Theoretical Physics Prof. Ruiz, UNC Asheville, doctorphys on YouTube Chapter Q Notes. Laplace Transforms. Q1. The Laplace Transform.

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1 Theoreical Phyic Prof. Ruiz, UNC Aheville, docorphy o YouTue Chaper Q Noe. Laplace Traform Q1. The Laplace Traform. Pierre-Simo Laplace ( ) Courey School of Mhemic ad Siic Uiveriy of S. Adrew, Scolad The Laplace raform i defied a follow:, where F( ) f ( ) e d >. Thi i alo wrie wih he oio how elow. F( ) L{ f ( )} I he piri of our heoreical phyic coure, we would like o "derive" hi formula. We will follow Prof. Muck' "derivio," oe h he give i hi Differeial Equio coure MIT. A hi poi, do o e cocered wih ay applicio of he Laplace raform. We will do h ler. Prof. Arhur Muck, MIT Differeial Equio Prof. Muck ue he rick x e l x o how how o arrive he Laplace raform from a ifiie erie. We ar wih he power erie A( x) ax, o where we wrie A(x) ice he coefficie are give y he lile a. Michael J. Ruiz, Creive Commo Ariuio-NoCommercial-ShareAlike 3. Upored Licee

2 Sice each i oe more ha he previou oe, 1 ad we wrie. o A( x) a x Now o go o a coiuou variale, we chage hi o a iegral. Rememer our hree ep: 1)chage he dela o "d", )rip off he idex ad replace wih a fucio of your promoed coiuou variale, ad 3)ur he ummio ig io a "ake" (a iegral ig). A( x) a( ) x d Everyoe love he ural ae e, o we ue he rick a l x l x A( e ) a( ) e d x l x e ad wrie he aove To icreae our chace h hi iegral will o "low up" i our face, we reric l x o h i i egive. Therefore, we w l x. Thi occur whe x 1, viualized from he plo elow. Image Courey The Auralia Learig ad Teachig Coucil, School of Phyic, The Uiveriy of New Souh Wale, Auralia. Therefore, l x, where >. We he have A( e ) a( ) e d, which i ( A e ) a( ) e d Michael J. Ruiz, Creive Commo Ariuio-NoCommercial-ShareAlike 3. Upored Licee

3 Now wo hig look rage. Fir, h e i he argume. So we fix hi y wriig ( A e ) F( ) a( ) e d Nex, h look rage ice we ypically reerve "" for ieger. So le' ue he variale "" iead. The, F( ) a( ) e d Fially, ice we origially ared wih "A" mched o he lile "a," le' replace a() wih f() o h "F" i mched wih lile "f." Thi i he Laplace raform. F( ) f ( ) e d Q. Evaluig Laplace Traform. where >. Sill, do o cocer yourelf wih ay applicio. Le' ecome familiar wih he Laplace raform y evaluig Laplace raform for ome commo fucio f(). The fucio f ( ) 1 F( ) f ( ) e d e d e 1 F( ) F( ) 1, where >. Michael J. Ruiz, Creive Commo Ariuio-NoCommercial-ShareAlike 3. Upored Licee

4 f ( ) 1 F( ) 1/ Image made Fucio Grapher ad Calculor. Thik of hee wo graph a reidig i wo epare pace, like wo world. The imple co fucio i -pace appear a 1/ i -pace. See he aalogy elow where we raform from he real igh ky o Va Gogh' Sarry Nigh (1889). Normal Space Va Gogh Space The fucio f ( ) e ( a ) F( ) e e d e d. We mu have a ( a ) e 1 1 F( ) a a a for > a >. Michael J. Ruiz, Creive Commo Ariuio-NoCommercial-ShareAlike 3. Upored Licee

5 Summary: F( ) 1 a, where a >. The fucio g( ) e f ( ) ad he Laplace Traform Shifig Propery ( a) G( ) e f ( ) e d f ( ) e d G( ) F( a), where > a. PQ1 (Pracice Prolem). Fid he Laplace raform for f ( ) e hifig propery G( ) F( a), where g( ) e f ( ) wih f ( ) 1. y uig he The fucio f ( ) co ad i i We ue he Real-Imagiary Trick ad ake f ( ) co + ii e. The imagiary umer "i" keep he wo he fucio epare for u. Ue L{ e } 1 a wih a i. L{ e } 1 i 1 + i + i L{ e } i + i + L{co } L{i } ad + + Michael J. Ruiz, Creive Commo Ariuio-NoCommercial-ShareAlike 3. Upored Licee

6 The coie ad ie fucio ca alo ee worked wih i he radiioal maer. The fucio f ( ) co (he radiioal way). Le' ue he ackward Euler formula: co e + e i i 1 1 i i F( ) e e d e e d + Now ue our former reul L{ e } 1 a where a >. F( ) i + i F( ) 1 + F( ) + PQ (Pracice Prolem). Ue he radiioal way o how h he Laplace raform for f ( ) i i F( ). + Michael J. Ruiz, Creive Commo Ariuio-NoCommercial-ShareAlike 3. Upored Licee

7 PQ3 (Pracice Prolem). Ue he hifig propery h he Laplace raform of g( ) e f ( ) i G( ) F( a) o how h he Laplace raform of f ( ) e co i + a F( ) ( a). + + PQ4 (Pracice Prolem). Ue he hifig propery o how h he Laplace raform of f ( ) e i i The fucio f ( ) F( ) ( ). + a + F( ) e d d F( ) e d d d F( ) L{1} d F( ) d d 1! F( ) 1 + By he way, oe alo h he Laplace raform i a liear operio Michael J. Ruiz, Creive Commo Ariuio-NoCommercial-ShareAlike 3. Upored Licee

