17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides


 Grace Rice
 1 years ago
 Views:
Transcription
1 7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion f() is piecewise coninuous on he inerval I = [a,b] if i is defined and coninuous on his inerval excep, probably, a finie number of poins,,,..., k, a each of which he lef and righ limis of his funcion exis (i.e., all he disconinuiies of he firs ype, or jump disconinuiies). Example. Consider he funcion he graph of which is given in he figure..5. f() =, 4,, > 4, f() Figure : Piecewise coninuous funcion f() from Example You can see ha here is one poin of disconinuiy, namely = 4, a which f() has he disconinuiy of he firs ype, or jump disconinuiy. We have lim 4 =, lim =, f() +f() hey boh exis and are no equal o each oher. The value of he jump is. Wha is he Laplace ransform of f()? We can find i using he definiion: L f} = = 4 4 f()e s d ( )e s d+ = s e s 4 s e s 4 = s e 4s s + s e 4s = s (e 4s ). 4 e s d MATH66: Inro o ODE by Arem Novozhilov, Fall 3
2 Example 3 (Heaviside funcion). The mos imporan example for us will be he Heaviside funcion, defined as, <, u() =, >. Noe ha L u()} = s, he same as for L }, because we are only ineresed in funcions on [, ). This yields ha for any funcion f() defined on [, ) i is rue ha L f()} = L u()f()}. Addiionally o he Heaviside funcion, we consider shifed Heaviside funcion u( a) for a nonnegaive a (see he figure). u() u( a) a Figure : Heaviside funcion u() (lef) and shifed Heaviside funcion u( a) (righ) I is a simple exercise o check, using he definiion, ha L u( a)} = e as. Now I can sae he final propery of he Laplace ransform ha we will use (here are many more acually): 6 Time shifing. Le L f()} = F(s), hen To prove i, consider L u( a)f( a)} = L u( a)f( a)} = e as F(s). u( a)f( a)e s d = a f( a)e s d, because u( a) = when < a. Now make he subsiuion ξ = a, hen d = dξ and if = a hen ξ =. We obain f(ξ)e s(ξ+a) dξ = e sa L f}.
3 Le us find he Laplace ransform of he funcion in Example. Example 4 (Coninue Example ). Noe ha using he shifed Heaviside funcion we can consruc for any a < b he funcion u( a) u( b), such ha his funcion is equal o when (a,b) and zero oherwise (hink his ou!) This means u( a) u( b) a b Figure 3: u( a) u( b) for b > a ha for any f() he funcion f() ( u( a) u( b) ) equals f(), when (a,b), and oherwise. Now consider again, 4, f() =, > 4. This, using he previous, can be represened as Now, using Propery 6, f() = ( u() u( 4) ) + u( 4) = u()+u( 4). exacly as we already found in Example. L f} = L u()+u( 4)} = s +e 4s s Example 5. Find he Laplace ransform for, <, f() =, < <,, >. Again, using he properies of he Heaviside funcion, we can wrie f() = ( u( ) u( ) ) = u( ) u( ). However, we canno apply Propery 6 direcly o his expression since we need he expressions of he form f( a)u( a). To deal wih i, consider = ( +) = ( ) +( )+., 3
4 Similarly, Hence we have = ( +) = ( ) +4( )+4. which implies ha f() = ( ( ) +( )+ ) u( ) ( ( ) +4( )+4 ) u( ), L f} = ( s 3 + s + ) ( e s + s s s + 4 ) e s. s Now we are ready o sar solving ODE wih piecewise coninuous righ hand side. Example 6. Solve where We have y +y +y = f(), y() =, y () =, f() =, <,, >. f() = u() u( ). Applying he Laplace ransform o boh sides, we have (s +s+)y s = s ( e s ), or, afer some rearrangemen, Since, using he parial fracions, Y = s s(s+) e s. hen we find s(s+) = s s+ (s+), y() = L Y} = ( e ( ) e ( ) ( ) ) u( ), using Propery 6. Noe ha we can rewrie our soluion as, <, y() = e ( ), >. The graph of he soluion in given in he figure below. Example 7. Solve where f() as in he figure below. y +y +y = f(), y() =, y () =, 4
