UNIT I FOURIER SERIES T
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1 UNIT I FOURIER SERIES PROBLEM : Th urig mom T o h crkh o m gi i giv or ri o vu o h crk g dgr T Epd T i ri o i. Souio: L T = i + i + i +, Sic h ir d vu o T r rpd gc o T T i T i T i To = [m vu o T i ] = [ 59.7 ] = = [m vu o T i ] = [ ] = = [m vu o T i ] = = i i PROBLEM : Az hrmoic h giv ow d pr i Fourir ri upo h hird hrmoic.
2 Souio: Sic h vu o i rpiio o h ir, o h i vu wi ud. Th gh o h irv i. L = + co + i + co + i + co + i + co i co i co i = [m vu o ] = =.9 = [m vu o co] = 6 [ ] = -.7 = [m vu o i] = 6 [ ] =.7 = [m vu o co]
3 = 6 [ ] = -. = [m vu o i] = 6 [ ] = -.6 = [m vu o co] = 6 [ ] =. = [m vu o i] = Hc = co +.7 i. co +.6 i +. co. PROBLEM : Fid h Fourir ri pio or h ucio = i i < < d dduc h SOLUTION: Th Fourir ri i co i... Giv = i = d = i d = [ - co -- i] = - = -
4 = cod = i cod = [ i i ]d [ SiACoB SiA B SiA B] co co i i co co = [ ] [ ] [ ] - = whr = co d = i co d = i d co i = co = [ - = - co [ = i d
5 = i i d = [co co ]d [ SiASiB CoA B CoA B] i i co co = co co co co = = = whr = = = = = i d i i d i d co d co d i co = [ ] =
6 Suiu i, w g = co co i Dducio Pr: Pu [ i poi o coiui] [= i poi o coiui] = i = i = co co - + = co + = co = co - - = = [...] [ ] [ ]
7 Prom: Souio: Giv = Fid h Fourir ri or = i Thi i v ucio. = i i - < < = Co... = d = Si d = i d = [ co ] = - co co = - [ ] [--] = = cod = Si cod = i cod = [i i ] d co co = [ ]
8 co co = [ ] = [ ] = [ ] = [ ] = = = [ ] [ ] [ ] [ ] = i. = co d = Si co d = i co d = i d co = [ ] = [ ] = Suiu i quio, w g [ ] = co
9 Hc = Prom:5 [ ] co Epd = Souio: co i Fourir ri i h irv -, Giv = co. Thi i v ucio, = = Co... = d = = = = co d / [ co d cod] / / [ co d co d] / [i / i / ] = [ ] = = cod = co cod
10 / = [ co cod co cod] / / = co cod co cod / / = [co co ]d [co co ]d - / = i i i i [ ] [ ] / / Hc i / i / i / i / = [ ] [ ] i / i / = [ ] i / / i / / = [ ] co co = [ ] = co [ ] = co [ ] = co [ ] = co [ ] = [ ] i i v = i i odd [ ] = co
11 Comp Form o Fourir ri: Prom 6: Fid h comp orm o h Fourir ri o h ucio = wh - < < d + =. Souio: W kow h = C C = = i i i = i = i i d d i d i = i i = [ ] i i i = [ ] i i i co ii co ii i i = [ ] i = [ ] i
12 i C = ih = i ih i i.., ih = i i Prom7: Fid h comp orm o h Fourir ri o = Co i -, whr i ihr zro or igr. Souio: Hr c = or c =. W kow h = C C = i i i d = Co d = [ i i co i ] i i = [ i co i i co i i i i i = [i co i ] = [i co ii i co ] C = i
13 Hc com Co = i i Prom8: Fid h comp orm o h Fourir ri o = - i - Souio: W kow h = C C = d i = d i = d = = = = = i i [ ] i i i i i co ii co ii i i = i i = ih Hc com - = i i ih
14 UNIT- FOURIER TRANSFORMS. Fid h Fourir rorm o, i So: Th Fourir rorm o h ucio F B Prv idi F d. Dduc h i F co ii d i d co d i co d co d d d d d d d d d i d i d.
