www.vidrhipu.com TRANSFORMS & PDE MA65 Quio Bk wih Awr UNIT I PARTIAL DIFFERENTIAL EQUATIONS PART-A. Oi pri diffri quio imiig rirr co d from z A.U M/Ju Souio: Giv z ----- Diff Pri w.r. d p > - p/ q > q/ Suiuig i, w g p q 4z. Form h pri diffri quio imiig h rirr co & from z A.U Apri/M Souio: Giv z ----- Diff pri w.r. d z z p, q, p, q Suiuig i w g pq 4z. Fid h PDE of p hvig qu ircp o h d i. Souio: Th quio of p i z... Diff. w.r.. d pri rpciv p... q... Sovig & w g, pq 4. Oi h pri diffri quio imiig rirr co d from z. Souio: z - Diffriig pri w.r. d w g zp zp - zq zq - Suiuig & i w g z p z q z i.., z p q S. Joph Cog of Egirig & Pg No. ISO 9:8 S. Joph Iiu of Tchoog www.udfocu.com www.vidrhipu.com

5. Eimi h rirr fucio f from z f d form h PDE. Nov/Dc Souio: z ' p f... z q f '... From & p q p q 6. Form h pri diffri quio imiig h rirr co & from z Souio: Giv z ----- Diff pri w.r. d z z p, q, Suiuig i w g z pq. 7. Form h pri diffri quio imiig f from z f og. Souio: www.vidrhipu.com L z f og - Diffri pri w.r. d z ' p f og z ' q f og - - Eimiig ' f from & p p q p q q 8. From h pri diffri quio imiig g from g z z Souio: W kow h if gu, v h u f v z f z - ---- Diffriig pri w.r. d W g,. S. Joph Cog of Egirig & Pg No. ISO 9:8 S. Joph Iiu of Tchoog www.udfocu.com www.vidrhipu.com

' zp f z p ----- ' zq f z q.. Divid & zp p zq q q zp zpq p zq zpq z p z q 9. Fid h comp igr of pq pq Souio: Giv pq pq ----- Thi i of h p fp,q L z c h ouio for Pri Diff w.r. d w g p d q > > / Thrfor z / c i h comp ouio.. Fid h igur igr of z p q pq Souio: Thi of Ciru p d h comp ouio i z Diff w.r. d w g z, z! & i igur igr.. Fid h comp igr of pq Souio: Giv p q ----- Thi i of h p fp,q L z c h ouio for Pri Diff w.r. d w g p d q > > Thrfor z c h comp ouio.. Sov p q qz. Souio: p q qz - www.vidrhipu.com Thi quio i of h form f z, p, q z g h ouio L u z g u dz dz p q du du rduc o dz dz z dz z dz z dz z dz du du du du du du du z S. Joph Cog of Egirig & Pg No. ISO 9:8 S. Joph Iiu of Tchoog www.udfocu.com www.vidrhipu.com

Igrig og z u i.., og z i h comp ouio.. Sov p og q Souio: Th quio c rducd o h p F,p F,q! p og q! p og q/ k k! p k ; q 4. Sov! z pd qd! z k d k d k z k c k ' D D D z Nov/Dc Souio: ' D D D z Hr c, c -, m, m Gr ouio z f f 5. Sov p q Souio:. Thi i of h form f p, q. Th ouio i giv z c whr Th comp ouio i 6. Sov Souio: z c DD DD z. Thi i of h form DmD... C DmD C DmD C Z Hc m c m c c Hc h C.F. i z φ φ P.I. DD DD Hc, h comp ouio i φ φ 7. Sov Souio: www.vidrhipu.com D DD D Z. S. Joph Cog of Egirig & Pg No. 4 ISO 9:8 S. Joph Iiu of Tchoog Z www.udfocu.com www.vidrhipu.com

www.vidrhipu.com Suiuig D m, & D Th Auiir quio i m - m m,, - Compimr fucio i φ φ φ i.., z φ φ φ 8. Fid h pricur igr of D Souio: DD' 5D' P.I D DD' 5D' Rpc D d ' D,w g D D' z z z z z 9. Fid h gr ouio of 4 8 Souio: Auiir Equio i m m 4m8! m,, f f Gr ouio z. Fid h Pricur igr of Souio: f D D ' DD' D D ' z P.I D D ' Rpc D d D,w g P.I 4 PART - B. Fid h gr ouio of - z p - z q z - Souio: Th quio i of h form PpQqR whr P - z,q - z,r z - [A.U Apr/ M 8 ] Th uiir quio i d z d dz z z Uig h muipir,, z, w g ch rio i qu o d d zdz z z z i.. d d zdz Igrig w g z c S. Joph Cog of Egirig & Pg No. 5 ISO 9:8 S. Joph Iiu of Tchoog www.udfocu.com www.vidrhipu.com