5 y() Figure 4: Soluion y() in Example 6 f() π Figure 5: f() in Example 7 π We find ha f() = u( π) u( π). Applying he Laplace ransform o he ODE, we find from where s Y s+sy +Y = s (e πs e πs ), Y = s+ s +s+ + s(s +s+) (e πs e πs ). Noe ha using parial fracion decomposiion, we have herefore, finally we have s(s +s+) = s s+ s +s+, Y(s) = s (e πs e πs )+ s+ s +s+ ( e πs +e πs ). Now our ask is o find he inverse Laplace ransform for Y. We have L s} = u(), hence } L s (e πs e πs ) = u( π) u( π). 5
6 For he second erm in Y(s) consider firs herefore s+ s +s+ = s+/+/ (s+/) +( 3/) = s+/ (s+/) +( 3/) + 3/ 3 (s+/) +( 3/), } ( s+ 3 L s = e / cos +s+ + sin 3 3. Finally, using Propery 6, we have y() = L Y} = L s (e πs e πs )+ s+ } s +s+ ( e πs +e πs ) = 3 = u( π) u( π)+e (cos / + 3 sin u() 3 3( π) e (cos ( π)/ + 3( π) sin u( π)+ 3 3( π) +e (cos ( π)/ + 3( π) sin u( π). 3 Below is he graph of he soluion. y() π π Figure 6: Soluion y() depending on ime in Example 7 Example 8. Solve y +y = f() = cos, < π/,, > π/, y() = 3, y () =. We can rewrie he righ hand side as f() = cos ( u() u( π/) ). Noe ha o invoke Propery 6 we need he funcion wih he same argumen as he argumen of he Heaviside funcion. f() = u()cos cos( π/+π/)u( π/) = u()cos+sin( π/)u( π/), 6
7 where I used or, cos(a+b) = cosacosb sinasinb. Therefore, afer applying he Laplace ransform, I will find (s +)Y = 3s + s s + + s + e π/, Y(s) = 3s s + s s + + s (s +) + (s +) e π/. To find he inverse Laplace ransform, we will need } } L s (s +), L (s +). For his, using he propery of he differeniaion of he frequency, noe ha L sin} = s (s +), L cos} = s (s +). Hence For he second one hence Finally we find } L s (s +) = sin. (s +) = ( ) s + s (s +), } L (s +) = (sin cos). y() = L Y} = 3cos sin+ sin+ ( sin( π/) ( π/)cos( π/) ) u( π/). By noing ha sin( π/) = cos and cos( π/) = sin, we can simplify his expression as 3cos sin+ y() = sin, <, 5 π 4 cos+ 4 cos, >. Insead of conclusion I said ha he examples in his lecure are he main reason we need he Laplace ransform. This is indeed rue, bu only parially. There are wo more cases when Laplace ransform becomes indispensable heoreical and compuaional ool: Firs, when f() is an arbirary periodic funcion differen from simple sine and cosine funcions. Second, when we consider soluions of he equaions wih impulsive righ hand side (hink of an insananeous impac). Unforunaely, due o ime limiaion, we do no cover hese exciing opics, and I refer you o he exbook. 7
The Transport Equation
The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be
More informationDynamic Contracting: An Irrelevance Result
Dynamic Conracing: An Irrelevance Resul Péer Eső and Balázs Szenes Sepember 5, 2013 Absrac his paper considers a general, dynamic conracing problem wih adverse selecion and moral hazard, in which he agen
More informationAND BACKWARD SDE. Nizar Touzi nizar.touzi@polytechnique.edu. Ecole Polytechnique Paris Département de Mathématiques Appliquées
OPIMAL SOCHASIC CONROL, SOCHASIC ARGE PROBLEMS, AND BACKWARD SDE Nizar ouzi nizar.ouzi@polyechnique.edu Ecole Polyechnique Paris Déparemen de Mahémaiques Appliquées Chaper 12 by Agnès OURIN May 21 2 Conens
More informationFollow the Leader If You Can, Hedge If You Must