15 .Show h h Fourir rorm o ; ; i i co. Hc dduc h co i d. So: Th Fourir rorm o h ucio i d F i. Giv i d ohrwi. co i i co i co i co i co i co F d d i d d i d F i B ivr Fourir rorm d F i
16 i co i d i co co i i d i co co d i i co co d Pu &, w g i co d i. Rpc, w g i co d i co d i co i d.fid h Fourir rorm o ; i ; i. Hc dduc h i d d i d. So: Giv Th Fourir rorm o h ucio i d ohrwi. i F i d.
17 co co co i co i co i co F d d i d d i d F i B ivr Fourir rorm d F i co co i co co co i co co co d d i d d i d i Pu, w g. i. co d i d L, h d d
18 i i d d B Prv idi d d F co d d co d d co d d co d d co d i co d d L, h d d i i i d d d.fid h Fourir rorm o. Hc prov h i rciproc. So:
19 Th Fourir rorm o h ucio i d F i. d d d d d F i i i i i Pu i, h d d F d Pu, h F F i rciproc wih rpc o Fourir rorm.
20 5. Fid Fourir i d coi rorm o FC d hc prov h So: W kow h Pu i, i co ii d cod i i d d d i i i cod i i d cod i i d cod i i d cod i i d co ii co ii co ii co i i Equig R d imgir pr, w g co d co i d i
21 S C F F d d i co i i co co Pu i h ov ru, w g C c o F C co F C C F F Thror i rciproc wih rpc o Fourir coi rorm. 5. Fid Fourir i rorm d Fourir coi rorm o,. Hc vu d d d. So: Th Fourir i rorm o h ucio i i d F S.
22 i d F S Th Fourir coi rorm o h ucio i co d F C. co d F C B prv idi d d F S d d d d d d d d d d Rpc, w g d L d g B prv idi d g d g F F C S
23 .. d d i d d d d d d Rpc, w g d 6. Vri Prv horm o Fourir rorm or h ucio ; ;. So: B Prv horm d F d Giv ; ; i d d d F i i i i
24 i F i i i i i i i i F i i L. H. S. o d d d R. H. S. o d d d i d i d i d F
25 d F L. H. S o = R. H. S o Hc Prv horm vriid. 7. Sov or rom h igr quio co d. So: co d d co Ivr Fourir coi rorm i i.co co d
26 UNIT- PARTIAL DIFFERENTIAL EQUATIONS Prom o Lgrg quio. Sov p q z Souio: p q z Thi i o h orm Pp + Qq = R Hr P, Q, R z Suidir quio r d d dz P Q R d d dz z Groupig h ir wo mmr Igrig d d d d d d c c u Groupig h ohr wo mmr
27 d dz z Igrig z z z z c c v z Th gr ouio o i u,v =, z. Oi h gr ouio o pz + qz =. Souio: Giv pz + qz = Thi i o h orm Pp + Qq = R Hr P = z, Q= z, R = Suidir quio r d d dz P Q R d d dz z z Groupig h ir wo mmr d d z z Igrig
28 d d og og og og og u Groupig h ohr wo mmr d dz z d dz z d dz z d z dz rom Igrig z c z c z c. z z v z Th gr ouio i u,v =, z
29 . Sov z p z q z Souio: Giv z p z q z Thi i o h orm Pp + Qq = R Hr P z, Q z, R z Suidir quio r d d dz P Q R d d dz z z z Choo h o muipir,,z d z d z dz z z d d z dz z z d d z dz d d z dz Igrig w g z c z c u z coidr ohr o muipir Ech mmr o,, z
30 Igrig w g d d z d z d d dz z dz z d dz z og + og + og z = og ogz = og z = v = z Th gr ouio i u,v = z, z. Sov z p z q z Souio: Giv z p z q z Thi i o h orm Pp + Qq = R Hr P z, Q z, R z Suidir quio r d d dz P Q R d d dz z z z Ech o
31 d d d d z d d z z z z z z d d z d z i.., z z z z z d d z d z i.., z z Groupig h mmr d d z z Igrig w g Choo muipir,, z og og z og c og og c z c z u z d d z dz Ech o z z Choo ohr o muipir,, d d z dz Ech o z z z