. uig h ohr of muipir,, z d d dz z w g ch of h ov rio i qu o z z i.. d d dz z Igrig w g og z og c Th ouio of h ir p.d. i ϕ u, v i.. ϕ z, z Sov D DD' 6D ' z Souio: Th uiir quio i m m 6 m, C.F φ φ P.I D DD' 6D' D D' D DD' 6D' D 6 D D ' D D' D 6 D D ' 5 4 5 6 www.vidrhipu.com 9 6 Th gr Souio i φ φ Sov z p z q z 4 5 5 6 Thi i Lgrg ir quio of h form Pp Qq R Th uidir quio r d d dz z z z - d d d dz d dz z z z z z z i.., z d d z d z z z z z S. Joph Cog of Egirig & Pg No. 6 ISO 9:8 S. Joph Iiu of Tchoog www.udfocu.com www.vidrhipu.com

i.., www.vidrhipu.com d d z d z z z z z z - i.., d d z d z z z - og og z og Tkig h fir wo rio, d igrig z - 4 Simir kig h wo rio of w g, z z - 5 Bu d z r o idpd ouio for z z z rciproc of h cod ouio. giv z z which i h Thrfor ouio giv 4 d 5 r o idpd. Hc w hv o rch for ohr idpd ouio. d d zdz Uig muipir,, z i quio ch rio i z z d d dz Uig muipir,, ch rio i z zz d d zdz d d dz z z z zz d z d z z z zz d z z d z Hc Igrig z z k z z k z z z z k z zz S. Joph Cog of Egirig & Pg No. 7 ISO 9:8 S. Joph Iiu of Tchoog www.udfocu.com www.vidrhipu.com

www.vidrhipu.com i.., z z h gr ouio i φ z z, z Sov Souio: p q z Th giv quio c wri zp zq... Thi i of h p of f, z m pf, z m q Pu z Z d P zp, Q zq P zp & zq Equio c wri P Q 4 Q [A.U Nov/ Dc 7]! P - 4 Q - 4 Th quio i of h form f,p f,q!! P - 4 Q - 4 4 ; P d W kow h dz Pd Qd P Q - 4 dz d d Igrig w g Z ih 4 d Q - 4 4 coh. ih z coh Fid h igur ouio of h quio z p q p q Souio: Th quio i of h form z pqfp,q For comp igr rpc p d q,w g! Comp igr z ----- z Diff w.r. d w g z z...... Sovig w g...4 &...5 S. Joph Cog of Egirig & Pg No. 8 ISO 9:8 S. Joph Iiu of Tchoog www.udfocu.com www.vidrhipu.com

www.vidrhipu.com Th quio 4 c wri...6 / / Suiuig 6 i 5 w g...7 d...8 Suiuig 7 & 8 i w g 6z 7 which i igur igr. 4. Sov D 5DD' D' z 5i Souio: Th uiir quio i m 5m m, C.F φ φ P.I 5i D 5DD' D' 5i D 5DD' D' D 5DD' D' 5 i 8 5 5 i 4D 5D' 9 4 D 5D' 4D 5D' 5 i 4D 5D' 4D 5D' 9 4 D 5D' 5 i 6D 5D' 9 5 9 Th Gr Souio i [ 8 co 5 co ] 9 5 φ 8 co 5 co 9 Sov z z p z q z. φ [ ] Souio: 9 Thi i Lgrg ir quio of h form Pp Qq R Th uidir quio r d d dz z z z z Tkig,, z muipir, w hv ch frcio d d zdz d d zdz S. Joph Cog of Egirig & Pg No. 9 ISO 9:8 S. Joph Iiu of Tchoog www.udfocu.com www.vidrhipu.com

