APACE TRANSFORMS Ingrl rnform i priculr kind of mhmicl opror which ri in h nlyi of om boundry vlu nd iniil vlu problm of clicl Phyic. A funcion g dfind by b rlion of h form gy) = K x,y f x dx i clld h ingrl rnform of fx). Th funcion Kx, y) i clld h krnl of h rnform. Th vriou yp of ingrl rnform r plc rnform, Fourir rnform, Mllin rnform, Hnkl rnform c. plc rnform i widly ud ingrl rnform in Mhmic wih mny pplicion in phyic nd nginring. I i nmd fr Pirr-Simon plc, who inroducd h rnform in hi work on probbiliy hory. Th plc rnform i ud for olving diffrnil nd ingrl quion. In phyic nd nginring i i ud for nlyi of linr im-invrin ym uch lcricl circui, hrmonic ocillor, opicl dvic, nd mchnicl ym. Dfiniion Th plc rnform of funcion f), dfind for ll rl numbr, i h funcion F), dfind by: - {f )} = F f d. Th prmr i complx numbr. i clld h plc rnformion opror. Trnform of lmnry funcion. {} =, >. Proof: - {}.d = if > n! n+ whn n =,,,... = Γn+ ohrwi n+ n. Proof:
- n = d = + - n - n n n- n n- n n- n- = =...=... =, > -. Proof: - n! n+ if n i poiiv ingr n n- =... = n+ ohrwi n+ = d = - - d -- = - =, > - - in = +, Proof:. in, co = +, Proof:. co in co in d - - -co+in cod +, + inh = -, 6.
Proof: inh, coh =, - 7. Proof: coh, Propri of plc rnform. inriy propry. If, b, c b ny conn nd f, g, h b ny funcion of hn f +bg -ch = f +b g -c h Bcu of h bov propry of, i i clld linr opror.. Fir hifing propry. If {f )} = F) hn f = F -. Proof: -- f = f d = f d - -r = f d whr r = -. = Fr) = F -. Applicion of hi propry ld o h following rul.
= n. - n! n+. whn n =,,,... ohrwi b inb = - +b. - cob = - +b. inhb = - b. - b - cohb = - -b whr in ch c >. Γ n+ - n+ Exmpl: Find h plc rnform of h following.. Sin Soluion: in 6 in in in in in 6 6 8.. in in in 6 6 6. inco Soluion: inco in + in 9
. co Soluion: co co., f ),, Soluion: - - - f)..d.d d Exrci: Find h plc rnform of. in.. coh in. f in,,. Trnform of ingrl If f) = F hn fu)du = F Proof:
φ = f u du, hn φ = f nd φ =. \ φ = φ -φ Hnc φ = φ i.. f udu = F. Muliplicion by n = F If f n n n d hn f = - F n, n =,,,... d Proof: - W hv f d = F d d Byibniz' rul for diffrniion undr h ingrl ign d - d Conidr f d = F - f d = - d f d = - F d F d d which prov h horm for n =. Now um h horm o ru for n = my), o h m - m d f d = - F m d m m+ d - m d Thn f d = - F m+ d d By ibniz' rul, m m+ - m+ d f d = - F m+ d m
Thi how h if h horm i ru for n = m, i i lo ru for n = m+. Bu i i ru for n =. Hnc i i ru for n = + =, n = + = nd o on. Thu h horm i ru for ll poiiv ingrl vlu of n. Diviion by If f = F hn f = Fd Proof: - W hv F = f)d - Conidr F d f)d d - f)dd f) Exmpl: Find h plc rnform of. co Soluion: co = + d co d. - in Soluion: chnging h ordr of ingrion - f) d d i indpndn of - + +
in 9 d 6 in d 9 9 Now uing fir hifing propry,w g 6 6 in 9. co-cob Soluion: co-cob = - + +b co-cob = - d + +b +b = log + Exrci: Find h plc rnform of h following funcion.. in. - in in.. -cob. - co Priodic funcion: Suppo h h funcion Fi ) priodic wih priod. Thn i plc rnform i givn by F ) F ) d F ) d. u pu n. n) n n Thn F ) xp n ) F n) d. n
