t=a z=t z=a INTEGRAL EQUATIONS t=z z=x dz D.C. Sharma z=a z=a z t=z z=x t=a M.C. Goyal z=a

Transcription

1 z t= B(,) z=t z= z g t=z dt t= A(,) z B(,) z=t z= z= t A(,) t t=z z= B(,) dz D.C. Shrm M.C. Goyl z= A(,) t t= z e r dz z z= i B(,) t= C op yr ig ht ed M t er t IL i l INTEGRAL EQUATIONS A(,) PH dt

2 Itegrl Equtios D.C. Shrm Hed, Deprtmet of Mthemtics De, School of Mthemtics, Sttistics d Computtiol Scieces Cetrl Uiversity of Rjsth, Kishgrh, Ajmer M.C. Goyl De (Acdemics), Professor d Hed of Deprtmet of Mthemtics Rjsth College of Egieerig for Wome, Jipur PHI Lerig Copyrighted Mteril Delhi

3 Cotets Prefce... i Ackowledgemets... i 1. Bsic Cocepts Itroductio 1 1. Ael s Prolem Iitil Vlue Prolem d Boudry Vlue Prolem 1.4 Itegrl Equtio Specil Kids of Kerels Clssifictio of Itegrl Equtio Iterted Kerels Reciprocl Kerl or Resolvet Kerel Eigevlues d Eigefuctios Solutio of Itegrl Equtio 8 Eercise Applictios to Ordiry Differetil Equtios Itroductio 1. Method of Coversio of Iitil Vlue Prolem to Volterr Itegrl Equtio 1 Eercise Alterte Method of Trsformig the Iitil Vlue Prolem ito Volterr Itegrl Equtio 18 Eercise. 1.4 Boudry Vlue Prolem d its Coversio to Fredholm Itegrl Equtio 1 Eercise.3 7 PHI Lerig Copyrighted Mteril v

4 vi Cotets 3. Solutio of Homogeeous Fredholm Itegrl Equtios of the Secod Kid Itroductio 9 3. Chrcteristic Vlue (or Eigevlue) d Chrcteristic Fuctio (or Eigefuctio) Solutio of Homogeeous Fredholm Itegrl Equtio of the Secod Kid with Seprle (or Degeerte) Kerel Orthogolity of Two Fuctios Orthogolity of Eigefuctios Rel Eigevlues 34 Eercise Fredholm Itegrl Equtios with Seprle Kerels Itroductio Solutio of Fredholm Itegrl Equtio of the Secod Kid with Seprle (or Degeerte) Kerel 41 Eercise Itegrl Equtios with Symmetric Kerels Itroductio Symmetric Kerl Regulrity Coditio Ier or Sclr Product of Two Fuctios Orthogol System of Fuctios Fudmetl Properties of Eigevlues d Eigefuctios of Symmetric Kerels Hilert Schmidt Theorem Schmidt s Solutio of No-homogeeous Fredholm Itegrl Equtio of the Secod Kid 63 Prctice Questios with Itermedite Results 77 Eercise Solutio of Itegrl Equtios of the Secod Kid y Successive Approimtio PHI Lerig Copyrighted Mteril 6.1 Itroductio Iterted Kerel or Fuctio Resolvet Kerel or Reciprocl Kerel Solutio of Fredholm Itegrl Equtio of the Secod Kid y Successive Sustitutio Solutio of Volterr Itegrl Equtio of the Secod Kid y Successive Sustitutios 87