Igrig i.., www.vidrhipu.com z c z C - Agi, kig h wo mmr, w hv d dz z z i.., d dz z z z d z dz d zd dz zdz d zd dz zdz d d z zdz z Igrig w g z z C - From & h gr ouio i z z z φ, Sov D D' D D' z 7 Nov/ Dc 9 Souio: Th giv quio i D D' D D' z Thi c fcorizd D - D'-D--D'- z I i of h form D m D' c D md' c whr m c,m - c. C.F f f 7 P.I D - D'D D'- D D' D D' 7 D D' D D' D D' DD' 7 D D' 9 7 D D ' D D ' D D' 7 D D' D 7 D' 7 9 DD' 7... 7 S. Joph Cog of Egirig & Pg No. ISO 9:8 S. Joph Iiu of Tchoog www.udfocu.com www.vidrhipu.com

S. Joph Cog of Egirig & Pg No. ISO 9:8 S. Joph Iiu of Tchoog 9 7 D ' D 9 7 D D' D 9 7 D D' D 9 7... ' ' D D D D D 9 7 D D ' 9 7 D 9 7 D D d D 9 7 9 7 D d 9 7 6 6 9 7 6 9 65 6 9 65 f f z 5. Fid h igur ouio of h quio q pq q p p z Souio: Th quio i of h form z p q fp,q For comp igr rpc p d q,w g z! Comp igr z ----- Diff w.r. d w g...... z z Sovig w g...4 uiuig i w g...5 Suiuig 5 i 4 w g...6 Suiuig 5 & 6 i w g - - z - which i igur igr. www.vidrhipu.com www.vidrhipu.com www.udfocu.com

Sov D DD' D ' D D ' z 4 Souio: D DD' D ' D D ' z D D' D D' z 4 D D' D D' z 4 C.F φ φ 4 4 P.I D DD' D' www.vidrhipu.com D D' 4 D DD' D' D D' D 4 9 6 6 D D' 4 8 8 DD ' D ' D D ' UNIT II FOURIER SERIES PART-A. S Dirich codiio for h ic of Fourir ri of f i h irv. Nov./Dc.,Rg 8 A fucio f c pdd Fourir ri i h irv ifid. i f i priodic, ig vud d fii i ii f h fii umr of fii dicoiuii d o ifii dicoiui i iii f h fii umr of mim d miim i. Do po Fourir pio?. do o po Fourir pio cu h fucio dicoiui h poi. If 4 co,dduch... i [, ] 6 Pu h 4 co 4... 6 if h foowig codiio r 4. Th coi ri for f i i < < i giv i co co Dduc h. h h ifii S. Joph Cog of Egirig & Pg No. ISO 9:8 S. Joph Iiu of Tchoog www.udfocu.com www.vidrhipu.com

Giv i co co Pu, poi of coiui > i co co i.. co i.. 5. Fid h vu of h Fourir ri for f. Nov./Dc.,Rg. f f i poi of dicoiui, vu of Fourir ri of f i. f- im f h im h. h h f. f i.5 6. Fid h co rm of h Fourir ri for h fucio f, < < Sic f i v fucio i < < f d d www.vidrhipu.com Co rm 7. To which vu, h hf rg i ri corrpodig o f prd i h irv, 5 covrg 5? A h d poi hf rg i ri covrg o zro. Th hf rg i ri corrpodig o f covrg o zro 5. 8. Wh ou m Hrmoic i? Th proc of fidig h hrmoic i h Fourir ri pio of fucio umric i kow hrmoic i. 9. Fid h Fourir co for i i < <, wh prd Fourir ri. Souio: f i f i i f Hr f i v fucio. Fid h roo m qur vu of h fucio f i. RMS vu RMS vu [ f ] d. Oi h co rm of h Fourir coi ri of f i,6 uig h foowig. : 4 5 : 4 8 5 7 6 o M vu of Σ/6 4 Co rm 7 Fid h vu of i h coi ri pio of f K i h irv, f co d K co d 5 S. Joph Cog of Egirig & Pg No. ISO 9:8 S. Joph Iiu of Tchoog www.udfocu.com www.vidrhipu.com

