And the solution to the PDE problem must be of the form Π 1
|
|
- Rosemary Taylor
- 6 years ago
- Views:
Transcription
1 5. Self-Similar Soluions b Dimensional Analsis Consider he diffusion problem from las secion, wih poinwise release (Ref: Bluman & Cole, 2.3): c = D 2 c x + Q 0δ(x)δ() 2 c(x,0) = 0, c(±,) = 0 Iniial release wihin infiniel narrow neighborhood of x = 0, such ha Π (x)/d = δ (x) and L / d. Noe Q 0 has differen dimension as he previous Q because of he cross-secional area S and ime conained in δ (). 1. Dimensional analsis: {c}=ml -3, {D}=L 2 T -1, {Q 0 }=ML -2 (Mass releases per uni cross-secional area) {x}=l, {}=T Thus, we expec 2 Pi groups: = D Q 0 c, Π 2 = x D And he soluion o he PDE problem mus be of he form = f (Π 2 ) or c = Q 0 D f x D Normall we expec dimensional analsis o reduce he number of variables and parameers. Bu here we reduced he number of independen variables from 2 o 1! 2. Transformaion of PDE o ODE: Now we can plug his form back ino he PDE. Firs, he parial derivaives: c = Q 0 2 D f Q x 0 2D 2 f ', c x = Q 0 D f ', 2 c x 2 = Q 0 (D) 3/2 f " For > 0, here is no more injecion: δ () = 0. Afer insering he above ino he PDE: f 2 x 2 D f ' = f " or f "+ ξ 2 f '+ f = 0, (1) 2 where ξ = x D is our new independen variable. We have successfull ransformed he PDE ino an ODE. How abou he iniial and boundar condiions? Noe ha = 0 and x= boh correspond o ξ =, so ha he iniial and boundar condiions can be rolled ino one: 1
2 f( ± ) = 0. (2) Bu we need anoher condiion on f, one ha reflecs he amoun of iniial injecion. This is obained b inegraing he PDE over he following inervals: d [PDE]dx, where = 0 - means jus before = 0. 0 Now he lef-hand-side is c d dx c = dx 0 d = [c(x,) c(x, 0)]dx 0 = c(x,)dx and he wo erms on he righ-hand-side are: 2 c d x dx = d Now we have c 0 x d = 0 because c = 0 a ± ; Q 0 δ()δ(x)dx = Q 0 b virue of he definiion of he dela funcion. c(x,)dx = Q 0, which can be ransformed, using he variable ξ, ino f (ξ)dξ = 1 (3) ODE (1), along wih condiions (2) and (3) will uniquel deermine f(ξ), from which we ge c(x,). We are no concerned wih he acual soluion of he new ODE problem. Raher, he ineresing quesion is: How did we manage o urn a PDE o an ODE? 3. Discussion a. The problem admis a self-similar soluion: if x is scaled b he diffusion lengh (D) 1/2, hen he c(x,) profiles a differen imes can be collapsed ono each oher if c is scaled b Q 0 /(D) 1/2. b. This means ha x and are no reall 2 independen variables; as far as c is concerned, he can be rolled ino one independen variable ξ. c. Similari soluions are happ coincidences in phsical processes. Can we alwas find hem for an PDE s? No. This problem is special in ha here is no inheren lengh scale. Thus, we are no able o form dimensionless groups for each of he variables x,, and c; insead, we have o combine hem and end up wih onl 2 Pi groups. Tha s how we ended up wih he ODE. If we had he release lengh ds or he domain lengh L, he self-similari will be ruined. d. Can we alwas find similari soluions b dimensional analsis? No. Bu we will sud anoher example nex, and hen inroduce he general sreching ransformaion idea for deecing similari soluions. 2
3 6. Similari Soluions b Sreching Transformaion I is rare ha similari soluions can be obained from dimensional analsis. In his secion we inroduce he idea of sreching ransformaion which is a more general procedure for seeking ou similari in PDE problems. The maerials are based on Barenbla ( 5.2) and Bluman & Cole ( 2.5). As a concree example, we will ake Prandl s boundar laer equaion for flow over a fla semi-plane. Afer he boundar laer approximaion (ha viscosi acs onl wihin a hin laer, ha he gradien in he flow direcion (x) is much smaller han in he ransverse direcion (), and ha he pressure is consan in he direcion), he governing equaions are u u x + v u = ν 2 u 2 u x + v = 0 u(x,0) = 0, v(x,0) = 0 u(x, ) = U, u(0, ) = U where U is he free-sream veloci, and ν is he kinemaic viscosi. If ou recall our fluid mechanics, his problem does have a similari soluion (Blasius soluion), and he PDE can be reduced o ODE. (Tr o disinguish he veloci v from he viscosi ν. We could use differen smbols bu hese are he convenional ones.) 1. Would dimensional analsis work? Le s wrie ou he dimensions of all he variables and parameers: {u}={v}={u }=L/T, {ν}=l 2 /T, {x}={}=l There are 2 independen dimensions involved (L and T), and we can consruc 4 dimensionless groups ou of hese. For insance: = u U, Π 2 = v, Π 3 = U x U ν and we expec soluions such as = f (Π 3,Π 4 ), Π 2 = g(π 3,Π 4 )., Π 4 = U ν Plugging hese back ino he equaions, and we will see ha we have NOT achieved a reducion of he number of independen variable. Dimensional analsis has failed o give us he similari soluion. Wh? Even hough he problem has no inrinsic ime or lengh scales, similar o he diffusion problem in he las secion, here are onl 2 independen dimensions (L and T) insead of 3. Thus, i is possible for x and o form heir own Pi groups; he don have o be forced ino a single one. I urns ou ha in his paricular example, a rivial manipulaion can cure he above problem. This is no a general echnique, bu neverheless is fun o illusrae here. We will 3