5 Cotets vii 6.6 Solutio of Fredholm Itegrl Equtio of the Secod Kid y Successive Approimtios: Itertive Method (Itertive Scheme) Neum Series Resolvet Kerel of Fredholm Itegrl Equtio Illustrtios Bsed o the Solutio of Fredholm Itegrl Equtio y Successive Approimtios (Itertive Method) 9 Eercise Eercise Reciprocl Fuctios Aother Approch to Solve Fredholm Itegrl Equtio of the Secod Kid (Volterr Solutio) Solutio of Volterr Itegrl Equtio of the Secod Kid y Successive Approimtios: Itertive Method (Neum Series) Resolvet Kerel d Volterr Itegrl Equtio Illustrtios to Epli the Solutio of Volterr Itegrl Equtio y Successive Approimtios (or Itertive Method) 11 Eercise Clssicl Fredholm Theory Itroductio Fredholm s First Theorem Workig Rule for Evlutig the Resolvet Kerel d Solutio of Fredholm Itegrl Equtio of the Secod Kid y Usig Fredholm s First Theorem Fredholm s Secod Fudmetl Theorem Fredholm s Third Theorem 15 Eercise Itegrl Trsform Methods Itroductio Sigulr Itegrl Equtio Lplce Trsform Some Importt Properties of Lplce Trsform Iverse Lplce Trsform Some Importt Properties of Iverse Lplce Trsform Covolutio of Two Fuctios The Heviside Epsio Formul The Comple Iversio Formul Itegrl Equtios i Specil Forms 163 PHI Lerig Copyrighted Mteril

6 viii Cotets 8.11 Applictio of Lplce Trsform to Fid the Solutios of Volterr Itegrl Equtio Covolutio Type Kerels of Volterr Itegrl Equtio: Workig Procedure Resolvet Kerel of Volterr Itegrl Equtio y Usig Lplce Trsform Solutio of Itegrl Equtios of the Type y Usig Lplce Trsform: Workig Procedure Fourier Trsforms d Their Importt Properties Applictio of Fourier Trsform to Determie the Solutio of Sigulr Itegrl Equtios 179 Eercise Ide PHI Lerig Copyrighted Mteril

7 Prefce Itegrl Equtios re eig used s essetil effective tool y the mthemticis, egieers d theoreticl physicists to uderstd d solve reserch prolems i their field. Two-poit iitil vlue prolems d oudry vlue prolems with fied d vrile oudry re ofte ecoutered y reserchers. Itegrl equtios hve ow ee estlished s fr effective d highly useful rch i lmost ll disciplies of kowledge. Hece, this course hs ee tke s idispesle prt of sylli of ll Idi uiversities t postgrdute level. Not oly tht, i competitive emitios like NET/SET, this course iherits importce. Therefore, it ecomes ecessry for studets d techer like to follow the cocepts d opertios of itegrl equtios. Lookig to these requiremets, we hve tried to provide the cocepts d priciples quite clerly d i orgised mer. I order to mke the ook user-friedly, we hve preseted the mtter i simple wy d it is complete i ll respects from the emitio poit-of-view. By presetig sufficiet umer of ssorted emples, we hve tried to iculcte the hit d crete cofidece i studets to try to solve more d more prolems o their ow. Becuse of uthor s log eperiece of techig d lso eig ctively egged i reserch, it is epected tht the ook shll prove immesely eeficil to the studets for whom it is met. All suggestios for the improvemet of the ook shll e thkfully ckowledged. PHI Lerig Copyrighted Mteril D.C. Shrm M.C. Goyl i

8 CHAPTER Applictios to Ordiry Differetil Equtios.1 INTRODUCTION The quest for estlishig represettio formul to replce ordiry differetil equtio (with iitil vlue prolem or oudry vlue prolem) lwys leds to itegrl equtio ore speciclly it is foud tht iitil vlue prolem is coverted to olterr itegrl equtio while oudry vlue prolem is coverted to redholm itegrl equtio illy it eses our wor of dig the solutio of itegrl equtio thus otied ecllig oce more i iitil vlue prolem the oudry coditios re for the sme vlue of idepedet vrile while i the cse of oudry vlue prolem the oudry coditios re for differet vlues of idepedet vrile.. METHOD OF CONVERSION OF AN INITIAL VALUE PROBLEM TO A VOLTERRA INTEGRAL EQUATION Let ordiry differetil equtio of order e 1 d y d y d y 1( ) 1 ( ) ( ) ( ) y d d d φ (.1) with the iitil coditios ( 1) y( ) q, y'( ) q1,, y ( ) q 1 (.) PHI Lerig Copyrighted Mteril where 1 () ()... () f() re deed d re cotiuous i. Let u() e ukow fuctio such tht d y u ( ) d (.3) Itegrtig Eq. (.3) from to we get 1

