Appendix: The Laplace Transform

Transcription

1 Appedix: The Laplace Trasform The Laplace trasform is a powerful method that ca be used to solve differetial equatio, ad other mathematical problems. Its stregth lies i the fact that it allows the trasformatio of a differetial equatio to a algebraic equatio. The oe-sided Laplace trasform is defied as follows L x( t) X s x( t)e st dt, (I.) where the variable s is defied as cotaiig both a real ad imagiary part, i.e., s σ + iω with σ such that e st remais fiite as t. Referrig to equatio (I.) we say that X ( s) is the Laplace trasform of x( t). We also assumed that the time variable t starts at, but this could be chaged to ay other value (e.g., t ). For example, we calculate the Laplace trasform of a few simple fuctios [ ] Ae st dt L A A s e st A s, s > t L e at e at e st dt e s+a, s > a. (I.) A particularly importat trasform is that of a impulse of time duratio τ defied as x( t) τ, < t < τ, t > τ (I.3) with τ X s τ τ st e e st dt sτ If we ow take the limit of equatio (I.4) whe τ, we get lim X s τ sτ sτ ( e ). (I.4) sτ ( [ sτ ] ). (I.5) The limit of a fuctio such as, defied i equatio (I.3), whe the duratio of the impulse is take to be ifiitely small while keepig the area of the impulse costat is I

2 called a Dirac or delta fuctio. It is usually simply writte as δ t property that δ t for t ad δ ( t) for t, but, ad has the δ ( t t )dt. (I.6) Equally importat is the trasform of a step or Heaviside fuctio, represeted by H ( t), t >, t <. (I.7) Sice our defiitio of the Laplace trasform trucates ay fuctios that are o-zero for t <, the Laplace trasform of the step fuctio was evaluated i equatio (I.) ad foud to be L H ( t) s. (I.8) The list of trasforms appearig i Table I. ca be similarly verified. Table I. Laplace trasform pairs L Aδ ( t) A, s > L A H ( t) A s, s > L e at H ( t), L t H ( t) s > a!, s >,,, 3,... + s! +, s > a,,,3,... L t e at H ( t) L si( ωt) H ( t) ω s + ω, s > L cos( ωt) H ( t) s s + ω, s > ω L e at si( ωt) H ( t) ( ) + ω, s > a L e at cos( ωt) H ( t) ( ) + ω, s > a. II

3 The Laplace trasform also possesses other importat properties, some of which are (assumig that every time fuctio is zero for t < i what follows) I. Liearity. If A ad B are costats L A II. Trasform of derivatives. + By( t) AX s + BY ( s) (I.9) L d dt d e st dt x( t)e st dt + s sx ( s) x x( t)e st dt (I.) where we itegrated by parts, ad x the trasform of higher derivatives ca be show to give is the iitial coditio x( t). Similarly, L d dt L d dt s X s s X s III. Trasform of primitive of fuctios. L t sx dx( t) dt t d k x( t) s k k dt k x( τ )dτ t { x( τ )dτ } e st dt t x ( τ s { )dτ } e st + s X ( s) + t x ( τ s s )dτ { } t t x( t)e st dt (I.) (I.) where we agai itegrated by parts. The trasform of higher primitives is give by L... x( τ )( dτ ) X s + s k {... s x( τ )( dτ ) } k k+ t (I.3) IV. Time shiftig. Sice x( t) for t <, we ca write III

4 L x( t τ ) x( t τ )e st dt x( t τ )e st dt x( λ)e s( λ +τ ) dλ e sτ x( λ)e sλ dλ e sτ X s τ (I.4) where we made the substitutio λ t τ. V. Multiplicatio by a expoetial. L e at x( t) e at x( t)e st dt X x( t)e ( s+a)t dt (I.5) The residue theorem Oce a fuctio or a equatio has bee trasformed i the Laplace domai, the modified for oe purpose or aother, it will evetually eed to be trasformed back to the time domai. Although a iverse Laplace trasform ca be mathematically defied, it is always more coveiet ad easier to use the so-called residue theorem to go from the Laplace to the time domai. This theorem is stated as follows. Give a fuctio X ( s), m for which the deomiator ca be writte as a product of factors of the type j (where a j is called a pole of order m), we ca write L X ( s) ( m )! lim d m s a j ( ds m j ) m X ( s)e st, t > j (I.6), ad the quatity i betwee where is the umber of poles i the deomiator of X s the curly braces is called the residue of X ( s)e st at the pole a j of order m. Let s cosider a few examples IV

