Chapter 6. First Order PDEs. 6.1 Characteristics The Simplest Case. u(x,t) t=1 t=2. t=0. Suppose u(x, t) satisfies the PDE.


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1 Chaper 6 Firs Order PDEs 6.1 Characerisics The Simples Case Suppose u(, ) saisfies he PDE where b, c are consan. au + bu = 0 If a = 0, he PDE is rivial (i says ha u = 0 and so u = f(). If a = 0, i reduces o u + cu = 0 where c = b/a. (6.1) We know from 5.4 ha he soluion is f( c). This represens a wave ravelling in he direcion wih speed c, and wih consan shape. u(,) =0 =1 =2 c We now ake a new and more general poin of view. The equaion (6.1) can be wrien u + d d u = 0, or d u + d u = 0 (6.2) where d/d = c. Then inegraing gives = ξ + c where ξ is an inegraion consan such ha = ξ when = 0. Page 57. c Universiy of Brisol This maerial is copyrigh of he Universiy unless eplicily saed oherwise. I is provided
2 6.1. CHARACTERISTICS From he chain rule, u s = s u + s u = 0 (6.3) where s is a parameer lin. indep. of ξ. We hink of (ξ, s) being a change of variables from (, ) and so u(ξ, s) u(, ). Inegraing (6.3) gives u(ξ, s) = g(ξ) where g is an arbirary funcion. If we choose s = 0 on = 0, hen an I.C. of u(, 0) = f() is ransformed o u(ξ, 0) = f(ξ) and so mus have g f. I.e. soluion is u(ξ, s) = f(ξ) or, in erms of (, ), u(, ) = f(ξ) = f( c). Soluion is consan and equal o is iniial value along curves ξ = c = consan. Such curves are called Characerisic curves. ξ= consan(differen consan on each curve) Soluion consan along each curve in (,) space Each curve is a sraigh line passing hrough = 0 a = ξ wih slope 1/c. This agrees wih A more complicaed eample Consider a 1s order inhomogeneous linear PDE wih nonconsan coefficiens: u + u = sin wih I.C. u(, 0) = f(). As before, inroduce (ξ, s) s.. = ξ, s = 0 on = 0. By chain rule Maching erms, we have u s = s u + s u = sin s = 1, = s, (since s = 0 on = 0) s =, = ξes, (since = ξ on s = 0) Also, u = sin = sin s, wih u(ξ, 0) = f(ξ) s Inegrae up, using I.C. o ge u(ξ, s) = cos s+1+ f(ξ) Page 58. c Universiy of Brisol This maerial is copyrigh of he Universiy unless eplicily saed oherwise. I is provided
3 6.1. CHARACTERISTICS Invering variables: s = and ξ = e s = e. So soluion is, in erms of (, ) u(, ) = 1 cos + f(e ) In his eample, characerisics are no sraigh lines; given by ξ = e = consan. Or = log(/ξ) for differen consan values of ξ. ξ= consan Firs Order Quasilinear PDEs We consider he PDE u + g(u)u = 0 (6.4) where g is a given funcion of one variable. The equaion is called quasilinear, because i is linear in u and u, bu may be nonlinear in u. Also, have I.C. u(, 0) = f() As before, le = ξ and s = 0 on = 0. Then we have s = 1, s = g(u), u s = 0 The firs one gives = s, he las one gives u(ξ, s) = f(ξ). Then which we can inegrae up o give s = g( f(ξ)) = sg( f(ξ))+ξ so ha = ξ on s = 0 as required. Using s =, we have = g( f(ξ))+ξ (6.5) as he equaion defining characerisic curves. These are sraigh lines in he (, ) plane wih slope 1/g( f(ξ)). Soluion is consan along characerisics, wih I.e. soluion deermined by daa a = 0. Noe: Since u = f(ξ), can eliminae ξ: u(, ) = f(ξ) = f( g( f(ξ))). u(, ) = f( g(u)) Page 59. c Universiy of Brisol This maerial is copyrigh of he Universiy unless eplicily saed oherwise. I is provided
4 6.2. EXAMPLES OF NONLINEAR WAVE PROBLEMS which is an implici equaion defining u(, ). Remark: The mehod of characerisics works sraighforwardly for quasilinear equaions of he form a(u,, )u + b(u,, )u = c(u,, ) in erms of ODEs for he variables u, and, which may in general be coupled. The ICs may be defined along any curve no angen o a characerisic curve. In his case s = 0 on he IC curve and ξ is an arbirarily chosen parameer along he curve. More deails (also for second order problems) in he Mehods course. 6.2 Eamples of Nonlinear Wave Problems Eamples wih Shock Waves Eample 1. Suppose wih g(u) = 1+u, so u +(1+u)u = 0. (6.6) 1 for 0 u(, 0) = f() = 1 for 0 < < 1 0 for 1 A simple physical inerpreaion of he soluion goes as follows. The wave speed c = 1+u = 1+iniial heigh. So for < 0, c = 1+1 = 2 and ravels faser han for > 1, where c = 1+0 = 1. speed = 2 speed = 1 In he I.C. we can inerchange wih ξ. So For ξ < 0, g( f(ξ)) = 1+u(ξ, 0) = 1+1 = 2. So slope of char. is 1/2 and equaion of char. is = 2+ξ (from (6.5)). For ξ 1, g( f(ξ)) = 1+u(ξ, 0) = 1+0 = 1. So slope of char is 1 and equaion of char. is = +ξ For 0 < ξ < 1, g( f(ξ)) = 1+1 ξ = 2 ξ. So slope of char is 1/(2 ξ) and equaion of char is = (2 ξ)+ξ. Skech of chars: Page 60. c Universiy of Brisol This maerial is copyrigh of he Universiy unless eplicily saed oherwise. I is provided
