Boyce/DiPrim 9 h e, Ch 6.6: The Convoluion Inegrl Elemenry Differenil Equion n Bounry Vlue Problem, 9 h eiion, by Willim E. Boyce n Richr C. DiPrim, 9 by John Wiley & Son, Inc. Someime i i poible o wrie Lplce rnform H H = FG, where F n G re he rnform of known funcion f n g, repecively. In hi ce we migh expec H o be he rnform of he prouc of f n g. Th i, oe H = FG = L{f }L{g} = L{fg}? On he nex lie we give n exmple h how h hi equliy oe no hol, n hence he Lplce rnform cnno in generl be commue wih orinry muliplicion. In hi ecion we exmine he convoluion of f n g, which cn be viewe generlize prouc, n one for which he Lplce rnform oe commue.
Obervion Le f = n g = in. Recll h he Lplce Trnform of f n g re Thu n L f L, L g L in L L f g L in f L g Therefore for hee funcion i follow h f g L f L g L
Theorem 6.6. Suppoe F = L{f } n G = L{g} boh exi for >. Then H = FG = L{h} for >, where h f g f g The funcion h i known he convoluion of f n g n he inegrl bove re known convoluion inegrl. Noe h he equliy of he wo convoluion inegrl cn be een by mking he ubiuion u = -. The convoluion inegrl efine generlize prouc n cn be wrien h = f *g. See ex for more eil.
Theorem 6.6. Proof Ouline h L g f e g f e f g e u f e g u u f e g g e u u f e G F u u
Exmple : Fin Invere Trnform of Fin he invere Lplce Trnform of H, given below. Soluion: Le F = / n G = / +, wih Thu by Theorem 6.6., in G L g F L f H h H L in
Exmple : Soluion h of We cn inegre o implify h, follow. in in in ] [co co co co co in in in h h H L in
Exmple : Iniil Vlue Problem of 4 Fin he oluion o he iniil vlue problem y 4 y g, y 3, y Soluion: or L { y } 4 L { y } L { g } L { y } y y 4 L { y } G Leing Y = L{y}, n ubiuing in iniil coniion, 4 Y 3 G Thu 3 G Y 4 4
We hve Thu Noe h if g i given, hen he convoluion inegrl cn be evlue. g y in in 3co Exmple : Soluion of 4 4 4 4 3 4 4 3 G G Y
y 4 y g, y 3, y Exmple : Lplce Trnform of Soluion 3 of 4 Recll h he Lplce Trnform of he oluion y i 3 G Y Φ Ψ 4 4 Noe epen only on yem coefficien n iniil coniion, while epen only on yem coefficien n forcing funcion g. Furher, = L - { } olve he homogeneou IVP y 4 y, y 3, y while = L - { } olve he nonhomogeneou IVP y 4 y g, y, y
Exmple : Trnfer Funcion 4 of 4 Exmining more cloely, G Ψ H G, where H 4 4 The funcion H i known he rnfer funcion, n epen only on yem coefficien. The funcion G epen only on exernl exciion g pplie o yem. If G =, hen g = n hence h = L - {H} olve he nonhomogeneou iniil vlue problem y 4 y, y, y Thu h i repone of yem o uni impule pplie =, n hence h i clle he impule repone of yem.
Inpu-Oupu Problem of 3 Conier he generl iniil vlue problem Thi IVP i ofen clle n inpu-oupu problem. The coefficien, b, c ecribe properie of phyicl yem, n g i he inpu o yem. The vlue y n y ' ecribe iniil e, n oluion y i he oupu ime. Uing he Lplce rnform, we obin or y by cy g, y y y, y Y y y b Y y cy G b y y G Y Φ Ψ b c b c
y by cy g, y y y, y Lplce Trnform of Soluion of 3 We hve b y y G Y Φ Ψ b c b c A before, epen only on yem coefficien n iniil coniion, while epen only on yem coefficien n forcing funcion g. Furher, = L - { } olve he homogeneou IVP y by cy, y y y, y while = L - { } olve he nonhomogeneou IVP y by cy g, y, y
Trnfer Funcion 3 of 3 Exmining more cloely, G Ψ H G, b c where H b c A before, H i he rnfer funcion, n epen only on yem coefficien, while G epen only on exernl exciion g pplie o yem. Thu if G =, hen g = n hence h = L - {H} olve he nonhomogeneou IVP y by cy, y, y Thu h i repone of yem o uni impule pplie =, n hence h i clle he impule repone of yem, wih H G h g L