τ,y)dydτ≤‖¯fi‖L∞+n∑h=1Cη∫t0fh(τ,γi(τ,t,x))∫Rn∑k=1fk(τ,y)dydτ≤‖¯f‖∞+Cη∫t0N(τ)n∑h=1fh(τ,γi(τ,t,x))dτ≤‖¯f‖∞+CηN0∫t0n∑h=1‖fh(τ)‖L∞dτ≤‖¯f‖∞+nCηN0∫t0‖f(τ)‖∞dτ, |
and thus:
‖f(t)‖∞≤‖¯f‖∞+nCηN0∫t0‖f(τ)‖∞dτ. |
Using the Gronwall's lemma we find:
‖f(t)‖∞≤‖¯f‖∞exp(nCηN0t), |
which gives us the bound for
Theorem 3.7 (Global existence and uniqueness). Let
We conclude discussion about well-posedness of Cauchy problem (1), with the following theorem which gives countinuous dependence on initial data.
Theorem 3.8. Let
‖f(t)−g(t)‖X≤eLt‖¯f−¯g‖X, |
for all
Proof. Fixed
f(t)∈Bδ(0),g(t)∈Bδ(0), for all t∈[0,b], |
Reminding that
‖f(t)−g(t)‖X≤‖¯f−¯g|‖X+∫t0‖J(f(τ))−J(g(τ))‖Xdτ≤‖¯f−¯g|‖X+L∫t0‖f(τ)−g(τ)‖Xdτ, |
using Gronwall's inqueality we conclude the proof.
To the best of our knowledge the most general result so far available concerning the well-posedness of (1) is in [3], where the Authors study the initial value problem:
{∂tfi(t,x)+vi∂xfi(t,x)=χ(ρ(t,x)≤ρc)Ji(f)(t,x),fi(0,x)=¯fi(x),i=1,…,n. | (24) |
Here the interaction terms
χ(ρ≤ρc)={1 if ρ≤ρc,0 if ,ρ>ρc | (25) |
where
The aim of this section is to show, by a numerical example, that situations can occur in which the assumption that the interactions freeze at high density seems to be restrictive. In the sequel we furnish a toy model in which, given an arbitrary
‖¯f‖∞≤ρc, ¯ρ(x)=n∑i=1¯fi(x)≤ρc, ∀x∈R, |
for which there exist
B1hk=(1110), B2hk=(0001),h,k=1,2. |
and
{∂tf1(t,x)=f2(t,x)∫Dxf1(t,y)dy,∂tf2(t,x)+∂xf2(t,x)=−f2(t,x)∫Dxf1(t,y)dy. | (26) |
We solve numerically the Cauchy problem for (26) relative to initial data
1.05 | 4.5425 | 5.1890 |
1.00 | 4.5885 | 5.2214 |
0.95 | 4.6365 | 5.2660 |
0.90 | 4.6855 | 5.3269 |
0.85 | 4.7365 | 5.4092 |
0.80 | 4.7925 | 5.5192 |
0.75 | 4.8555 | 5.6652 |
0.70 | 4.9255 | 5.8575 |
0.65 | 5.0025 | 6.1090 |
0.60 | 5.4595 | 6.4658 |
0.55 | 5.4985 | 6.9726 |
0.50 | 5.5406 | 7.5731 |
0.45 | 5.5786 | 8.3056 |
0.40 | 5.6106 | 9.2196 |
0.35 | 5.6346 | 10.3606 |
0.30 | 5.6536 | 11.7122 |
0.25 | 5.6706 | 13.0508 |
0.20 | 5.6846 | 13.7962 |
0.15 | 5.6926 | 13.3582 |
In Figure 2, the final configurations for different values of
In summary, the well-posedness results of Section. 3 complete those of [3], furnishing global existence and uniqueness of solutions to (1) in its general form, i.e., in any traffic condition. Along the lines of what the Authors of [3] point out in their concluding remarks, in the present paper we succeeded in overcoming the constraint assumption that the r.h.s. of (1) disappears when the density is higher than a prescribed value
Finally, we point out the generalization of the proof method used in this paper toward the qualitative analysis of models based on the so-called kinetic theory of active particles (KTAP) [6], a more general framework though leading to mathematical structures similar to those here considered, as a challenging research perspective, in view of the ability of KTAP to design models for a wider class of complex systems. However, it is worth to advise that such a goal is far from being a trivial adaptation, due to additional difficulties such as stronger nonlinearity and nonlinear additivity of interactions [10], all aspects related to the inherent complexity of the above mentioned phenomena.