9 Applictios to Ordiry Differetil Equtios 1 EXERCISE. 1. Covert the followig iitil vlue prolems ito itegrl equtios. () () d y y g( ), y() 1, y () d d y y, y(), y () d. Covert y"( ) 3 y'( ) y( ) 4si with iitil coditios y() 1, y () ito olterr itegrl equtio. lso d the origil iitil vlue prolem from the otied itegrl equtio. 3. Trsform y y 1, y(), y(1) 1 ito itegrl equtio. 4. Trsform the oudry vlue prolem y y, y(), y (1) to Fredholm itegrl equtio. Aswers: 1. () () y( ) 1 ( t)[ g( t) t y( t)] dt y ( ) ( t) ytdt ( ). y() = 1 4 si {3 ( t)}y(t) dt o 1 1 (1 ), t(1 t), t t t y ( ) (1 ) K (, t) ytdtk ( ) ; (, t) 3 1 t, t y ( ) K (, t) ytdt ( ) ; 6 Kt (, ), t.4 BOUNDARY VALUE PROBLEM AND ITS CONVERSION TO FREDHOLM INTEGRAL EQUATION Whe ordiry differetil equtio is give with the coditios ivolvig depedet vrile d its derivtives t two differet vlues of idepedet vriles the prolem uder cosidertio is sid to e oudry vlue prolem. The method of coversio of oudry vlue prolem to Fredholm itegrl equtio c e mde cler y the emples give hereuder. PHI Lerig Copyrighted Mteril EXAMPLE.6: Reduce the followig oudry vlue prolem ito itegrl equtio: du lu with u(), u( l) d

10 CHAPTER 3 Solutio of Homogeeous Fredholm Itegrl Equtios of the Secod Kid 3.1 INTRODUCTION I the previous chpters, we hve lert the sic termiology d suitility of itegrl equtios over differetil equtios, d the, the clssictio of the itegrl equtios. Although, there hve ee my developmets i the theory, the sic divisio s iitil vlue prolems ito volterr itegrl equtios d oudry vlue prolems ito Fredholm itegrl equtios is must to follow. As we pproch to simplify these equtios, it is foud coveiet to focus the kerel, kid d the homogeeous ture of these equtios. I this chpter, we shll restri ourselves oly for the homogeeous Fredholm itegrl equtios of the secod kid. The chpter egis with the discussio of essetil prt clled eigevlue, eigefuctio, d the relted theorems re eplied. 3. CHARACTERISTIC VALUE * (OR EIGENVALUE) AND CHARACTERISTIC FUNCTION (OR EIGENFUNCTION) We cosider the followig homogeeous Fredholm itegrl equtio of the secod kid: u ( ) l K (, tutdt ) ( ) (3.1) It is cler tht u() = will lwys stisfy Eq. (3.1), d we cll such u() = s the trivil solutio of Eq. (3.1). The vlues of prmeter l for which the itegrl equtio [Eq. (3.1)] hs o-trivil (o-zero) solutios [u() ] re kow s eigevlues of Eq. (3.1) or chrcteristic vlues of kerel K(, t). If for u(), there eists cotiuous fuctio f() i the itervl [, ] such tht PHI Lerig Copyrighted Mteril * It is lso clled chrcteristic umer. φ( ) l K(, t) φ ( t) dt (3.) 9