4 ake his lile deour before marching ino he general echnique ha is he focus of his secion. Based on he phsical insigh ha hings happen a differen scales along he x and direcions, which is he fundamenal idea behind he boundar laer approximaion, we assign wo differen dimensions o x and, L and H, and for he momen preend ha he are differen dimensions. Now he lis of variables and unknowns are scaled as such: {u}={u }= L/T, {v}= H/T, {ν}=h 2 /T, {x}= L, {}= H. There are now 3 independen dimensions involved (L, H and T), and we can consruc onl 3 dimensionless groups ou of hese: = u U, Π 2 = v νu / x, Π 3 = νx /U = ζ. Now we expec a similari soluion in his form: u = U f (ζ ), v = νu x g(ζ). Plugging his ino he original PDE will show ha indeed, we have reduced he PDE problem o a couple of ODEs, whose soluion is deailed in Fluid Mechanics exbooks. For anoher example of such ingenious dimensional analsis, see he Raleigh problem analzed in he nex secion (see also Bluman & Cole, p. 195). We picall seek o increase he number of independen dimensions (as done above) or decrease he number of dimensional parameers (as done in Bluman & Cole s example). 2. Sreching ransformaion The ingenious dimensional analsis mehod is specific o he problems. There is, however, a general scheme for seeking ou possible similari soluions. The scheme someimes goes b he name of renormalizaion groups or invarian ransformaion groups, and is based on raher formalisic mahemaical manipulaions. We will skip he proofs and focus on he echnique iself. Since he essence of similari is ha he soluion is invarian afer cerain scaling of he independen and dependen variables, we consider he following sreching ransformaion, and see if such ransformaions will leave he PDE and he boundar condiions invarian. Consider: U = α a u, V = α b v, X = α c x, Y = α d where α is a posiive number. Under his ransformaion, we have u x = α c a U X, u = α d a U Y, v = α d b V Y, 2 u = α 2d a 2 U 2 Y. 2 Plugging hese ino he original PDE s and boundar condiions, we ll see wha choices of a, b, c, d ma mainain he invariance of he problem. The coninui equaion ields: c a = d b. 4
5 The hree erms of he momenum equaion requires: c 2a = d a b = 2d a. Noe ha he firs equaion above is idenical o he preceding equaion, and hus he momenum equaion adds onl 1 addiional consrain on he power indices. Finall he boundar condiions require a = 0 because for he problem in he new variables o be invarian, he non-homogeneous BC should remain as U (X, ) = U. Now we have 3 equaions consrain he 4 indices, and we rewrie he ransformaion as: U = u, V = v ε, where ε = α d. X = ε 2 x, Y = ε This ransformaion will leave he problem he same as before, in he new sreched and scaled variables. The fac ha his one-parameer famil of ransformaions will mainain he invariance of he PDE problem reveals he inrinsic self-similari of he problem. In oher words, if we srech he coordinae b a facor ε, hen we mus srech x b ε 2 and he veloci componen v b 1/ ε in order o collapse he veloci profiles. From his argumen, we recognize ha u, v x, x shall remain he same no maer how we srech he coordinaes. These are known as he invarians of he ransformaion, and immediael sugges he following similari soluion: u = f (ζ ) v = 1 x g(ζ), wih he similari variable ζ = x. This is he same form as obained from he ingenious dimensional analsis, aside from a few consan facors. Noe ha we reached he conclusion here no hrough dimensional consideraions, bu hrough he idea of invariance under general sreching ransformaions. Now i s a simple maer o plug hese forms ino he original PDE problem, and ransform i ino he following ODE problem: ζ ν f "+ f ' 2 f g = 0, ζ f ' 2g' = 0,, f ( ) = U, f (0) = 0, g(0) = 0 he soluion of which will no be of immediae ineres o us here. Noe ha he wo BC s a x = 0 and = boh projec ono ζ =. 5
6 3. Similari Soluion for he Raleigh Problem The Raleigh problem is anoher classical example wih a self-similar soluion. Consider he ransien moion in a viscous fluid induced b a fla plae moving in is own plane. Iniiall boh he plae and he fluid are a res. Saring a = 0, he plae moves wih a consan veloci U 0. The Navier-Sokes equaions, simplified for his problem, along wih he iniial and boundar condiions, can be wrien as: u = ν 2 u 2 u(,0) = 0, u(0,) = U 0, u(,) = 0 u(,) U 0 (a) Dimensional analsis. From he following dimensions: {u}={u 0 }= L/T, {ν}=l 2 /T, {}= T, {}= L, we can make 3 Pi groups, sa u/u 0, U 0 /ν, U 02 / ν, and here is no reducion o ODE. Again, we can pla ricks here, b eiher increasing he number of independen dimensions, or decreasing he number of parameers, so as o reduce he number of Pi groups. Using he phsical observaion ha he viscous diffusion happens along he direcion, while he primar flow is in he x direcion, we can inroduce differen lengh scales: {u}={u 0 }= L/T, {ν}=h 2 /T, {}= T, {}= H. Now here are onl 2 Pi groups: = u U 0, Π 2 = ν and we can r a similari soluion of he form u(,) = U 0 f ν. Alernaivel, we can reduce he number of parameers b scaling u b U 0, and calling u(,) = u(,) /U 0 he new dependen variable. Now he problem has one less parameer, and again onl admis 2 Pi groups. In he following, however, le us carr ou he formal procedure of sreching ransformaion as an exercise. (b) Sreching ransformaion. Consider: U = α a u, Y = α b, T = α c, where α is a posiive number. Under his ransformaion, we have 6