Journal of Machine Learning Research 15 (2014) 12811316 Submied 1/13; Revised 1/14; Published 4/14 Follow he Leader If You Can, Hedge If You Mus Seven de Rooij seven.de.rooij@gmail.com VU Universiy and
More informationANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS
ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,
More informationA Simple Introduction to Dynamic Programming in Macroeconomic Models
Economics Deparmen Economics orking Papers The Universiy of Auckland Year A Simple Inroducion o Dynamic Programming in Macroeconomic Models Ian King Universiy of Auckland, ip.king@auckland.ac.nz This paper
More informationOptimal demand response: problem formulation and deterministic case
Opimal demand response: problem formulaion and deerminisic case Lijun Chen, Na Li, Libin Jiang, and Seven H. Low Absrac We consider a se of users served by a single loadserving eniy (LSE. The LSE procures
More informationOUTOFBAG ESTIMATION. Leo Breiman* Statistics Department University of California Berkeley, CA. 94708 leo@stat.berkeley.edu
1 OUTOFBAG ESTIMATION Leo Breiman* Saisics Deparmen Universiy of California Berkeley, CA. 94708 leo@sa.berkeley.edu Absrac In bagging, predicors are consruced using boosrap samples from he raining se
More informationBIS Working Papers. Globalisation, passthrough. policy response to exchange rates. No 450. Monetary and Economic Department
BIS Working Papers No 450 Globalisaion, passhrough and he opimal policy response o exchange raes by Michael B Devereux and James Yeman Moneary and Economic Deparmen June 014 JEL classificaion: E58, F6
More informationToday s managers are very interested in predicting the future purchasing patterns of their customers, which
Vol. 24, No. 2, Spring 25, pp. 275 284 issn 7322399 eissn 1526548X 5 242 275 informs doi 1.1287/mksc.14.98 25 INFORMS Couning Your Cusomers he Easy Way: An Alernaive o he Pareo/NBD Model Peer S. Fader
More informationA Working Solution to the Question of Nominal GDP Targeting
A Working Soluion o he Quesion of Nominal GDP Targeing Michael T. Belongia Oho Smih Professor of Economics Universiy of Mississippi Box 1848 Universiy, MS 38677 mvp@earhlink.ne and Peer N. Ireland Deparmen
More informationCostSensitive Learning by CostProportionate Example Weighting
CosSensiive Learning by CosProporionae Example Weighing Bianca Zadrozny, John Langford, Naoki Abe Mahemaical Sciences Deparmen IBM T. J. Wason Research Cener Yorkown Heighs, NY 0598 Absrac We propose
More informationWhich Archimedean Copula is the right one?
Which Archimedean is he righ one? CPA Mario R. Melchiori Universidad Nacional del Lioral Sana Fe  Argenina Third Version Sepember 2003 Published in he YieldCurve.com ejournal (www.yieldcurve.com), Ocober
More informationFIRST PASSAGE TIMES OF A JUMP DIFFUSION PROCESS
Adv. Appl. Prob. 35, 54 531 23 Prined in Norhern Ireland Applied Probabiliy Trus 23 FIRST PASSAGE TIMES OF A JUMP DIFFUSION PROCESS S. G. KOU, Columbia Universiy HUI WANG, Brown Universiy Absrac This paper
More informationThe Macroeconomics of MediumTerm Aid ScalingUp Scenarios
WP//6 The Macroeconomics of MediumTerm Aid ScalingUp Scenarios Andrew Berg, Jan Goschalk, Rafael Porillo, and LuisFelipe Zanna 2 Inernaional Moneary Fund WP//6 IMF Working Paper Research Deparmen The
More informationAsymmetry of the exchange rate passthrough: An exercise on the Polish data 1
Asymmery of he exchange rae passhrough: An exercise on he Polish daa Jan Przysupa Ewa Wróbel 3 Absrac We propose a complex invesigaion of he exchange rae passhrough in a small open economy in ransiion.