32 d d z dz d d z dz z z z z z d d z dz d d z dz i.., z z z z z z z d d z dz d d dz z z d z d d z dz Igrig w g z z z z z z z z c v z z Th grouio i, z z. z 5. Sov mz p + z q = m Souio: Giv mz p + z q = m Thi i o h orm Pp + Qq = R Hr P = mz, Q = z, R = m Th Suidir quio r d d dz P Q R d d dz m z z m Choo o muipir,,z Ech o d d z dz m z z z m d d z dz d d z dz Igrig w g + + z = v = + + z
33 Choo o muipir, m, Ech o d m d dz m z m z m d m d dz d m d dz Igrig w g m z v m z Th gr ouio i u,v = z, m z Equio rduci o drd p: Tp V: Equio o h orm m p, q = or m p, q, z = whr m d r co Thi p o quio c rducd o Tp I or Tp III h oowig rormio. C i : I m, Pu m = X, = Y z z X m z p m P, whr P X X m i.., p m P z z Y z q Q, whr Q Y Y i.., q Q m m Th quio p, q rduc o m P, Q which i Tp I q. Th quio p, q,z rduc o m P, Q,z which i Tp III q. C ii : I m =, = h p, q = or p, q, z =
34 Pu og = X, og = Y. z z X z z p P, whr P X X X i.., p P z z Y z z q Q, whr Q Y Y Y i.., q Q Th quio p, q = rduc o P, Q = which Tp I q. Th quio p, q, z = rduc o P, Q, z = which Tp III q. Tp VI : Equio o h orm z k p, z q = or z k p, q,, = Whr k i co, Thi c rducd o Tp I or Tp IV quio h oowig uiuio. C i : I k Pu Z = z k + Z Z z k P k z p z k P z p k Z Z z k Q k z q z k Q z q k k k P Q Th quio z p, z q rduc o, which i Tp I q. k k k k P Q Th quio z p, z q,, rduc o,,, which i Tp IV q.. k k
35 C ii : I k = p q p q, or,,, z z z z pu z og z z Z z P p. z z p P z z Z z Q q. z z q Q z p q Th q, rduc o Tp Iq. z z p q Th q,,, rduc o Tp IV q. z z Z Z z k P k z p z k P z p k Z Z z k Q k z q z k Q z q k k k P Q Th quio z p, z q rduc o, which i Tp I q. k k k k P Q Th quio z p, z q,, rduc o,,, which i Tp IV q.. k k
36 H r m,. Pu X m X = Y og X Y z z p q z X z Y X Y p P q Q p P q Q
37 Rduc o P Q z Thi i o h orm p, q, z which i TpIII q. L z X Y ri Souio. z u u X Y u u, P z z Q X Y z u z u u X u Y P z z Q u u r d u c o d z d z z d u d u d z z d u d z d u d z z d u d z d u z z.
38 Igrig dz du z og z u c ogz X Y c og ogz c X, Y og which i h comp i gr. Sov z p q Souio : Giv z p q z p z q k k Thi i o h orm z p, z q,, Tp VI Hr k Pu Z z Z z
39 P Z z Q Z z Z z z z P z p Q z q z p P Q z q Eq rduc o P Q i.., P Q Thi i o Sdrd Tp IV, p, q L P Q c P c Q c P c Q c P c Q c P c Q c W kow Z Z dz d d Z dz P d Q d P, Q dz c d c d Z
40 dz c d c d c c Z c ih c coh C C z c c ih c ccoh Z z C C Emp:.Form h p d imiig h rirr ucio rom h rio z = + Souio: Giv z = Diriig pri w. r.. d,
41 z ' p., z ' q. p q p q p q i h rquird pri diri quio..form h p d imiig h rirr ucio d g rom h rio z = +c + g -c. Souio : Giv z = +c + g -c D i r i i g p r i w i h r p c o z ' c g ' c, A g i d i r i i g p r i w i h r p c o z " c g " c
42 D i r i i g p r i w i h r p c o z c ' i c g ' i, A g i d i r i i g p r i w i h r p c o z c " i c g " i z c z z UNIT-IV Appicio o Pri Diri Equio. A igh rchd rig wih id d poi d i iii i poiio giv, i. I i rd rom r rom hi poiio. Fid h dipcm im ''. So: O dimio wv quio Th codiio r i, ii, iii, iv, i Th corrc ouio which iig h giv oudr codiio i, c co p c i p c co p c i p
43 App h oudr codiio i i, c c co p c i p wg c, h quio com, c i p c co p c i p App h oudr codiio ii i, c i p c co p c i p wg i p Th quio com p p, c i c co c i Diri Pri w.r.o, c i c i c co App h oudr codiio iii i wg c, c i c, h quio com, c i c co c i co B h uprpoiio pricip, h mo gr ouio i, c App h oudr codiio iv i 5 whr c c c i co