Bu F n) F ), w g F ) xp n ) xp ) F ) d xp ) xp ) F ) d w Thorm If ) n n xp ) F ) d. F h plc Trnform nd if n F ) d F ) F ), F ). w Exmpl ), c Find h rnform of h funcion, c) ; c, c), c)., c c c d, ). c c c ) Exmpl b) Find h rnform of h qur-wv funcion, c Q, c) ; Q c, c) Q, c)., c c Q c c c d d c c c / c ) ) c, ) nh. c c c c / ) ) Exrci:. Dfin ringulr-wv funcion T, c ), c T, c) ; T c, c) T, c). c, c c plc Trnform. by Skch T, c) nd find i
G by G ), c; G c) G ),. Skch h. Dfin h funcion ) grph of G ) nd find i plc Trnform.. Dfin h funcion ) grph of Snd ) find i plc Trnform. S by S ), ; S ) S ),. Skch h, dcribd blow, nd find i. Skch hlf-wv rcificion of h funcion in in, rnform. F ) ; F F ). w, w F whr F ) for nd F ) F ).. Find ) A Sp Funcion: Applicion frqunly dl wih iuion h chng bruply pcifid im. W nd noion for funcion h will uppr givn rm up o crin vlu of nd inr h rm for ll lrgr. Th funcion w r bou o inroduc ld u o powrful ool for conrucing invr rnform., u dfin funcion ) by Th dfiniion y h ) )., i zro whn h rgumn i ngiv nd ) i uniy whn h rgumn i poiiv or i zro. I follow h, c c)., c Th plc Trnform of c) F c) i c) F c) F c) d. c Now pu c v in h ingrl o obin cv) c c) F c) F v) dv F ). Thorm f ) F ), If if c, nd if ) c, f ) F c) c). Exmpl ): c Fb ignd vlu no mr wh on) for
, whr y ). 6, y ) ) ) 6 ) ) 6 ) ) Find y) nd ) y 6 ). Exmpl b) : Find nd kch funcion g ) for which g ). g ) ) ) ) ), 7,, g) Exrci:. Skch h grph of h givn funcion for. - i) ) ) ). ii). Expr ) ). Fin rm of h funcion nd find ) i), F )., ii), F ),. 7, iii) in, F )., F.
. Find nd kch n invr plc rnform of. Evlu ). ) ). If Fi ) o b coninuou for nd F ), vlu F), F), F ).. INVERSE APACE TRANSFORMS Inroducion: {f)} = F). Thn f) i dfind h invr plc rnform of F) nd i dnod by - {F)}. Thu - F) = f)..) - i known h invr lplc rnform opror nd i uch h In h invr problm ), F) i givn known) nd f) i o b drmind. Propri of Invr plc rnform For ch propry on plc rnform, hr i corrponding propry on invr plc rnform, which rdily follow from h dfiniion. ) inriy Propry - {F)} = f) nd - {G)} = g) nd nd b b ny wo conn. Thn ) Shifing Propry - [ F) b G)] = - {F)} b - { G)} If - {F)}=f) hn - [F-)]= - {F)} ) Invr rnform of driviv If - {F)}=f) hn - {F n )}= ) n n { F )}
) Diviion by If - {F)}=f) hn F) f ) d Tbl of Invr plc Trnform of om ndrd funcion F) f ) F ),,, Co Sin,,, Sin h Co h, n n =,,,,..., n n > - n n! n n Exmpl i). Find h invr plc rnform of h following: b ii) 9 iii) 9
Soluion: ) i b b b ii in co ) 9 9 9 8 ) iii h h in co in co Exrci: Find h invr plc rnform of h following 6 ) i ii) 6 ) iii) 8 iv) 8 Evluion of - F ) W hv, if { f)} = F), hn [ f)] = F ), nd o - F ) = f) = - F) Exmpl Evlu:. - - Givn Uing h formul g w nd n king nd n n n,!