7 u = α c a U T, 2 u = α 2b a 2 U 2 Y. 2 To mainain invariance of he PDE, we require c a = 2b a, or c = 2 b. The boundar condiion u ( 0, ) = U 0 requires a = 0. Thus, we have he following ransformaion ha renders he problem invarian: U = u, Y = ε, T = ε 2, which ε = α b. This ransformaion dicaes ha and be ransformed in a coordinaed wa. Thus u and ζ = / shall be our new variables ha remain unchanged for an sreching α or ε: u = f = f (ζ ). This reduces he original PDE ino he following ODE problem: 2ν f "+ ζ f ' = 0, f (0) = U, f ( ) = 0 which can be inegraed analicall o give: ζ f = c 1 exp z2 0 4ν dz + c 2. Noing ha exp z2 0 4ν dz = 2 ν exp( ξ 2 ) dξ = πν, he wo consans of 0 inegraion are deermined: c 1 = U 0 / πν and c 2 = U 0. Finall he soluion can be wrien in erms of he complemenar error funcion: ζ f = U 0 erfc 2 ν = U erfc 0 wih erfc(x)=1-2 exp z 2 π dz. 0 x ( ) 4ν, 7
Chapter 2. First Order Scalar Equations
Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.
More informationTwo Coupled Oscillators / Normal Modes
Lecure 3 Phys 3750 Two Coupled Oscillaors / Normal Modes Overview and Moivaion: Today we ake a small, bu significan, sep owards wave moion. We will no ye observe waves, bu his sep is imporan in is own
More informationdt = C exp (3 ln t 4 ). t 4 W = C exp ( ln(4 t) 3) = C(4 t) 3.
Mah Rahman Exam Review Soluions () Consider he IVP: ( 4)y 3y + 4y = ; y(3) = 0, y (3) =. (a) Please deermine he longes inerval for which he IVP is guaraneed o have a unique soluion. Soluion: The disconinuiies
More informationSecond Order Linear Differential Equations
Second Order Linear Differenial Equaions Second order linear equaions wih consan coefficiens; Fundamenal soluions; Wronskian; Exisence and Uniqueness of soluions; he characerisic equaion; soluions of homogeneous
More informationMath 334 Test 1 KEY Spring 2010 Section: 001. Instructor: Scott Glasgow Dates: May 10 and 11.
1 Mah 334 Tes 1 KEY Spring 21 Secion: 1 Insrucor: Sco Glasgow Daes: Ma 1 and 11. Do NOT wrie on his problem saemen bookle, excep for our indicaion of following he honor code jus below. No credi will be
More informationMath 333 Problem Set #2 Solution 14 February 2003
Mah 333 Problem Se #2 Soluion 14 February 2003 A1. Solve he iniial value problem dy dx = x2 + e 3x ; 2y 4 y(0) = 1. Soluion: This is separable; we wrie 2y 4 dy = x 2 + e x dx and inegrae o ge The iniial
More informationLecture 2-1 Kinematics in One Dimension Displacement, Velocity and Acceleration Everything in the world is moving. Nothing stays still.
Lecure - Kinemaics in One Dimension Displacemen, Velociy and Acceleraion Everyhing in he world is moving. Nohing says sill. Moion occurs a all scales of he universe, saring from he moion of elecrons in
More informationdy dx = xey (a) y(0) = 2 (b) y(1) = 2.5 SOLUTION: See next page
Assignmen 1 MATH 2270 SOLUTION Please wrie ou complee soluions for each of he following 6 problems (one more will sill be added). You may, of course, consul wih your classmaes, he exbook or oher resources,
More informationKEY. Math 334 Midterm I Fall 2008 sections 001 and 003 Instructor: Scott Glasgow
1 KEY Mah 4 Miderm I Fall 8 secions 1 and Insrucor: Sco Glasgow Please do NOT wrie on his eam. No credi will be given for such work. Raher wrie in a blue book, or on our own paper, preferabl engineering
More information23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
More informationSome Basic Information about M-S-D Systems
Some Basic Informaion abou M-S-D Sysems 1 Inroducion We wan o give some summary of he facs concerning unforced (homogeneous) and forced (non-homogeneous) models for linear oscillaors governed by second-order,
More informationMatlab and Python programming: how to get started
Malab and Pyhon programming: how o ge sared Equipping readers he skills o wrie programs o explore complex sysems and discover ineresing paerns from big daa is one of he main goals of his book. In his chaper,
More informationFinal Spring 2007
.615 Final Spring 7 Overview The purpose of he final exam is o calculae he MHD β limi in a high-bea oroidal okamak agains he dangerous n = 1 exernal ballooning-kink mode. Effecively, his corresponds o
More informationMath 2142 Exam 1 Review Problems. x 2 + f (0) 3! for the 3rd Taylor polynomial at x = 0. To calculate the various quantities:
Mah 4 Eam Review Problems Problem. Calculae he 3rd Taylor polynomial for arcsin a =. Soluion. Le f() = arcsin. For his problem, we use he formula f() + f () + f ()! + f () 3! for he 3rd Taylor polynomial
More informationChapter 6. Systems of First Order Linear Differential Equations
Chaper 6 Sysems of Firs Order Linear Differenial Equaions We will only discuss firs order sysems However higher order sysems may be made ino firs order sysems by a rick shown below We will have a sligh
More informationPhysics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle
Chaper 2 Newonian Mechanics Single Paricle In his Chaper we will review wha Newon s laws of mechanics ell us abou he moion of a single paricle. Newon s laws are only valid in suiable reference frames,
More informationLet us start with a two dimensional case. We consider a vector ( x,
Roaion marices We consider now roaion marices in wo and hree dimensions. We sar wih wo dimensions since wo dimensions are easier han hree o undersand, and one dimension is a lile oo simple. However, our