44 , c c i c i. i c i i i i c i... i i c, c, c, c c5... Th quio 5 com, i co i co. A igh rchd rig o gh h i d d d. Th midpoi o h rig i h k o high h d h rd rom r i h poiio. Oi prio or h dipcm o h rig uqu im. So: O dimio wv quio i Th codiio r i, ii, iii, iv, Y, h,,, X
45 Coidr h irv,, h d poi r,,, h Uig wo poi ormu or h righ i h h Coidr h irv,, h d poi r,,, h Agi wo poi ormu or h righ i h h h, h, h Th corrc ouio which iig h giv oudr codiio i, c co p c i p c co p c i p App h oudr codiio i i, c c co p c i p wg c, h quio com, c i p c co p c i p App h oudr codiio ii i, c i p c co p c i p wg i p p p, h quio com, c i c co c i Diri Pri w. r. o
46 , c i c i c co App h oudr codiio iii i W g c, c i c h quio com, c i c co c i co B h uprpoiio pricip, h mo gr ouio i, c i co App h oudr codiio iv i 5, c h, i. h, whr c c c To id h vu o c, pd i h rg Fourir i ri W kow h h rg Fourir i ri o i 5 i 7 B comprig 6 d 7, w g c To id c i i ough o id i d h i d h i d
47 c h h h h i 8 i 8 i co i co i co i co Th quio 5 com co i i 8, h. Th poi o ricio o rig r pud id hrough dic o oppoi id o h poiio o quiirium d h rig i rd rom r. Fid prio or h dipcm. So: O dimio wv quio i Th codiio r i, ii, iii, iv,
48 Equio o OA: i, Equio o AB: i, Equio o BC: i, i, i, i, Th corrc ouio which iig h giv oudr codiio i, c co p c i p c co p c i p App h oudr codiio i i, c c co p c i p wg c, h quio com, c i p c co p c i p App h oudr codiio ii i, c i p c co p c i p wg i p p p, h quio com, c i c co c i Diri Pri w. r. o
49 , c i c i c co App h oudr codiio iii i W g c, c i c h quio com, c i c co c i co B h uprpoiio pricip, h mo gr ouio i, c i co App h oudr codiio iv i 5, c whr c c c i, i. i, i, To id h vu o c, pd i h rg Fourir i ri W kow h h rg Fourir i ri o i i 7 B comprig 6 d 7, w g c To id c i i ough o id i d i d i d i d
50 8 i 6 i i odd i i i v Suiu vu i quio 5 w g h rquird ouio 6, i i co,. A igh rchd rig wih id d poi d i iii r i i quiirium poiio. I i i virig givig ch poi voci k. Fid h dipcm o h rig im. So: O dimio wv quio Th codiio r i, ii, iii, iv, k Th corrc ouio which iig h giv oudr codiio i, c co p c i p c co p c i p pp h oudr codiio i i, c c co p c i p wg c h quio com, c i p c co p c i p App h oudr codiio ii i wg i p, c i p c co p c i p p h quio com p,
51 , c i c co c i App h oudr codiio iii i hr c h quio com, c i c, c i c i c i i B h uprpoiio pricip, h mo gr ouio i, c Diri Pri w.r.o, c whr c c c i i i co App h oudr codiio iv i 5, c i. k To id h vu o c, pd i h rg Fourir i ri W kow h h rg Fourir i ri o i i B comprig 6 d 7, w g c c
52 .. co. co i co i i k k d k d odd i i k v i i k k k k c Th quio com,,5 i i 8, k. A rod o gh h i d A d B kp c d c rpciv ui d codiio prvi. I h mprur B i rducd o c d o whi h o A i miid, id h mprur diriuio o h rod. So: O dimio h quio u u Sd quio i u d h d ouio i u -- Th oudr codiio r u ii u i App i i, u