11 Solutio of Homogeeous Fredholm Itegrl Equtios of the Secod Kid 31 This is the eigevlue of Eq. (i), d usig Eq. (iii), the correspodig eigefuctio is g ( ) ce e e 1 c y tkig the costt s uity. e 1 EXAMPLE 3.: Show tht the homogeeous itegrl equtio 1 f( ) l (3 ) tf( t) dt hs o chrcteristic umer d o eigefuctio. Solutio: The give itegrl equtio is Let PHI Lerig Copyrighted Mteril 1 f( ) l(3 ) tf( t) dt (i) 1 c tf() t dt (ii) so tht f( ) l(3 ) c (iii) or f() t lc(3t ) The, 1 c tlc(3t ) dt 3 c lct t c Thus, f() = is the solutio of Eq. (i) d we do ot get y chrcteristic umer or eigefuctio. We lso see tht kerel K(, t) = (3 )t is ot symmetric, d i this cse, the kerel does ot possess y chrcteristic umer ecessrily. 3.3 SOLUTION OF HOMOGENEOUS FREDHOLM INTEGRAL EQUATION OF THE SECOND KIND WITH SEPARABLE (OR DEGENERATE) KERNEL Let the homogeeous Fredholm itegrl equtio of the secod kid e 1 u ( ) l Ktutdt (, ) ( ) (3.3) where kerel K(, t) is seprle, which mes it c e epressed s the sum of the product of terms, ech hvig fuctios of d t seprtely. Let The, Eq. (3.3) is Kt (, ) f( g ) ( t) (3.4) i i1 i u ( ) f i( g ) i( t l ) utdt ( ) (3.5) i1

12 CHAPTER 4 Fredholm Itegrl Equtios with Seprle Kerels 4.1 INTRODUCTION Like the previous chpter, this chpter is lso devoted to Fredholm itegrl equtios of the secod kid. But, ow the equtio is ot homogeeous, i.e., F() i geerl. Furthermore, the ture of kerel is seprle. This chpter shows tht the role of chrcteristic fuctio with determit D(l) is of vitl importce. All iterrelted possile cses re icluded with suitle emples. 4. SOLUTION OF FREDHOLM INTEGRAL EQUATION OF THE SECOND KIND WITH SEPARABLE (OR DEGENERATE) KERNEL We cosider the followig itegrl equtio: u ( ) F ( ) l Ktutdt (, ) ( ) (4.1) where, kerel K(, t) is seprle, d therefore, we epress i i1 which o sustitutig i Eq. (4.1) provides Kt (, ) f( ). g( t) (4.) PHI Lerig Copyrighted Mteril or u ( ) F ( ) f i( ). g i( t l ) utdt ( ) i1 u ( ) F ( ) l f ( ) g ( tutdt ) ( ) i i1 i (4.3) (upo iterchgig the order of itegrtio d summtio) 41 i

13 CHAPTER 5 Itegrl Equtios with Symmetric Kerels 5.1 INTRODUCTION I the previous chpters, Fredholm itegrl equtios of the secod kid hve ee cosidered for y give kerl K(, t) y hvig the eigevlues d correspodig eigefuctios. I this chpter, the sme equtio is the mi motive, ut ow, the kerel is symmetric. 5. SYMMETRIC KERNAL A kerel K(, t) is sid to e symmetric (lso comple symmetric or Hermiti) if K(, t) K( t, ) (5.1) where, the r deotes the comple cojugte. If the kerel is rel, the symmetry reduces to equlity. K(, t) K( t, ) (5.) Theorem: symmetric. If kerel is symmetric, the ll its iterted kerels re lso Proof: Let the kerel K(, t) e symmetric. The, y detio, K(, t) K( t, ) (5.3) By deitio, the iterted kerels K (, t), I re deed s PHI Lerig Copyrighted Mteril K 1 (, t) K(, t) [5.4()] K (, t) K(, z). K ( z, t) dz, =,3,... [5.4()] 1 Also, K(, t) K1(, z). K( z, t) dz, =, 3,... [5.4(c)] 56