More informationEffects of Coordinate Curvature on Integration
Effecs of Coordinae Curvaure on Inegraion Chrisopher A. Lafore clafore@gmail.com Absrac In his paper, he inegraion of a funcion over a curved manifold is examined in he case where he curvaure of he manifold
More information15. Bicycle Wheel. Graph of height y (cm) above the axle against time t (s) over a 6-second interval. 15 bike wheel
15. Biccle Wheel The graph We moun a biccle wheel so ha i is free o roae in a verical plane. In fac, wha works easil is o pu an exension on one of he axles, and ge a suden o sand on one side and hold he
More informationChallenge Problems. DIS 203 and 210. March 6, (e 2) k. k(k + 2). k=1. f(x) = k(k + 2) = 1 x k
Challenge Problems DIS 03 and 0 March 6, 05 Choose one of he following problems, and work on i in your group. Your goal is o convince me ha your answer is correc. Even if your answer isn compleely correc,
More informationEchocardiography Project and Finite Fourier Series
Echocardiography Projec and Finie Fourier Series 1 U M An echocardiagram is a plo of how a porion of he hear moves as he funcion of ime over he one or more hearbea cycles If he hearbea repeas iself every
More information= ( ) ) or a system of differential equations with continuous parametrization (T = R
XIII. DIFFERENCE AND DIFFERENTIAL EQUATIONS Ofen funcions, or a sysem of funcion, are paramerized in erms of some variable, usually denoed as and inerpreed as ime. The variable is wrien as a funcion of
More informationMath 334 Fall 2011 Homework 11 Solutions
Dec. 2, 2 Mah 334 Fall 2 Homework Soluions Basic Problem. Transform he following iniial value problem ino an iniial value problem for a sysem: u + p()u + q() u g(), u() u, u () v. () Soluion. Le v u. Then
More informationMATH 4330/5330, Fourier Analysis Section 6, Proof of Fourier s Theorem for Pointwise Convergence
MATH 433/533, Fourier Analysis Secion 6, Proof of Fourier s Theorem for Poinwise Convergence Firs, some commens abou inegraing periodic funcions. If g is a periodic funcion, g(x + ) g(x) for all real x,
More information20. Applications of the Genetic-Drift Model
0. Applicaions of he Geneic-Drif Model 1) Deermining he probabiliy of forming any paricular combinaion of genoypes in he nex generaion: Example: If he parenal allele frequencies are p 0 = 0.35 and q 0
More informationt is a basis for the solution space to this system, then the matrix having these solutions as columns, t x 1 t, x 2 t,... x n t x 2 t...
Mah 228- Fri Mar 24 5.6 Marix exponenials and linear sysems: The analogy beween firs order sysems of linear differenial equaions (Chaper 5) and scalar linear differenial equaions (Chaper ) is much sronger
More informationSecond Order Linear Differential Equations
Second Order Linear Differenial Equaions Second order linear equaions wih consan coefficiens; Fundamenal soluions; Wronskian; Exisence and Uniqueness of soluions; he characerisic equaion; soluions of homogeneous
More informationLinear Response Theory: The connection between QFT and experiments
Phys540.nb 39 3 Linear Response Theory: The connecion beween QFT and experimens 3.1. Basic conceps and ideas Q: How do we measure he conduciviy of a meal? A: we firs inroduce a weak elecric field E, and
More informationThe fundamental mass balance equation is ( 1 ) where: I = inputs P = production O = outputs L = losses A = accumulation
Hea (iffusion) Equaion erivaion of iffusion Equaion The fundamenal mass balance equaion is I P O L A ( 1 ) where: I inpus P producion O oupus L losses A accumulaion Assume ha no chemical is produced or
More information4.6 One Dimensional Kinematics and Integration
4.6 One Dimensional Kinemaics and Inegraion When he acceleraion a( of an objec is a non-consan funcion of ime, we would like o deermine he ime dependence of he posiion funcion x( and he x -componen of
More information4.5 Constant Acceleration
4.5 Consan Acceleraion v() v() = v 0 + a a() a a() = a v 0 Area = a (a) (b) Figure 4.8 Consan acceleraion: (a) velociy, (b) acceleraion When he x -componen of he velociy is a linear funcion (Figure 4.8(a)),
More informationMA 214 Calculus IV (Spring 2016) Section 2. Homework Assignment 1 Solutions
MA 14 Calculus IV (Spring 016) Secion Homework Assignmen 1 Soluions 1 Boyce and DiPrima, p 40, Problem 10 (c) Soluion: In sandard form he given firs-order linear ODE is: An inegraing facor is given by
More information23.5. Half-Range Series. Introduction. Prerequisites. Learning Outcomes
Half-Range Series 2.5 Inroducion In his Secion we address he following problem: Can we find a Fourier series expansion of a funcion defined over a finie inerval? Of course we recognise ha such a funcion
More informationSection 3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients
Secion 3.5 Nonhomogeneous Equaions; Mehod of Undeermined Coefficiens Key Terms/Ideas: Linear Differenial operaor Nonlinear operaor Second order homogeneous DE Second order nonhomogeneous DE Soluion o homogeneous
More informationExam 1 Solutions. 1 Question 1. February 10, Part (A) 1.2 Part (B) To find equilibrium solutions, set P (t) = C = dp
Exam Soluions Februar 0, 05 Quesion. Par (A) To find equilibrium soluions, se P () = C = = 0. This implies: = P ( P ) P = P P P = P P = P ( + P ) = 0 The equilibrium soluion are hus P () = 0 and P () =..