53 Th quio com u App ii i u Th quio com u Now coidr h ud codiio. Th codiio r iii u, iv u, v u, Th corrc ouio which iig h giv oudr codiio i p u, c co p c i p c App h oudr codiio iii i u, c wg c p c h quio com u App h oudr codiio iv i 5 p u, c i p c wg i p p Th quio 5 com p, c i pc p, u, cc i c i whr B h upr poiio pricip, h mo gr ouio i c c c u, c i App h oudr codiio v i
54 u, c i To id c, pd i h rg i ri W kow h h rg Fourir i ri o i i From h quio 7 & 8 w g Now c i d i d co i co Th rquird ouio i u, i 5. Th d A d B o rod cm og hv hir mprur kp c d8 c, ui d codiio prvi. Th mprur o h d B i udd rducd o 6 c d h o A i icrd o c. Fid h mprur diriuio i h rod r im. So: u u O dimio h quio -----
55 u Sd quio i d h d ouio i u -- h oudr codiio r i u ii u 8 App i i, u Th h quio com u App ii i u 8 5 Th h quio com 5 u Now coidr h ud codiio. I ud h ui ouio which iig h giv oudr codiio i p u, c co p c i p c Th oudr codiio r 5 iii u, iv u, 6 v u, u Sic w hv o zro oudr codiio, w wri h mprur diriuio ucio u, u u, 5 To id u u, u, u Th ouio i u 6 Th oudr codiio r u 6, u, u u 6 6
56 h quio 6 com u 7 To id, u Giv h oudr codiio r i vi u vii u, u, u, u, u viii u, u, u I ud, h ui ouio which iig h giv oudr codiio u, c co p c i p c p App h oudr codiio vi i 8 8 u, c c p hr c h quio 8 com u App h oudr codiio vii i 9 p, c i pc 9 hr p u, cc i p i p p h quio 9 com u, c i c u, cc c i i B h upr poiio pricip, h mo gr ouio i whr c c c
57 i, c u App h oudr codiio viii i. i, c u To id c, pd i h rg i ri W kow h h rg Fourir i ri o i i From h quio & w g c. co i co i i d d c h quio com i, u Th h rquird mprur diriuio ucio i i, u
58 6. A rcgur p i oudd h i,,,,. I urc r iud. Th mprur og d r kp C d h ohr C. Fid h d mprur poi o h p. So: u u Two dimio h quio i L h gh o h qur p Giv h oudr codiio r i u, ii u, iii u, iv u, Aum h mprur diriuio u, u, u, To id u, Coidr h oudr codiio v u, vi u, vii u, viii u, Th ui ouio which iig h giv oudr codiio i A p p u, c c c co p c i p App h oudr codiio v i u, c c c co p c i p wg c c c c h h quio com u, c c p p c p p c c co p c co p c App h oudr codiio vi i i p i p p p u, c c wg c h h quio com
59 u p p, c c i p App h oudr codiio vii i p p u, c c i p wg i p p h h quio com p u, c c i c c ih i c ih i Th mo gr ouio i u whr c cc, c ih i App h oudr codiio viii i u, c ih i 5 To id c, pd i h rg i ri W kow h h rg Fourir i ri o i i From h quio 5 & 6 w g c ih c ih 6
60 c i d i d co co i i v i i odd ih h h quio 6 com u,,,5 To id u, i i odd ih i ih Coidr h oudr codiio i u, u, i u, ii u, d h ui ouio i hi c i u p p, c co p c i p c c 7 App h oudr codiio i i 7 p p u, c5 c6 c8 wg c 5 h h quio 8 com u p p, c i p c c 8
61 App h oudr codiio i 8 u, c6 i p c7 c8 wg c7 c8 c8 c7 h h quio 9 com u, c c 6 6 i p c c 7 7 i p p p c 7 p p App h oudr codiio i i 9 9 p p u, c6c7 i p wg i p p h h quio com p u, c6 c7 i c6 c7 ih i c ih i Th mo gr ouio i u whr c c6c7, c ih i App h oudr codiio ii i u, c ih i To id c, pd i h rg i ri H rg Fourir i ri o i i From h quio & w g
62 c c ih ih d d co i i odd i i v i i co c ih i i odd h h quio com u i ih ih,,,5 Th h quio A com u u u i ih ih i ih ih,,,,,5,,5
63
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