5 6.2. EXAMPLES OF NONLINEAR WAVE PROBLEMS The wo bounding curves are (A): = 2 and (B): = + 1. They cross when (, ) are he same, i.e. a 1+ = 2 or = 1, = 2. Inerior curves are given by = (2 ξ)+ξ for 0 < ξ < 1. Easy o see hey all pass hrough = 2, = 1. Soluion: u = f(ξ) so or, in erms of (, ), 1 for ξ 0 u(, ) = 0 for ξ 1 1 ξ for 0 < ξ < 1 u(, ) = 1 for 2 0 for 1+ ( ) 1+ for 2 < < 1+ 1 (las equaion comes from = (2 ξ)+ξ implies ξ(1 ) = 2 implies ξ = ( 2)/(1 ) and so 1 ξ = 1 (( 2)/(1 ))). Can see ha he soluion blows up a = 1. This is where he characerisics cross one anoher. Always he case: when characerisics cross he soluion breaks down. Indicaive of shock waves. Physically, in his problem, i is where he linear ramp becomes verical; infinie gradien implies derivaives don eis. Very ineresing... bu no here. Eample 2. The same PDE bu a smooh iniial condiion: 1 2 [1 anh 3( 1 2 )]. Here is is graph: u(, 0) Page 61. c Universiy of Brisol This maerial is copyrigh of he Universiy unless eplicily saed oherwise. I is provided
6 6.2. EXAMPLES OF NONLINEAR WAVE PROBLEMS I is a smoohedou version of Eample 1. Characerisics: Eample wih an Epansion Fan Eample 3: The same PDE u +(1+u)u = 0, bu his ime wih an iniial condiion which increases wih : 0 for 0 u(, 0) = f() = /a for 0 < < a where a > 0 1 for a The characerisics spread ou, and here are no shocks in his problem. gradien=1 gradien = 1/2 0 a In he limi as a 0 he soluion is called an epansion fan. gradien=1 gradien = 1/2 0 Page 62. c Universiy of Brisol This maerial is copyrigh of he Universiy unless eplicily saed oherwise. I is provided
7 6.3. TRAFFIC FLOW 6.3 Traffic Flow This heory was invened in Mancheser in 1955 by Sir James Lighhill and G. B. Whiham The Traffic Flow Equaion Le be disance along a road (no necessarily sraigh). Traffic densiy ρ(, ) on a road is defined as he number of cars (or oher vehicles) per uni disance a he poin and ime. Then he number of cars a ime in he region a < < b is ρ is really a suble kind of average. b a ρ(, ) d. Conservaion Law: No cars can be creaed or desroyed and so can use he conservaion law in ( suff is now cars). ρ = φ where φ(, ) is flu. In his cone, flu is he rae a which cars are crossing he fied poin. I.e. i is (densiy of cars) (speed of cars). where u(, ) is he Traffic speed. So we have φ(, ) = ρ(, )u(, ) 0 = ρ +(ρu) = ρ + ρ u+ρu (6.7) This one equaion involves wo dependen variables ρ and u. Need anoher equaion linking ρ and u o close he model. Insead of u = u(, ), propose u = u(ρ) = u(ρ(, )). I.e. ones speed is no dependen where you are on he road, or wha ime i is, only on he densiy of he raffic surrounding you. So φ = ρu = ρu(ρ) f(ρ), say. Now we ge 0 = ρ + f(ρ) = ρ + f (ρ)ρ (6.8) This is a quasilinear PDE of he ype already invesigaed in The Quadraic Model Assumpions: Page 63. c Universiy of Brisol This maerial is copyrigh of he Universiy unless eplicily saed oherwise. I is provided
8 6.3. TRAFFIC FLOW 1. When ρ = 0, u = u ma, he maimum speed a car will ravel a on an empy road (he speed limi?) 2. u = 0 when ρ = ρ c where ρ c = 1/spacing beween cars in a jam. Consider he simples model, in which u is a linear funcion of ρ: ( u = u ma 1 ρ ) for ρ ρ c (6.9) ρ c Now f(ρ) = ρu = u ma ρ(1 ρ/ρ c ) and Plug ino he conservaion eqn (6.8) o ge ρ + c(ρ c 2ρ)ρ = 0 (6.10) where c = u ma ρ c. (6.11) This is he ype of PDE considered in 6.2, bu wih a negaive coefficien for he quadraic erm ρρ. Wave speed = 1/(slope of characerisics) = c(ρ c 2ρ). Can be +ve or ve. Speed of raffic is c[ρ c ρ] > 0 from (6.9) and (6.11). So he wave speed < raffic speed. This means ha changes in densiy ravel more slowly han cars. So when you drive, you go faser han he changes in densiy; ha s why you have o slow down o avoid hickening of raffic. This is he oher way round from waer waves, where as you floa wih he wave, breakers come up behind you. The reason is he nonlinear erm ρρ in (6.10) has a minus sign, where he nonlinear erm in (6.6) has a plus sign. I is easy o verify ha if ρ(, 0) is an increasing funcion, you ge shock formaion, so when raffic sars o ge hicker (as when a moorway lane closes), disconinuiies end o develop. Anyone who has driven on a busy moorway knows ha raffic jams can form suddenly and for no obvious reason. They are shock waves formed by he seepening of iniially smooh changes of densiy. Remark: These quasilinear equaions also closely conneced o oher observable phenomena, such as glacier flows and sedimenaion in river delas. Page 64. c Universiy of Brisol This maerial is copyrigh of he Universiy unless eplicily saed oherwise. I is provided
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