More information6.2 Transforms of Derivatives and Integrals.
SEC. 6.2 Transforms of Derivaives and Inegrals. ODEs 2 3 33 39 23. Change of scale. If l( f ()) F(s) and c is any 33 45 APPLICATION OF s-shifting posiive consan, show ha l( f (c)) F(s>c)>c (Hin: In Probs.
More informationIntegration Over Manifolds with Variable Coordinate Density
Inegraion Over Manifolds wih Variable Coordinae Densiy Absrac Chrisopher A. Lafore clafore@gmail.com In his paper, he inegraion of a funcion over a curved manifold is examined in he case where he curvaure
More information( ) a system of differential equations with continuous parametrization ( T = R + These look like, respectively:
XIII. DIFFERENCE AND DIFFERENTIAL EQUATIONS Ofen funcions, or a sysem of funcion, are paramerized in erms of some variable, usually denoed as and inerpreed as ime. The variable is wrien as a funcion of
More informationPredator - Prey Model Trajectories and the nonlinear conservation law
Predaor - Prey Model Trajecories and he nonlinear conservaion law James K. Peerson Deparmen of Biological Sciences and Deparmen of Mahemaical Sciences Clemson Universiy Ocober 28, 213 Ouline Drawing Trajecories
More informationPhysics 127b: Statistical Mechanics. Fokker-Planck Equation. Time Evolution
Physics 7b: Saisical Mechanics Fokker-Planck Equaion The Langevin equaion approach o he evoluion of he velociy disribuion for he Brownian paricle migh leave you uncomforable. A more formal reamen of his
More informationA Special Hour with Relativity
A Special Hour wih Relaiviy Kenneh Chu The Graduae Colloquium Deparmen of Mahemaics Universiy of Uah Oc 29, 2002 Absrac Wha promped Einsen: Incompaibiliies beween Newonian Mechanics and Maxwell s Elecromagneism.
More information15. Vector Valued Functions
1. Vecor Valued Funcions Up o his poin, we have presened vecors wih consan componens, for example, 1, and,,4. However, we can allow he componens of a vecor o be funcions of a common variable. For example,
More informationBiol. 356 Lab 8. Mortality, Recruitment, and Migration Rates
Biol. 356 Lab 8. Moraliy, Recruimen, and Migraion Raes (modified from Cox, 00, General Ecology Lab Manual, McGraw Hill) Las week we esimaed populaion size hrough several mehods. One assumpion of all hese
More informationFinish reading Chapter 2 of Spivak, rereading earlier sections as necessary. handout and fill in some missing details!
MAT 257, Handou 6: Ocober 7-2, 20. I. Assignmen. Finish reading Chaper 2 of Spiva, rereading earlier secions as necessary. handou and fill in some missing deails! II. Higher derivaives. Also, read his
More informationChapter 15: Phenomena. Chapter 15 Chemical Kinetics. Reaction Rates. Reaction Rates R P. Reaction Rates. Rate Laws
Chaper 5: Phenomena Phenomena: The reacion (aq) + B(aq) C(aq) was sudied a wo differen emperaures (98 K and 35 K). For each emperaure he reacion was sared by puing differen concenraions of he 3 species
More informationSMT 2014 Calculus Test Solutions February 15, 2014 = 3 5 = 15.