. If f i prd Fourir ri i h irv -, o which vu hi ri covrg? f f 6 Sic i d poi h f covrg o 4 4. S Prv horm i h irv c,c. If h Fourir ri corrpodig o f covrg uiform o f i c,c c h [ f ] d c 5. Wri dow h comp form of Fourir Sri i c,c i f c whr c 6. If f i < <, fid. Souio: co d c i c f co i { }. 7. Drmi i h Fourir ri pio of f i < <. f i d i d co i 8. Fid h co rm i h prio of Souio: f i < <. Hc how h S. Joph Cog of Egirig & Pg No. 4 ISO 9:8 S. Joph Iiu of Tchoog d co Fourir ri i h irv,. co i co d d Co rm 9. Giv h prio for h Fourir ri coffici for h fucio f dfid i. Souio: f i d. Drmi h vu of & i h Fourir ri pio of f i < < Souio: f f f f i odd fucio. If www.vidrhipu.com PART B www.udfocu.com www.vidrhipu.com

i.... ii.... 4 6 Souio: W kow h www.vidrhipu.com f co i d 4 4 d 4 6 α co d 4 co d i co i 4 co 4 4 4 i d 4 co i co 4 S. Joph Cog of Egirig & Pg No. 5 ISO 9:8 S. Joph Iiu of Tchoog www.udfocu.com www.vidrhipu.com

4 f co. co co... Pu f... i p of dicoiui. f 4 4 4 >... 4... 4... 6 Pu f i Hr i p of coiui. f. www.vidrhipu.com >...... S. Joph Cog of Egirig & Pg No. 6 ISO 9:8 S. Joph Iiu of Tchoog www.udfocu.com www.vidrhipu.com

www.vidrhipu.com... Fid h Fourir ri for i. Souio: Giv Sic i v fucio S. Joph Cog of Egirig & S. Joph Iiu of Tchoog Pg No. 7 ISO 9:8 www.udfocu.com www.vidrhipu.com

Suiu i quio w g [ ] f co. Oi h Fourir ri o rpr h fucio f i < < d dduc h.... 5 8 Souio: Giv f f f Th giv fucio i v fucio. Hc www.vidrhipu.com f co d co d { } i co co if i v 4 if i odd 4 f co, 4 co co... Pu S. Joph Cog of Egirig & Pg No. 8 ISO 9:8 S. Joph Iiu of Tchoog www.udfocu.com www.vidrhipu.com

f 4... 5 Hr i p of coiui f 4... 5 4... 5.... 8 5 Fid h fir wo hrmoic of h Fourir ri of f giv 4 5 f 9 8 4 8 6 Souio: Hr h gh of h i v i 6, co i co i 9 9 9 8 9 5.7-9 5.6 4 4 -.9 - -.8 8-8 8 4 4 8 6 - -.6 -.6 5 5 7.6 - -7.4 5-5 -.4-7 f co i co i www.vidrhipu.com 5 4.66 6 6 S. Joph Cog of Egirig & Pg No. 9 ISO 9:8 S. Joph Iiu of Tchoog www.udfocu.com www.vidrhipu.com

co 8. 6 i.5 6 www.vidrhipu.com co. 6 i 6 4.66 f 8.co.co.5i.. Fid h Fourir ri for h fucio Souio: Giv Thrfor i odd fucio. Thrfor Suiu i w g Fid h fir wo hrmoic of h Fourir ri of f. Giv 4 5 f.4.9.7.5.. S. Joph Cog of Egirig & Pg No. ISO 9:8 S. Joph Iiu of Tchoog www.udfocu.com www.vidrhipu.com

Souio: www.vidrhipu.com Th vu of i rpiio of h fir; o h fir i vu wi ud Th vu of co, co, i, i ud f co i co i..4.5.866 -.5.866.9 -.5.866 -.5.866.7-4.5 -.5 -.866 -.5 -.866 5..5 -.866 -.5 -.866 co co.9,.7,. 6 6 6 i i.7,.6 6 6 4. Oi h Fourir ri for of priod d dfid foow Hc dduc h Souio: Giv Th Fourir ri i S. Joph Cog of Egirig & Pg No. ISO 9:8 S. Joph Iiu of Tchoog www.udfocu.com www.vidrhipu.com

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www.vidrhipu.com Souio: Giv Th coi ri i if i v 4 if i odd Suiuig i quio w g Dducio i B Prv idi S. Joph Cog of Egirig & S. Joph Iiu of Tchoog Pg No. ISO 9:8 www.udfocu.com www.vidrhipu.com

www.vidrhipu.com Dducio ii 4 S 6 96 5. 4 5 S 6 96 Fid h comp form of h Fourir ri of i Souio: Th comp form of h Fourir ri i i giv Whr Hc com S. Joph Cog of Egirig & S. Joph Iiu of Tchoog Pg No. 4 ISO 9:8 www.udfocu.com www.vidrhipu.com