5 L lim d s a! ds e at, t > L ( ) d lim s a! ds e st ( ) te at, t > e st (I.7) ad fially L ( ) + ω lim s a+iω d! ds + lim s a iω L ( s + ( a iω )) d! ds iω ( ) s + ( a + iω ) + ω est ( s + ( a + iω )) iωe ( a+iω )t iωe ( a iω )t + iω iω e at cos( ωt), t >. ( ) ( ) + ω est (I.8) These results ca be verified agaist the examples preseted i Table I.. Applicatio to the damped oscillator problem Let s ow solve a few cases ivolvig the equatio of motio of a damped oscillator with differet types of drivig iput. The equatio to solve is I. f ( t) Aδ ( t). x ( t) + β x ( t) + ω x( t) f ( t) (I.9) + β x ( t) + ω x( t) L L f t (I.) Usig the liearity property of the Laplace trasform ad Table I., we get ( s X ( s) sx x ) + β sx ( s) x + ω X s A, (I.) V

6 or X ( s) s ( + βs + ω ) A + x ( s + β) + x. (I.) I everythig that will follow, we will assume that x x. We ow solve equatio (I.) X ( s) A s + βs + ω A s + β β ( ( ω )) s + β + β ω ( ) (I.3) We ow use the residue theorem stated i equatio (I.6) β β ( ω ) t A e β + β ( ω ) t β ω e β ω e βt A β ω e β ω t e β ω t, t > (I.4) A close examiatio of equatio (I.4) shows that the respose of the damped oscillator to a Dirac fuctio is othig more tha the complemetary solutio of the equatio of motio. I the case of the uderdamped oscillator ( β < ω ), we fid that x( t) A e βt si( ω t), t > (I.5) ω with ω ω β. II. f ( t) A H ( t) I this case, we have (assumig that β < ω, ad ω ω β ) X ( s) s ( + βs + ω ) A s (I.6) VI

7 Figure I. Respose to a Dirac fuctio drivig iput. x( t) A s s + βs + ω A s s + ( β iω ) ( s + ( β + iω )) e ( β iω )t A ω + iω iω β A ω ω t + e β +iω ( iω + β) iω e βt β + ω cos( ω t φ), t > (I.7) with β φ ta. ω (I.8) Figure I. Respose to a step fuctio as drivig iput. VII

8 The Laplace trasform ca be systematically applied to more complicated types of problems ad drivig fuctios (periodic or ot). It is also importat to realize that the solutio to a give problem provided by the applicatio of the Laplace trasform icludes both the complemetary ad the particular solutios. The Two-sided Laplace Trasform It is geerally the case i physics that a fuctio is ot limited to t but ca exist for times both positive ad egative. We ca the geeralize the oe-sided Laplace trasform give i equatio (I.) with its two-sided versio X s Usig our previous otatio for the oe-sided trasform L s we ca write for the two-sided trasform e st dt. (I.9) e st dt (I.3) X s e st dt + x( t)e st dt L s + L x t s. t> t< (I.3) I equatio (I.3) we made the depedecy of the Laplace trasform o the parameter s explicit by addig ( s) to the left-had side. The last term o the right-had side of the secod of equatios (I.3) ca be ascertaied with (usig λ t ) L x t s x t e st dt x( λ)e sλ dλ x( λ)e sλ dλ. (I.3) For example, if we calculate the two-sided Laplace trasform of the followig fuctio x( t) e t, t < e 3t, t (I.33) we fid (usig Table I.) VIII

9 X ( s) s! + s<! s + 3 s> 3 5, for 3 < s <. s + s 5 (I.34) We must also determie the proper relatio to calculate iverse Laplace trasforms for two-sided fuctios. Usig equatio (I.3) ad the residue theorem for oe-sided fuctios (i.e., equatio (I.6)) we ca write L X ( s) x t ( m )! lim d m s a j ( ds m j ) m X ( s)e st, t >. j (I.35) We ca ow look at egative time values by chagig t t, ad x( t) ( m )! lim d m s a j ( ds m j ) m X ( s)e st, t <. (I.36) j We fially also chage s s to fid x( t) j j ( m )! lim s a j d s ( m )! lim d m s a j m X s m j d m ( ) m ds m ( ) m s a j e st m X s e st ( m )! lim d m s a s a j ( ds m j ) m X ( s)e st, t <. j (I.37) That is, the iverse Laplace trasform for a fuctio defied oly for times t < is similar to that for a fuctio defied for t >, except for the overall egative sig. For example, if we calculate the iverse Laplace trasform of equatio (I.34) we fid 5( s ) + 5( s + 3) s + s 5 et H t s + s 5 e 3t H t e t H ( t) e 3t H ( t), (I.38) which is the same as equatio (I.33). Fially, we ote that because the two-sided Laplace trasform cosists of a itegral performed over the domai < t < the depedecies o derivatives ad itegrals IX

10 evaluated at t (see equatios (I.) ad (I.3)) do ot appear i the results of calculatios. Notably we have dt d... x ( τ { )( dτ ) }e st dt e st dt s X ( s) X ( s). s (I.39) X