SMT Calculus Tes Soluions February 5,. Le f() = and le g() =. Compue f ()g (). Answer: 5 Soluion: We noe ha f () = and g () = 6. Then f ()g () =. Plugging in = we ge f ()g () = 6 = 3 5 = 5.. There is a
More informationME 391 Mechanical Engineering Analysis
Fall 04 ME 39 Mechanical Engineering Analsis Eam # Soluions Direcions: Open noes (including course web posings). No books, compuers, or phones. An calculaor is fair game. Problem Deermine he posiion of
More informationSOLUTIONS TO ECE 3084
SOLUTIONS TO ECE 384 PROBLEM 2.. For each sysem below, specify wheher or no i is: (i) memoryless; (ii) causal; (iii) inverible; (iv) linear; (v) ime invarian; Explain your reasoning. If he propery is no
More informationSolutions to Assignment 1
MA 2326 Differenial Equaions Insrucor: Peronela Radu Friday, February 8, 203 Soluions o Assignmen. Find he general soluions of he following ODEs: (a) 2 x = an x Soluion: I is a separable equaion as we
More informationGuest Lectures for Dr. MacFarlane s EE3350 Part Deux
Gues Lecures for Dr. MacFarlane s EE3350 Par Deux Michael Plane Mon., 08-30-2010 Wrie name in corner. Poin ou his is a review, so I will go faser. Remind hem o go lisen o online lecure abou geing an A
More informationSystem of Linear Differential Equations
Sysem of Linear Differenial Equaions In "Ordinary Differenial Equaions" we've learned how o solve a differenial equaion for a variable, such as: y'k5$e K2$x =0 solve DE yx = K 5 2 ek2 x C_C1 2$y''C7$y
More information!!"#"$%&#'()!"#&'(*%)+,&',-)./0)1-*23)
"#"$%&#'()"#&'(*%)+,&',-)./)1-*) #$%&'()*+,&',-.%,/)*+,-&1*#$)()5*6$+$%*,7&*-'-&1*(,-&*6&,7.$%$+*&%'(*8$&',-,%'-&1*(,-&*6&,79*(&,%: ;..,*&1$&$.$%&'()*1$$.,'&',-9*(&,%)?%*,('&5
More informationSolutions from Chapter 9.1 and 9.2
Soluions from Chaper 9 and 92 Secion 9 Problem # This basically boils down o an exercise in he chain rule from calculus We are looking for soluions of he form: u( x) = f( k x c) where k x R 3 and k is
More informationMath Week 14 April 16-20: sections first order systems of linear differential equations; 7.4 mass-spring systems.
Mah 2250-004 Week 4 April 6-20 secions 7.-7.3 firs order sysems of linear differenial equaions; 7.4 mass-spring sysems. Mon Apr 6 7.-7.2 Sysems of differenial equaions (7.), and he vecor Calculus we need
More informationAnnouncements: Warm-up Exercise:
Fri Apr 13 7.1 Sysems of differenial equaions - o model muli-componen sysems via comparmenal analysis hp//en.wikipedia.org/wiki/muli-comparmen_model Announcemens Warm-up Exercise Here's a relaively simple
More informationln 2 1 ln y x c y C x
Lecure 14 Appendi B: Some sample problems from Boas Here are some soluions o he sample problems assigned for Chaper 8 8: 6 Soluion: We wan o find he soluion o he following firs order equaion using separaion
More informationClass Meeting # 10: Introduction to the Wave Equation
MATH 8.5 COURSE NOTES - CLASS MEETING # 0 8.5 Inroducion o PDEs, Fall 0 Professor: Jared Speck Class Meeing # 0: Inroducion o he Wave Equaion. Wha is he wave equaion? The sandard wave equaion for a funcion
More informationChapter 3 Boundary Value Problem
Chaper 3 Boundary Value Problem A boundary value problem (BVP) is a problem, ypically an ODE or a PDE, which has values assigned on he physical boundary of he domain in which he problem is specified. Le
More informationKINEMATICS IN ONE DIMENSION
KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings move how far (disance and displacemen), how fas (speed and velociy), and how fas ha how fas changes (acceleraion). We say ha an objec
More informationTheory of! Partial Differential Equations!
hp://www.nd.edu/~gryggva/cfd-course/! Ouline! Theory o! Parial Dierenial Equaions! Gréar Tryggvason! Spring 011! Basic Properies o PDE!! Quasi-linear Firs Order Equaions! - Characerisics! - Linear and
More informationTraveling Waves. Chapter Introduction
Chaper 4 Traveling Waves 4.1 Inroducion To dae, we have considered oscillaions, i.e., periodic, ofen harmonic, variaions of a physical characerisic of a sysem. The sysem a one ime is indisinguishable from
More information2. Nonlinear Conservation Law Equations
. Nonlinear Conservaion Law Equaions One of he clear lessons learned over recen years in sudying nonlinear parial differenial equaions is ha i is generally no wise o ry o aack a general class of nonlinear
More informationVanishing Viscosity Method. There are another instructive and perhaps more natural discontinuous solutions of the conservation law
Vanishing Viscosiy Mehod. There are anoher insrucive and perhaps more naural disconinuous soluions of he conservaion law (1 u +(q(u x 0, he so called vanishing viscosiy mehod. This mehod consiss in viewing
More informationMath 4600: Homework 11 Solutions
Mah 46: Homework Soluions Gregory Handy [.] One of he well-known phenomenological (capuring he phenomena, bu no necessarily he mechanisms) models of cancer is represened by Gomperz equaion dn d = bn ln(n/k)
More informationModule 2 F c i k c s la l w a s o s f dif di fusi s o i n
Module Fick s laws of diffusion Fick s laws of diffusion and hin film soluion Adolf Fick (1855) proposed: d J α d d d J (mole/m s) flu (m /s) diffusion coefficien and (mole/m 3 ) concenraion of ions, aoms
More informationOutline of Topics. Analysis of ODE models with MATLAB. What will we learn from this lecture. Aim of analysis: Why such analysis matters?
of Topics wih MATLAB Shan He School for Compuaional Science Universi of Birmingham Module 6-3836: Compuaional Modelling wih MATLAB Wha will we learn from his lecure Aim of analsis: Aim of analsis. Some
More informationMath 10B: Mock Mid II. April 13, 2016
Name: Soluions Mah 10B: Mock Mid II April 13, 016 1. ( poins) Sae, wih jusificaion, wheher he following saemens are rue or false. (a) If a 3 3 marix A saisfies A 3 A = 0, hen i canno be inverible. True.