8 L{ α f ( ) + β g( )} α L{ f ( )} + β L{ g( )} PQ5 (Pracice Prolem). Explai or how why L {}. Our Laplace Traform Tale ( > a > ). f ( ) F( ) 1 1! 1 + e 1 a co + i + e co + a ( + a) + e i ( + a) + Michael J. Ruiz, Creive Commo Ariuio-NoCommercial-ShareAlike 3. Upored Licee

9 Q3. The Laplace Traform of a Derivive. df L{ f '( )} e d d We will ue iegrio y par. Alway hik of iegrio y par a eig reled o he produc rule for differeiio. d d df f e e fe d Therefore, d L{ f '( )} f e d + f e d d L{ f '( )} fe + L{ f ( )} I i impor h f ( ) < e o he iegral coverge. L{ f '( )} fe + L{ f ( )} f () + F( ) L{ f '( )} F( ) f () Nex, coider a Laplace raform of a ecod derivive. The rick i o coider he ecod derivive a a fir derivive of omehig. L{ f "( )} L{ g '( )} wih g( ) f '( ) L{ g '( )} G( ) g() wih G( ) F( ) f () L f F f f { "( )} ( ) () '() Michael J. Ruiz, Creive Commo Ariuio-NoCommercial-ShareAlike 3. Upored Licee

10 Q4. Differeial Equio: Radioacive Decay. Now come he applicio! 1. Take he Laplace Traform of Your Differeial Equio The Differeial Equio MELTS io a Algeraic Equio i -Space Normal Space Va Gogh Space. Solve he Algeraic Equio i -Space 3. Ue Your Laplace Traform Tale o Come Home o Regular Space Your "Laplace Traform Tale" i your porhole o reur home. Radioacive Decay The ifiieimal chage i he umer of radioacive decay paricle i proporioal o he produc of he umer of paricle remaiig ad he ifiieimal ime ierval. The miu ig idice a decreae i radioacive paricle remaiig. dn λnd wih λ > Therefore, he re of chage i proporioal o wh you have lef. dn d λn PQ6 (Pracice Prolem). Solve he aove differeial equio uig adard mehod. We will olve hi differeial equio uig Laplace raform. Michael J. Ruiz, Creive Commo Ariuio-NoCommercial-ShareAlike 3. Upored Licee

11 We will ue hi form for our equio. dn( ) d + λn( ) 1. Take he Laplace Traform dn( ) L{ } + L{ λn( )} L{} d Rememer h L{ f '( )} F( ) f (). Therefore.. Solve Your Algeraic equio F( ) N() + λf( ) F( ) + λf( ) N() F( )( + λ) N() F( ) N() + λ 3. Ue he Laplace Traform Tale o Ge Your Soluio N( ) L { F( )} N() L { } + λ e 1 a N( ) N() e λ Michael J. Ruiz, Creive Commo Ariuio-NoCommercial-ShareAlike 3. Upored Licee

12 Q5. The Damped Harmoic Ocillor. A Ma i Aached o Sprig. Courey David M. Harrio Deparme of Phyic, Uiveriy of Toroo Recall our ma o he prig. F kx v ma d x dx + + m kx d d Le' olve hi wih iiial codiio x() A ad dx() d v() A. m We will eed he Laplace raform for he fir ad ecod derivive: L{ f '( )} F( ) f () ad L{ f "( )} F( ) f () f '(). 1. Take he Laplace Traform d x dx + + m kx d d [ ] ( ) () () + ( ) () + ( ) A v() A ad we have m m F x v F x kf Wih our iiial codiio x() ( ) + + [ ( ) ] + ( ) m m F A A F A kf Michael J. Ruiz, Creive Commo Ariuio-NoCommercial-ShareAlike 3. Upored Licee

13 . Solve Your Algeraic equio F( ) m + + k Am A + A ( m + ) F( ) A m + + k 3. Ue he Laplace Traform Tale o Ge Your Soluio ( m + ) x L F L A m + + k 1 1 ( ) { ( )} { } From our ale, i appear we have omehig cloe o hi oe elow. Thi look promiig o complee he quare i he deomior. e co + a ( + a) + Fir divide y he ma m. F( ) A ( + ) m k + + m m F( ) ( + ) A m k + + m m m Michael J. Ruiz, Creive Commo Ariuio-NoCommercial-ShareAlike 3. Upored Licee

14 + a give u ( + a) + Mchig hi wih a m ad k m m wih he oluio x( ) Ae co Noe h if here i o fricio, i.e., for he friciole ocillor i defied a mechaic oe ofe defie Wih hee defiiio: β k, he m m k, i.e., m. Thi agular frequecy k m, callig hi he dampig coefficie.. I claical k m β ad m, a m β ad k. m m β Michael J. Ruiz, Creive Commo Ariuio-NoCommercial-ShareAlike 3. Upored Licee

15 Summary: For he iiial codiio x() A ad v(), m β x( ) Ae co β, where m ad β. Courey Uer LP, Wikimedia Commo Sice π π f, he period of he damped ocillio i T T π, i.e., T π β. Michael J. Ruiz, Creive Commo Ariuio-NoCommercial-ShareAlike 3. Upored Licee