14 Comiig Eq. (5.) d (5.3), we d Itegrl Equtios with Symmetric Kerels 63 Sice l +1, Eq. (5.4) gives ( hh, ) f( ) y ( ) ( h, K h) (5.4) ( 1) l1 f( ) y ( ) s (5.5) Now, we use the reltio f( ) y( ) f( ) y ( ) y ( ) y( ) (5.6) where y() is the limit of the series with prtil sum y. As show ove, the rst term o the R.H.S. of (5.6) teds to zero, d to show tht the secod term of R.H.S. of (5.6) lso teds to zero, we proceed s follows: Sice Eq. (5.18) coverges uiformly, we hve, for ritrrily smll d positive qutity, y ( ) y ( ), whe is sufcietly lrge. y ( ) y ( ) ( ) d hece y ( ) y ( ) Hece, Eq. (5.6) shows tht f() = y(), d thus, the result follows. 5.8 SCHMIDT S SOLUTION OF NON-HOMOGENEOUS FREDHOLM INTEGRAL EQUATION OF THE SECOND KIND Cosider y ( ) f( ) l K( t, ) ytdt ( ) (5.7) where K(, t) is cotiuous, rel d symmetric kerel d l is ot eigevlue. Sttemet of Hilert Schmidt theorem: Let F() e geerted from cotiuous fuctio y() y the opertor l K (, t ) y ( t ) dt, where K(, t) is cotiuous, rel d symmetric, PHI Lerig Copyrighted Mteril so tht F ( ) l Kt (, ) ytdt ( ) (5.8) The fuctio F() c e epressed over the itervl (, ) y lier comitio of the ormlised eigefuctios of homogeeous itegrl equtio y ( ) l K( t, ) ytdt ( ) (5.9) hvig K(, t) s its kerel. 1/

15 CHAPTER 6 Solutio of Itegrl Equtios of the Secod Kid y Successive Approimtio 6.1 INTRODUCTION I the previous chpters of the ook, the solutio of the itegrl equtios ws mily focused upo Fredholm itegrl equtio. I this chpter the developmet is for Volterr itegrl equtio lso. Moreover, where the solutio is ot possile i closed form, successive pproimte method is lso discussed. Aprt from the vrious theorems, the chpter icludes three prts () iterted kerels, () resolvet kerel, d (c) solutio of itegrl equtios y pplyig resolvet kerel. 6. ITERATED KERNEL OR FUNCTION 1. Let us cosider the followig Fredholm itegrl equtio of the secod kid: y ( ) f( ) l Kt (, ) ytdt ( ) (6.1) The, the iterted kerels K (, t), = 1,, 3,..., re deed s follows: K 1 (, t) K(, t) d K(, t) K(, z) K1( z, t) dz,,3,... or K(, t) K1(, z) K( z, t) dz,,3,... [6.()] PHI Lerig Copyrighted Mteril. Let the Volterr itegrl equtio of the secod kid e [6.()] y ( ) f( ) l Kt (, ) ytdt ( ) (6.3) The, the iterted kerels K (, t), = 1,, 3,..., re deed s follows: K 1 (, t) K(, t) [6.4()] 84