More informationCH.7. PLANE LINEAR ELASTICITY. Continuum Mechanics Course (MMC) - ETSECCPB - UPC
CH.7. PLANE LINEAR ELASTICITY Coninuum Mechanics Course (MMC) - ETSECCPB - UPC Overview Plane Linear Elasici Theor Plane Sress Simplifing Hpohesis Srain Field Consiuive Equaion Displacemen Field The Linear
More informationChapter 7 Response of First-order RL and RC Circuits
Chaper 7 Response of Firs-order RL and RC Circuis 7.- The Naural Response of RL and RC Circuis 7.3 The Sep Response of RL and RC Circuis 7.4 A General Soluion for Sep and Naural Responses 7.5 Sequenial
More informationElementary Differential Equations and Boundary Value Problems
Elemenar Differenial Equaions and Boundar Value Problems Boce. & DiPrima 9 h Ediion Chaper 1: Inroducion 1006003 คณ ตศาสตร ว ศวกรรม 3 สาขาว ชาว ศวกรรมคอมพ วเตอร ป การศ กษา 1/2555 ผศ.ดร.อร ญญา ผศ.ดร.สมศ
More information10. State Space Methods
. Sae Space Mehods. Inroducion Sae space modelling was briefly inroduced in chaper. Here more coverage is provided of sae space mehods before some of heir uses in conrol sysem design are covered in he
More informationODEs II, Lecture 1: Homogeneous Linear Systems - I. Mike Raugh 1. March 8, 2004
ODEs II, Lecure : Homogeneous Linear Sysems - I Mike Raugh March 8, 4 Inroducion. In he firs lecure we discussed a sysem of linear ODEs for modeling he excreion of lead from he human body, saw how o ransform
More informationSimulation-Solving Dynamic Models ABE 5646 Week 2, Spring 2010
Simulaion-Solving Dynamic Models ABE 5646 Week 2, Spring 2010 Week Descripion Reading Maerial 2 Compuer Simulaion of Dynamic Models Finie Difference, coninuous saes, discree ime Simple Mehods Euler Trapezoid
More informationCh.1. Group Work Units. Continuum Mechanics Course (MMC) - ETSECCPB - UPC
Ch.. Group Work Unis Coninuum Mechanics Course (MMC) - ETSECCPB - UPC Uni 2 Jusify wheher he following saemens are rue or false: a) Two sreamlines, corresponding o a same insan of ime, can never inersec
More informationWEEK-3 Recitation PHYS 131. of the projectile s velocity remains constant throughout the motion, since the acceleration a x
WEEK-3 Reciaion PHYS 131 Ch. 3: FOC 1, 3, 4, 6, 14. Problems 9, 37, 41 & 71 and Ch. 4: FOC 1, 3, 5, 8. Problems 3, 5 & 16. Feb 8, 018 Ch. 3: FOC 1, 3, 4, 6, 14. 1. (a) The horizonal componen of he projecile
More informationEECE251. Circuit Analysis I. Set 4: Capacitors, Inductors, and First-Order Linear Circuits
EEE25 ircui Analysis I Se 4: apaciors, Inducors, and Firs-Order inear ircuis Shahriar Mirabbasi Deparmen of Elecrical and ompuer Engineering Universiy of Briish olumbia shahriar@ece.ubc.ca Overview Passive
More informationPosition, Velocity, and Acceleration
rev 06/2017 Posiion, Velociy, and Acceleraion Equipmen Qy Equipmen Par Number 1 Dynamic Track ME-9493 1 Car ME-9454 1 Fan Accessory ME-9491 1 Moion Sensor II CI-6742A 1 Track Barrier Purpose The purpose
More informationDifferential Equations
Mah 21 (Fall 29) Differenial Equaions Soluion #3 1. Find he paricular soluion of he following differenial equaion by variaion of parameer (a) y + y = csc (b) 2 y + y y = ln, > Soluion: (a) The corresponding
More informationLAPLACE TRANSFORM AND TRANSFER FUNCTION
CHBE320 LECTURE V LAPLACE TRANSFORM AND TRANSFER FUNCTION Professor Dae Ryook Yang Spring 2018 Dep. of Chemical and Biological Engineering 5-1 Road Map of he Lecure V Laplace Transform and Transfer funcions
More informationt 2 B F x,t n dsdt t u x,t dxdt
Evoluion Equaions For 0, fixed, le U U0, where U denoes a bounded open se in R n.suppose ha U is filled wih a maerial in which a conaminan is being ranspored by various means including diffusion and convecion.
More informationCHEMICAL KINETICS: 1. Rate Order Rate law Rate constant Half-life Temperature Dependence
CHEMICL KINETICS: Rae Order Rae law Rae consan Half-life Temperaure Dependence Chemical Reacions Kineics Chemical ineics is he sudy of ime dependence of he change in he concenraion of reacans and producs.