More informationThe U.S. Treasury Yield Curve: 1961 to the Present
Finance and Economics Discussion Series Divisions of Research & Saisics and Moneary Affairs Federal Reserve Board, Washingon, D.C. The U.S. Treasury Yield Curve: 1961 o he Presen Refe S. Gurkaynak, Brian
More informationExchange Rate PassThrough into Import Prices: A Macro or Micro Phenomenon? Abstract
Exchange Rae PassThrough ino Impor Prices: A Macro or Micro Phenomenon? Absrac Exchange rae regime opimaliy, as well as moneary policy effeciveness, depends on he ighness of he link beween exchange rae
More informationAnchoring Bias in Consensus Forecasts and its Effect on Market Prices
Finance and Economics Discussion Series Divisions of Research & Saisics and Moneary Affairs Federal Reserve Board, Washingon, D.C. Anchoring Bias in Consensus Forecass and is Effec on Marke Prices Sean
More informationBoard of Governors of the Federal Reserve System. International Finance Discussion Papers. Number 1003. July 2010
Board of Governors of he Federal Reserve Sysem Inernaional Finance Discussion Papers Number 3 July 2 Is There a Fiscal Free Lunch in a Liquidiy Trap? Chrisopher J. Erceg and Jesper Lindé NOTE: Inernaional
More informationThe Simple Analytics of Helicopter Money: Why It Works Always
Vol. 8, 201428 Augus 21, 2014 hp://dx.doi.org/10.5018/economicsejournal.ja.201428 The Simple Analyics of Helicoper Money: Why I Works Always Willem H. Buier Absrac The auhor proides a rigorous analysis
More informationKONSTANTĪNS BEŅKOVSKIS IS THERE A BANK LENDING CHANNEL OF MONETARY POLICY IN LATVIA? EVIDENCE FROM BANK LEVEL DATA
ISBN 9984 676 20 X KONSTANTĪNS BEŅKOVSKIS IS THERE A BANK LENDING CHANNEL OF MONETARY POLICY IN LATVIA? EVIDENCE FROM BANK LEVEL DATA 2008 WORKING PAPER Lavias Banka, 2008 This source is o be indicaed
More informationParttime Work, Wages and Productivity: Evidence from Matched Panel Data
Parime Work, Wages and Produciviy: Evidence from Mached Panel Daa Alessandra Caaldi (Universià di Roma "La Sapienza" and SBSEM) Sephan Kampelmann (Universié de Lille, CLERSE, SBSEM) François Rycx (Universié
More informationWP/15/85. Financial Crisis, US Unconventional Monetary Policy and International Spillovers. by Qianying Chen, Andrew Filardo, Dong He, and Feng Zhu
WP/15/85 Financial Crisis, US Unconvenional Moneary Policy and Inernaional Spillovers by Qianying Chen, Andrew Filardo, Dong He, and Feng Zhu 2015 Inernaional Moneary Fund WP/15/85 IMF Working Paper European
More informationResearch Division Federal Reserve Bank of St. Louis Working Paper Series
Research Division Federal Reserve Bank of S. Louis Working Paper Series Wihsanding Grea Recession like China Yi Wen and Jing Wu Working Paper 204007A hp://research.slouisfed.org/wp/204/204007.pdf March
More informationWhen Should Public Debt Be Reduced?
I M F S T A F F D I S C U S S I ON N O T E When Should Public Deb Be Reduced? Jonahan D. Osry, Aish R. Ghosh, and Raphael Espinoza June 2015 SDN/15/10 When Should Public Deb Be Reduced? Prepared by Jonahan
More informationDoes Britain or the United States Have the Right Gasoline Tax?
Does Briain or he Unied Saes Have he Righ Gasoline Tax? Ian W.H. Parry and Kenneh A. Small March 2002 (rev. Sep. 2004) Discussion Paper 02 12 rev. Resources for he uure 1616 P Sree, NW Washingon, D.C.
More informationAre Under and Overreaction the Same Matter? A Price Inertia based Account
Are Under and Overreacion he Same Maer? A Price Ineria based Accoun Shengle Lin and Sephen Rasseni Economic Science Insiue, Chapman Universiy, Orange, CA 92866, USA Laes Version: Nov, 2008 Absrac. Theories
More informationThe concept of potential output plays a
Wha Do We Know (And No Know) Abou Poenial Oupu? Susano Basu and John G. Fernald Poenial oupu is an imporan concep in economics. Policymakers ofen use a onesecor neoclassical model o hink abou longrun
More informationBIS Working Papers No 365. Was This Time Different?: Fiscal Policy in Commodity Republics. Monetary and Economic Department
BIS Working Papers No 365 Was This Time Differen?: Fiscal Policy in Commodiy Republics by Luis Felipe Céspedes and Andrés Velasco, Discussion Commens by Choongsoo Kim and Guillermo Calvo Moneary and Economic
More information