Givn - Evlu:. - - - in co )Evlu : - h h in co Exrci Find h invr plc rnform of h following i) 6 ii) 9 7 iii) ) iv) ) ) INVERSE APACE TRANSFORM OF - F)
Evluion of - [ - F)] W hv, if {f)} = F), hn [f-) H-) = - F), nd o - [ - F)] = f-) H-) Exmpl ) Evlu : Hr, F ) Thrfor f ) F ) 6 Thu f ) H ) H 6 ) Evlu : Givn f H f H Hr f ) Now rlion ) rd Givn in f ) H co H co H in co co H ) Exrci: Find h invr plc rnform of h following i) coh ii)
iii) iv) INVERSE APACE TRANSFORM BY PARTIA FRACTION AND OGARITHMIC FUNCTION Exmpl ) ) ) F)= ) ) By pplying pril frcion w g ) ) = ) ) ) ) ) ) B A B A = ) ) B A pu = B B pu =- A A Thrfor F)= ) ) F)= ) ) Tking invr lplc rnform, w g )} { F ) ) = Evlu:. hv w C B A
Thn +- = A+) -) + B -) + C +) For =, w g A =, for =, w g C = nd for = -, w g B = -. Uing h vlu in ), w g Evlu:. Conidr C B A Thn + = A + ) + B + ) + ) + C + ) For = -, w g A =, for = -, w g C = - Compring h cofficin of, w g B + C =, o h B =. Uing h vlu in ), w g Hnc Evlu:. ) D C B A Hnc = A + ) + ) + B -) + )+C + D) )
For =, w g A = ¼; for = -, w g B = ¼; compring h conn rm, w g D = A-B) = ; compring h cofficin of, w g = A + B + C nd o C = ½. Uing h vlu in ), w g Tking invr rnform, w g co h co co Evlu:. Conidr in in in in h
Trnform of logrihmic nd invr funcion W hv, if { f)} d d d d F),hn f F Hnc. F f ) Exmpl ) Evlu : log b F ) log log b d Thn F d b So h Thu or f f d d F b b log b b b ) Evlu n F ) n Thn or f or d d f F d d in F in in o h Invr rnformof Sinc f F F d w hv F f d
Exmpl: : ) Evlu co in in ) d F Thn F f o h F dno u w g Uing hi, pr. ingrion by on, : ) - d d Hnc hv w Evlu CONVOUTION THEOREM AND.T. OF CONVOUTION INTEGRA
Th convoluion of wo funcion f) nd g) dnod by f) g) i dfind f) g) = f u) g u) du Propry: f) g) = g) f) Proof :- By dfiniion, w hv f) g) = f u) g u) du Sing -u = x, w g f) g) = f x) g x) dx) = g x) f x) dx g ) f ) Thi i h dird propry. No h h oprion i commuiv. CONVOUTION THEOREM: [f) g)] = {f)}.{g)} Proof: u dno f) g) = ) = f u) g u) du Conidr [ )] [ f u) g u)] d = f u) g u) du ) W no h h rgion for hi doubl ingrl i h nir r lying bwn h lin u = nd u =. On chnging h ordr of ingrion, w find h vri from u o nd u vri from o.
u u= =u = u= Hnc ) bcom [)] = u u u = g u) f u) g u) ddu u) f u) d du u u = g u) v f v) dv du, whr v = -u u = g u) du v f v) dv = g). f) Thu f). g) = [f) g)] Thi i dird propry. Exmpl:. Vrify Convoluion horm for h funcion f) nd g) in h following c : i) f) =, g) = in i) Hr, ii) f) =, g) = f g = f u) g u) du = u in u) du Employing ingrion by pr, w g o h f g = in
[f g] = ) ) Nx conidr f). g) = ) ) From ) nd ), w find h [f g] = f). g) Thu convoluion horm i vrifid. ii) Hr f g = u u du Employing ingrion by pr, w g o h f g = [f g] = ) ) Nx f). g) = ) ) From ) nd ) w find h [f g] = f). g) Thu convoluion horm i vrifid.. By uing h Convoluion horm, prov h f ) d f ) u dfin g) =, o h g-u) = Thn f ) d f ) g u) d [ f g]
= f). g) = f). Thu f ) d f ) Thi i h rul dird.. Uing Convoluion horm, prov h u in u) du ) ) u dno, f) = - g) = in, hn u u) du f u) g in u) du = f). g) = ) ) = ) ) Thi i h rul dird.
) Employ plc Trnform mhod oolv hingrl quion. uin u f) l f du Tking plc rnform of h givn quion, w g f) f u in u du By uing convoluion horm, hr, w g f) f ) in f ) Thu f ) or f ) Thi i h oluion of h givn ingrl quion. Exrci: Solv h following problm. Vrify convoluion horm for h following pir of funcion: i) f) = co, g) = cob ii) f) =, g) = - iii) f) = g) = in. Uing h convoluion horm, prov h following: u i) u) co udu ) ) u ii) u) u du )
) f ) f u u du ) f ' ) f u co u du, f ) Invr rnform of F) by uingconvoluion horm W hv, if ) F) nd g) G), hn f) g) f ) g ) F ) G ) nd o F ) G ) f ) g ) f ugu Thi xprion i clld du h convoluion horm for invr plc rnform Exmpl Employ convoluion horm o vlu h following : ) b u dno F), G ) b Tking hinvr, w g f) -, g) -b Thrfor, by convoluion horm, - b u bu b du u du b b b b