More informationTheory of! Partial Differential Equations-I!
hp://users.wpi.edu/~grear/me61.hml! Ouline! Theory o! Parial Dierenial Equaions-I! Gréar Tryggvason! Spring 010! Basic Properies o PDE!! Quasi-linear Firs Order Equaions! - Characerisics! - Linear and
More informationChapter 7: Solving Trig Equations
Haberman MTH Secion I: The Trigonomeric Funcions Chaper 7: Solving Trig Equaions Le s sar by solving a couple of equaions ha involve he sine funcion EXAMPLE a: Solve he equaion sin( ) The inverse funcions
More informationThe Arcsine Distribution
The Arcsine Disribuion Chris H. Rycrof Ocober 6, 006 A common heme of he class has been ha he saisics of single walker are ofen very differen from hose of an ensemble of walkers. On he firs homework, we
More informationInventory Analysis and Management. Multi-Period Stochastic Models: Optimality of (s, S) Policy for K-Convex Objective Functions
Muli-Period Sochasic Models: Opimali of (s, S) Polic for -Convex Objecive Funcions Consider a seing similar o he N-sage newsvendor problem excep ha now here is a fixed re-ordering cos (> 0) for each (re-)order.
More informationHOMEWORK # 2: MATH 211, SPRING Note: This is the last solution set where I will describe the MATLAB I used to make my pictures.
HOMEWORK # 2: MATH 2, SPRING 25 TJ HITCHMAN Noe: This is he las soluion se where I will describe he MATLAB I used o make my picures.. Exercises from he ex.. Chaper 2.. Problem 6. We are o show ha y() =
More informationu(x) = e x 2 y + 2 ) Integrate and solve for x (1 + x)y + y = cos x Answer: Divide both sides by 1 + x and solve for y. y = x y + cos x
. 1 Mah 211 Homework #3 February 2, 2001 2.4.3. y + (2/x)y = (cos x)/x 2 Answer: Compare y + (2/x) y = (cos x)/x 2 wih y = a(x)x + f(x)and noe ha a(x) = 2/x. Consequenly, an inegraing facor is found wih
More informationBasic Circuit Elements Professor J R Lucas November 2001
Basic Circui Elemens - J ucas An elecrical circui is an inerconnecion of circui elemens. These circui elemens can be caegorised ino wo ypes, namely acive and passive elemens. Some Definiions/explanaions
More information2.7. Some common engineering functions. Introduction. Prerequisites. Learning Outcomes
Some common engineering funcions 2.7 Inroducion This secion provides a caalogue of some common funcions ofen used in Science and Engineering. These include polynomials, raional funcions, he modulus funcion
More informationMath 116 Practice for Exam 2
Mah 6 Pracice for Exam Generaed Ocober 3, 7 Name: SOLUTIONS Insrucor: Secion Number:. This exam has 5 quesions. Noe ha he problems are no of equal difficuly, so you may wan o skip over and reurn o a problem
More informationEcon107 Applied Econometrics Topic 7: Multicollinearity (Studenmund, Chapter 8)
I. Definiions and Problems A. Perfec Mulicollineariy Econ7 Applied Economerics Topic 7: Mulicollineariy (Sudenmund, Chaper 8) Definiion: Perfec mulicollineariy exiss in a following K-variable regression
More information18 Biological models with discrete time
8 Biological models wih discree ime The mos imporan applicaions, however, may be pedagogical. The elegan body of mahemaical heory peraining o linear sysems (Fourier analysis, orhogonal funcions, and so
More informationFITTING EQUATIONS TO DATA
TANTON S TAKE ON FITTING EQUATIONS TO DATA CURRICULUM TIDBITS FOR THE MATHEMATICS CLASSROOM MAY 013 Sandard algebra courses have sudens fi linear and eponenial funcions o wo daa poins, and quadraic funcions
More informationGround Rules. PC1221 Fundamentals of Physics I. Kinematics. Position. Lectures 3 and 4 Motion in One Dimension. A/Prof Tay Seng Chuan
Ground Rules PC11 Fundamenals of Physics I Lecures 3 and 4 Moion in One Dimension A/Prof Tay Seng Chuan 1 Swich off your handphone and pager Swich off your lapop compuer and keep i No alking while lecure
More informationIn this chapter the model of free motion under gravity is extended to objects projected at an angle. When you have completed it, you should
Cambridge Universiy Press 978--36-60033-7 Cambridge Inernaional AS and A Level Mahemaics: Mechanics Coursebook Excerp More Informaion Chaper The moion of projeciles In his chaper he model of free moion
More informationSecond quantization and gauge invariance.
1 Second quanizaion and gauge invariance. Dan Solomon Rauland-Borg Corporaion Moun Prospec, IL Email: dsolom@uic.edu June, 1. Absrac. I is well known ha he single paricle Dirac equaion is gauge invarian.
More informationSecond-Order Differential Equations
WWW Problems and Soluions 3.1 Chaper 3 Second-Order Differenial Equaions Secion 3.1 Springs: Linear and Nonlinear Models www m Problem 3. (NonlinearSprings). A bod of mass m is aached o a wall b means
More informationMath 2214 Solution Test 1A Spring 2016
Mah 14 Soluion Tes 1A Spring 016 sec Problem 1: Wha is he larges -inerval for which ( 4) = has a guaraneed + unique soluion for iniial value (-1) = 3 according o he Exisence Uniqueness Theorem? Soluion
More informationThe role of the error function in three-dimensional singularly perturbed convection-diffusion problems with discontinuous data
The role of he error funcion in hree-dimensional singularl perurbed convecion-diffusion problems wih disconinuous daa José Luis López García, Eser Pérez Sinusía Depo. de Maemáica e Informáica, U. Pública
More informationThe expectation value of the field operator.
The expecaion value of he field operaor. Dan Solomon Universiy of Illinois Chicago, IL dsolom@uic.edu June, 04 Absrac. Much of he mahemaical developmen of quanum field heory has been in suppor of deermining
More information