Fid h hf rg Fourir i ri for f i h irv, d dduc h... 5 Souio: www.vidrhipu.com Th i ri i f i f i d i d co i co 4 f Dducio: [ ] 8,,,... 8 i if if i v i odd Pu i poi of coiui 8 f 5... 4 8... 5 5... UNIT- III - APPLICATIONS OF PARTIAL DIFFERENTIAL EQUATIONS PART A. S wo umpio md i h drivio of h o dimio wv quio. i. Th rig i prfc ic d do o offr ric o dig. ii. Dfcio d op d/d vr poi of h rig r m, o h hir highr powr m gcd. S. Joph Cog of Egirig & Pg No. 5 ISO 9:8 S. Joph Iiu of Tchoog www.udfocu.com www.vidrhipu.com

www.vidrhipu.com. Wri dow h pri diffri quio govrig o dimio wv quio? C whr, i h dipcm of h rig.. I h wv quio C, wh do C d for? C T/m, whr T i h io d m i h m of h rig. 4. Wri poi ouio of h rvr virio of h rig i o dimio. i, A p B p C p D p ii, A co p Bi p C co p i p. iii, A B C D 5. Wri dow h ppropri ouio of h virio of rig quio. How i i cho? Th ppropri ouio of h virio of rig quio i, A co p B i p C co p D i p. Wh w d wih h virio of ic rig, rprig h dipcm of h rig poi, i mu priodic i. Hc h ov ouio which coi of priodic fucio i i h propr ouio of h prom o virio of rig. 6. A igh rchd rig wih fid d poi d i iii r i i quiirium poiio. If i i virig givig ch poi voci -, wri dow h oudr d iii codiio of h ov prom., d, for,, d λ 7. S Fourir w of h coducio. for Th r which h fow hrough r i joi proporio o h r d o h mprur grdi orm o h r. 8. Wri dow h pri diffri quio h rpr o dimio h quio. u u α, whr α k ρc 9. Wh r h vriou poi ouio of o dimio diffuio quio? i u, A p B p C α p α p ii u, A co p B i p C iii u, A B. p p α p Wh c u, A B C h corrc ouio i ovig o dimio h quio? A, u. I i o poi.. I d codiio driv h ouio of o dimio h fow quio. S. Joph Cog of Egirig & Pg No. 6 ISO 9:8 S. Joph Iiu of Tchoog www.udfocu.com www.vidrhipu.com

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S. Joph Cog of Egirig & Pg No. 5 ISO 9:8 S. Joph Iiu of Tchoog ] [, F f F whr d f d F d d d d d d d 5 4 4 4 4 4 5 co i 8 co i 8 co i 8 co i. 5 d co i 5 d co i, d, wh, d d, Pu 5 d co i 5 5 8 5 8 d co i 5 5 5 5 5 5 5 5 Fid h Fourir i rform of whr >. [Nov/Dc, 8] i d f f F S Diffriig oh id w.r.o i co co co d df d d d df www.vidrhipu.com www.vidrhipu.com www.udfocu.com

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F [ f ] F www.vidrhipu.com i d co d i i co co co ii d i d i B ivr Fourir rform f F d f f Pu, w g co co co d i i d co co d co i d co d co co co i i d i. i co d d. L, h d d d i d i B Prv idi F d d f d co d d co d d co d d S. Joph Cog of Egirig & Pg No. 5 ISO 9:8 S. Joph Iiu of Tchoog www.udfocu.com www.vidrhipu.com

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www.vidrhipu.com 4 Fid Z Souio: Z{ } Z { } { } { } Z Z Z z z z 4z z z z z z 4 z z z 4z z z z z z z 6z 4 4 4. Fid Z r co Souio: θ d hc dduc Z co iθ iθ z Z Z z iθ iθ z z iθ z Z z z z iθ iθ iθ iθ iθ z z θ z z coθ iiθ z zcoθ iziθ z z co coθ z iθ θ θ z z z co θ ii θ i z zco z zco quig r d imgir pr, S. Joph Cog of Egirig & Pg No. 65 ISO 9:8 S. Joph Iiu of Tchoog www.udfocu.com www.vidrhipu.com

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