16 Solutio of Itegrl Equtios of the Secod Kid y Successive Approimtio 85 d or K (, t) K(, z) K ( z, t) dz,,3, t 1 1 K (, t) K (, z) K( z, t) dz,,3, t [6.4()] 6.3 RESOLVENT KERNEL OR RECIPROCAL KERNEL We cosider the solutio of Fredholm itegrl equtio of the secod kid. d let it tke the followig form y ( ) f( ) l Kt (, ) ytdt ( ) (6.5) y ( ) f( ) l Rt (, ; l) f( tdt ) [6.6()] or y ( ) f( ) l ( t, ; l) f( tdt ) [6.6()] Here, R(, t; l) or (, t; l) is kow s resolvet kerel of Eq. (6.5). Alogously, we hve the resolvet kerel for Volterr itegrl Eq. (6.3). We cosider the followig theorem without proof: Theorem: The m th iterted kerel K m (, t) stises the reltio m r mr K (, t) K (, y) K ( y, t) dy where, r is y positive iteger less th m. 6.4 SOLUTION OF FREDHOLM INTEGRAL EQUATION OF THE SECOND KIND BY SUCCESSIVE SUBSTITUTION Theorem: Let y ( ) f( ) l Kt (, ) ytdt ( ) (6.7) e the give Fredholm itegrl equtio of the secod kid. Let 1. K(, t) e rel d cotiuous i the rectgle R, for which, t. Also, ssume K(, t) Mi R.. f() e rel d cotiuous i the itervl I for which. Also, ssume f() N i I. 3. l e costt such tht l < 1/M( ). The (6.5) hs uique solutio i I d this solutio is give y the solutely d uiformly coverget series * y ( ) f( ) l Kt (, ) ftdt ( ) l Kt (, ) Ktt (, ) ftdtdt ( ) PHI Lerig Copyrighted Mteril (6.8) * A coverget iite series hs sum. But, if we differetite (or itegrte) its ll the terms, the this sum my ot e equl to the derivtive (or itegrl) of sum. However, if the series is solutely d uiformly coverget, the it is possile. To check this chrcteristics, we check if the modulus of the th term is less th deite qutity.

17 Solutio of Itegrl Equtios of the Secod Kid y Successive Approimtio 89 where, R 1 t t 1 ( ) l K(, t) K( t, t ) K( t, t ) y( t ) dt dt dt (6.3) We ow cosider the followig iite series: l t (6.31) y ( ) l Kt (, ) ftdt ( ) Kt (, ) Ktt (, ) ft ( ) dt Now, due to coditios (1) d (), ech term of Eq. (6.31) is cotiuous i I. It mkes Eq. (6.31) lso cotiuous i I, provided it coverges uiformly i I. Let V () e the geerl term of Eq. (6.31), give y t t ( ) V ( ) l NM V ( ) l K(, t) K( t, t ) K( t, t ). f( t ) dt dt dt The, (6.3)! [Here, we hve pplied coditios 1 d over mod of Eq. (6.3)] ( ) or V ( ) l NM,! or V ( ) l N[ M( )] /! (6.33) Now, from Eq. (6.3), it is cler tht Eq. (6.31) is coverget for ll l, N, M, ( ); thus, from Eq. (6.33), it is followed tht Eq. (6.31) is coverget osolutely d uiformly. So, if Eq. (6.1) hs cotiuous solutio, it must e epressed y Eq. (6.9). If y() is cotiuous i I, y() must hve mimum vlue, sy Y. or y() Y (6.34) 1 Now, from Eq. (6.3), 1 1 ( ) R1( ) l Y M ( 1)! R ( ) l Y M PHI Lerig Copyrighted Mteril 1 ( ) ( 1)! Hece, lim R1 ( ) It thus follows tht fuctio y() stisfyig Eq. (6.9) is the cotiuous fuctio give y Eq. (6.4) or Eq. (6.31). It thus proves the theorem. 6.6 SOLUTION OF FREDHOLM INTEGRAL EQUATION OF THE SECOND KIND BY SUCCESSIVE APPROXIMATIONS: ITERATIVE METHOD (ITERATIVE SCHEME) NEUMANN SERIES We cosider the followig Fredholm itegrl equtio of the secod kid: y ( ) f( ) l Kt (, ) ytdt ( ) (6.35)

18 CHAPTER 7 Clssicl Fredholm Theory 7.1 INTRODUCTION The solutio of the Fredholm itegrl equtio of the secod kid, i.e., g ( ) f( ) l Ktgtdt (, ) ( ) (7.1) hs ee discussed i Chpter 6 s uiformly coverget power series i prmeter l for suitly smll vlue of l. As mtter of fct, Fredholm otied the solutio of Eq. (7.1) i geerl form, which is vlid for ll vlues of prmeter l. These solutios re cotied i three theorems, which re kow s Fredolms rst seod d tird fudmetl teorems. I this chpter, we shll study Eq. (7.1) whe the fuctios f() d kerel K(, t) re y itegrle fuctios. Moreover, the preset method eles us to get eplicit formule for the solutio i terms of certi determits. 7. FREDHOLM S FIRST THEOREM Sttemet: The o-homogeeous Fredholm itegrl equtio of the secod kid g ( ) f( ) l Ktgtdt (, ) ( ) (7.) where the fuctios f() d K(, t) re itegrle, hs uique solutio PHI Lerig Copyrighted Mteril g ( ) f( ) l Rt (, ; l) ftdt ( ) (7.3) where the resolvet kerel R(, t; l) is meromorphic fuctio of prmeter l deed y Dt (, ; l) Rt (, ; l) ; [D(l) ] (7.4) D( l) 11