[1] | [ G. Ajmone Marsan,N. Bellomo,A. Tosin, null, Complex Systems and Society: Modeling and Simulation, Springer, 2013. |
[2] | [ L. Arlotti,E. De Angelis,L. Fermo,M. Lachowicz,N. Bellomo, On a class of integro-differential equations modeling complex systems with nonlinear interactions, Appl.Math. Lett., 25 (2012): 490-495. |
[3] | [ N. Bellomo,A. Bellouquid, Global solution to the Cauchy problem for discrete velocity models of vehicular traffic, J. Differ. Equations, 252 (2012): 1350-1368. |
[4] | [ N. Bellomo,V. Coscia,M. Delitala, On the mathematical theory of vehicular traffic fow Ⅰ -Fluid dynamic and kinetic modeling, Math. Mod. Meth. Appl. Sci., 12 (2002): 1801-1843. |
[5] | [ N. Bellomo,C. Dogbé, On the modelling of traffic and crowds -a survey of models, speculations and perspectives, SIAM Rev., 53 (2011): 409-463. |
[6] | [ N. Bellomo,D. Knopoff,J. Soler, On the difficult interplay between life, "complexity", and mathematical sciences, Math. Mod. Meth. Appl. Sci., 23 (2013): 1861-1913. |
[7] | [ N. Bellomo, B. Piccoli and A. Tosin, Modeling crowd dynamics from a complex system viewpoint Math. Mod. Meth. Appl. Sci. 22 (2012), 1230004, 29pp. |
[8] | [ A. Bellouquid, E. De Angelis and L. Fermo, Towards the modeling of vehicular traffic as a complex system: A kinetic theory approach Math. Mod. Meth. Appl. Sci. 22(2012), 1140003, 35pp. |
[9] | [ A. Bellouquid,M. Delitala, null, Mathematical Modeling of Complex Biological Systems. A Kinetic Theory Approach, Birkhäuser, Boston, 2006. |
[10] | [ A. Benfenati,V. Coscia, Nonlinear microscale interactions in the kinetic theory of active particles, Appl. Math. Lett., 26 (2013): 979-983. |
[11] | [ V. Coscia,M. Delitala,P. Frasca, On the mathematical theory of vehicular traffic flow Ⅱ: Discrete velocity kinetic models, Int. J. Non-Linear Mech., 42 (2007): 411-421. |
[12] | [ V. Coscia,L. Fermo,N. Bellomo, On the mathematical theory of living systems Ⅱ: The interplay between mathematics and system biology, Comput. Math. Appl., 62 (2011): 3902-3911. |
[13] | [ M. Delitala,A. Tosin, Mathematical modeling of vehicular traffic: A discrete kinetic theory approach, Math. Mod. Meth. Appl. Sci., 17 (2007): 901-932. |
[14] | [ L. Arlotti, N. Bellomo, E. De Angelis and M. Lachowicz, Generalized Kinetic Models in Applied Sciences World Scientific, New Jersey, 2003. |
[15] | [ J. Banasiak and M. Lachowicz Methods of Small Parameter in Mathematical Biology Birkhauser, 2014. |
[16] | [ S. Kaniel,M. Shinbrot, The Boltzmann equation. Ⅰ. Uniqueness and local existence, Math. Phys., 58 (1978): 65-84. |
[17] | [ B. S. Kerner, The Physics of Traffic, Empirical Freeway Pattern Features Engineering Applications and Theory, Springer, 2004. |
[18] | [ P. Lax, Hyperbolic Partial Differential Equations Courant Lecture Notes, 2006. |
1.05 | 4.5425 | 5.1890 |
1.00 | 4.5885 | 5.2214 |
0.95 | 4.6365 | 5.2660 |
0.90 | 4.6855 | 5.3269 |
0.85 | 4.7365 | 5.4092 |
0.80 | 4.7925 | 5.5192 |
0.75 | 4.8555 | 5.6652 |
0.70 | 4.9255 | 5.8575 |
0.65 | 5.0025 | 6.1090 |
0.60 | 5.4595 | 6.4658 |
0.55 | 5.4985 | 6.9726 |
0.50 | 5.5406 | 7.5731 |
0.45 | 5.5786 | 8.3056 |
0.40 | 5.6106 | 9.2196 |
0.35 | 5.6346 | 10.3606 |
0.30 | 5.6536 | 11.7122 |
0.25 | 5.6706 | 13.0508 |
0.20 | 5.6846 | 13.7962 |
0.15 | 5.6926 | 13.3582 |
1.05 | 4.5425 | 5.1890 |
1.00 | 4.5885 | 5.2214 |
0.95 | 4.6365 | 5.2660 |
0.90 | 4.6855 | 5.3269 |
0.85 | 4.7365 | 5.4092 |
0.80 | 4.7925 | 5.5192 |
0.75 | 4.8555 | 5.6652 |
0.70 | 4.9255 | 5.8575 |
0.65 | 5.0025 | 6.1090 |
0.60 | 5.4595 | 6.4658 |
0.55 | 5.4985 | 6.9726 |
0.50 | 5.5406 | 7.5731 |
0.45 | 5.5786 | 8.3056 |
0.40 | 5.6106 | 9.2196 |
0.35 | 5.6346 | 10.3606 |
0.30 | 5.6536 | 11.7122 |
0.25 | 5.6706 | 13.0508 |
0.20 | 5.6846 | 13.7962 |
0.15 | 5.6926 | 13.3582 |