) u dno F), G ) Thn f) in, g) co Hnc by convoluion horm, ) - Hr in uco u du in in u du, co u u in by uing compound ngl in formul F), G ) Thrfor f), g) in By convoluion horm, w hv - u in u du u in u co u in co in co By mploying convoluion horm, vlu h following:
) ) ) b ) ) 6), b Applicion of plc rnform plc rnform i vry uful for olving linr diffrnil quion wih conn cofficin nd wih givn iniil condiion. W k h plc rnform of h diffrnil quion nd hn mk u of h iniil condiion, which rnform h diffrnil quion o n lgbric quion. Solv for plc rnform from hi lgbric quion nd h rquird oluion i obind by king h invr of hi rnform. plc rnform of driviv: f ' f f ). ''. f f f f ') ''' '. Problm: f f f f f '') nd o on.. Solv y '' y co givn ' y, y whn. Soluion: Tking plc rnform, On uing h iniil condiion, ' Y y y Y Y Tking invr,
y in. Solv '' ' ', givn y y y y 6y Soluion: Tking plc rnform, On uing h iniil condiion, ' Y y y Y y 6 Y Y 8 8 6 Rolving ino pril frcion, Y Tking h invr plc rnform, y ) y Exrci :- Solv h following: '' ' '. y y y y y,. y ''' y '' y ' y y y ' y '' Elcricl circui:, Conidr impl circui comprid of n inducnc of mgniud hnry ), rinc of mgniud R ohm), nd cpcinc of mgniud C frd) conncd in ri. If E i h mf vol) pplid o n RC circui, hn h currn i mpr) in h circui im i govrnd by h diffrnil quion, di q Ri E d C
dq Hr q i h chrgcoulomb) i rld o i hrough h rlion i. If q) hn h d bov quion cn b rwrin, di Ri i d E dx C Exmpl:. A R circui crri n mf of volg E E in, whr E nd r conn. Find h currn i in h circui if iniilly hr i no currn in h circui. Soluion: Th diffrnil quion govrning h currn i i, di Ri E in d or, di E R i in d Tking plc rnform on boh id, E I i) I, whr R Applying h iniil condiion, i), w g, I E By rolving ino pril frcion, E I Tking invr plc rnform, E i in co. A rinc R in ri wih n inducnc i conncd wih mf E ). Th currn i di i givn by, Ri E). Th wich i conncd im nd diconncd d
im Find h currn i in rm of, givn h h mf i conn whn h wich i on. Soluion: E E ) Hr conidr E conn. E ) E H ) H ), Th govrning quion bcom, di R E i H ) H ) d Tking plc rnform nd pplying h iniil condiion, i), E I I, whr E E I ) R Tking invr plc rnform, ) R E i H ) R E i R E R Exrci:. A volg E i pplid o circui of inducnc nd rinc R. Show E R h currn im i R.. A impl lcricl circui coni of rinc R nd inducnc in ri wih conn mf E. If h wich i clod whn, find h currn ny im. M pring ym
Conidr pring of lngh x, id on nd o uppor nd h ohr nd i id o fixd m m which i fr. If F i h forc cing on h objc, hn from Nwon cond lw of moion, d x m kx d Suppo h mdium hrough h up i workd i riing wih vlociy of x' ) w hv, d x m kx cx ' d mx '' ) cx" ) kx ) Th uxiliry quion i givn by, md cd k Th roo of h qudric quion r, c c mk D m Th vlu of c mk drmin h niiviy of h mdium.... c c c mk impli moion i undr-dmpd. mk impli h moion i criiclly dmpd. mk impli h moion i ovr-dmpd. Hr k i h iffn of h pring nd cn b givn by h wigh of h objc pr uni ol w lngh of h pring, k, whr b i h lngh of h pring nirly nd w mg b. A pring cn xnd cm whn.kg of m i chd o i. I i upndd vriclly from uppor nd ino vibrion by pulling i down cm nd impring vlociy of cm / vriclly upwrd. Find h diplcmn from i quilibrium. Soluion: x ) b h diplcmn from i quilibrium. Hr b cm, m gm, x) cm, x') cm / W cn hu clcul h vlu of k, i.. k dy / cm Th quion of moion i, x'' ) x ) x'' ) 9 x ) Tking plc rnform,
X x) x') 9X X 9 9 Tking h invr plc rnform, x ) co7 in 7 7 7. A pring of iffn k h m m chd o on nd. I i cd upon by xrnl forc Ain. Dicu i moion in gnrl. Soluion: W know from h Nwon lw of moion, x) x, x') v k x'' ) x ) Ain m Tking plc rnform, k A X x) x') X m m By pplying h iniil condiion, k m A X x v m k m Tking h invr plc rnform, k m d x m kx in, d k in Ain k ) co m A x x v m k m k k m m Exrci:. A pring i rchd 6 inch by pound wigh. h wigh b chd o h pring nd pulld down inch blow h quilibrium poin. If h wigh id rd wih n upwrd vlociy of f pr cond, dcrib h moion. No dmping or imprd forc i prn.
. A pring i uch h lb wigh rch i 6 inch. An imprd forc co8 i cing on h pring. If h pound wigh i rd from h quilibrium poin wih n upwrd vlociy of f pr cond, dcrib h moion.