19 CHAPTER 8 Itegrl Trsform Methods 8.1 INTRODUCTION The itegrl trsform methods provide useful tool for the solutio of itegrl equtios of vrious specil forms. Let the followig doule itegrl eist: g( ) F(, t ) K( t, t) g( t) dt dt (8.1) This doule itegrl c e evluted s iterted itegrl. If we tke f( ) K(, t) g( t) dt, (8.) the from Eq. (8.1), we hve g ( ) Ft (, ) ftdt ( ) (8.3) Thus, if Eq. (8.1) is regrded s itegrl equtio i g, solutio is give y Eq. (8.), wheres if Eq. (8.3) is regrded s itegl equtio i f, solutio is give y Eq. (8.). It is covetiol to refer, oe of these fuctios s the trsform of the secod fuctio, d to the secod fuctio, s iverse trsform of the rst. Some emples * re give here. 1. The most well-kow doule itegrl Eq. (8.1) is the Fourier itegrl 1 is it g() s e e g() t dt d p PHI Lerig Copyrighted Mteril which results i the reciprocl reltios s: 1 ist f() s e g() t dt p * Refer to Chpter 6 of Itegrl Trsforms pulished y RBD. 157

20 158 Itegrl Equtios d hs 1 ist gs () e f() t dt p The fuctio f(s) is kow s Fourier trsform of g(t) d g(s) s the iverse Fourier trsform of f(t), d vice vers.. Cosiderig the doule itegrl gs () (sis si t) gt () d p, we d tht this leds to the sie trsform f() s (si st) g() t dt p d its iverse s, gs () (si st) f() t dt p, respectively. 8. SINGULAR INTEGRAL EQUATION Sigulr itegrl equtios occur frequetly i mthemticl physics d possess very uusul properties; hece, their solutios re quite essetil. A itegrl equtio is clled sigulr if either the rge of itegrtio is iite or the kerel is discotiuous. For emple, the sigulr itegrl equtios of the rst kid re: PHI Lerig Copyrighted Mteril f( ) si( t) g( t) dt (8.4) t f( ) e g( t) dt (8.5) I Eqs. (8.4) d (8.5), the rge of itegrtio is iite, while i itegrl gt () f( ) dt t (8.6) the rge of itegrtio is ite, ut the kerel is discotiuous. The Ael s Itegrl Equtio Oe of the simplest form of sigulr itegrl equtio, which ppers i mechics, is the Ael s itegrl equtio, give s, gt () f( ) dt, 1 (8.7) ( t) where g(t) is ukow fuctio to e determied d f() is kow fuctio.

21 Itegrl Trsform Methods LAPLACE TRANSFORM * Let f() e fuctio deed for >. The, the Lplce trsform of f(), deoted y L{f(); p} or F(p), is deed with the help of the followig itegrl: p L{ f( ); p} F( p) e f( ) d (8.8) provided tht the itegrl eists. It is to e recollected tht the Lplce trsform of f() eists if the itegrl i Tle 8.1 shows Lplce trsform for sme elemetry fuctios. Eq. (8.8) is coverget for some desigted vlues of p. Tle 8.1 Lplce Trsform for Some Elemetry Fuctios S.No. f() L{(); p} F(p) /p, p >., is positive iteger!/p +1, p > 3., > 1 ( + 1)/p +1, p > 4. e 1/(p ), p > 5. si /(p + ), p > 6. cos p/(p + ), p > 7. sih /(p ), p > 8. cosh p/(p ), p > 9. J () 1/ p + 1. J () { p p} PHI Lerig Copyrighted Mteril p 11. d( ) e p 1. erf ( ) 1/{ p p+ 1} 13. (1 )p SOME IMPORTANT PROPERTIES OF LAPLACE TRANSFORM 1. If for i {1,,..., }, c i re costts d f i () re fuctios with Lplce trsforms F i (p), respectively, the L{ c f ( ) c f ( ); p} c L{ f ( ); p} c L{ f ( ); p} Lcf { ( ) c f( ); p} cf( p) cf( p) * For detils of Lplce trsform d its iverse, the reder c refer to the recet ook o Itegrl Trsforms pulished y RBD.

22 8.5 INVERSE LAPLACE TRANSFORM Itegrl Trsform Methods 161 If the Lplce trsform of fuctio f() is F(p), i.e., L{ f(); p} = F(p) the f() is clled iverse Lplce trsform of F(p), d we epress 1 f( ) L { F( p); } where L 1 is kow s the iverse Lplce trsformtio opertor. Tle 8. shows iverse Lplce trsform of some elemetry fuctios. Tle 8. Iverse Lplce Trsform * of Some Elemetry Fuctios S.No. F(p) L 1 {F(p); } f() 1. 1/p 1. 1/p +1 ( is positive iteger) /! 3. 1/p +1 {Re() > 1} /( + 1) 4. 1/(p ) e 5. 1/(p + ) si / 6. p/(p + ) cos 7. 1/(p ) (sih )/ 8. p/(p ) cos h 9. 1/ ( p ) J () 1. 1/{ p ( p 1)} erf ( ) 8.6 SOME IMPORTANT PROPERTIES OF INVERSE LAPLACE TRANSFORM If for ll i {1,,..., }, c i re costts d F i (p) re the Lplce trsforms of f i (), respectively, the L { c F ( p) c F ( p); } c L { F ( p); } c L { F ( p); } 1 L { c F ( p) c F ( p); } c f ( ) c f ( ) If L 1 {F(p); } = f(), the 1 1 L { F( p); } f ( / ),. PHI Lerig Copyrighted Mteril If L 1 {F(p); } = f(), the 1 1 L { F( p); } e f( ) e L { F( p); }. * The reder is dvised to refer to Tle 8.1 simulteously.

23 Itegrl Trsform Methods 163 The covolutio theorem (or covolutio property) Let f() d g() e two fuctios of clss A d let L 1 {F(p); } = f() d L 1 {G(p); } = g(). The, i.e., 1 L { F( p) G( p); } f( u) g( u) du f * g. Oe useful form of the ove is L{f *g) = F(p) G(p) L f() u g( u) du L f( u) g() u du F() p G(). p 8.8 THE HEAVISIDE EXPANSION FORMULA Let F(p) d G(p) e polyomils i p, where F(p) hs degree less th G(p). Now, if G(p) hs distict zeros r, r = 1,...,, the 1 F ( p ) F( ) ; r r L e. Gp ( ) G'( r ) r1 8.9 THE COMPLEX INVERSION FORMULA If f() hs cotiuous derivtive d is of epoetil order g for lrge positive vlues of, where g > d if F(p) = L{f(); p}, the d f() = ; <. 1 1 g i L { F( p); } f( ) e F( p) dp, pi g i 8.1 INTEGRAL EQUATIONS IN SPECIAL FORMS 1. The itegrl equtio PHI Lerig Copyrighted Mteril p g ( ) f( ) K ( t) gtdt ( ) wherei kerel K( t) is fuctio of the differece oly, is kow s itegrl equtio of covolutio type. Applyig the deitio of covolutio, it c e epressed s g() = f() + K() * g(). A itegrl equtio i which derivtives of the ukow fuctio g() re lso preset is sid to e itegrodifferetil equtio. For emple,