TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS

Transcription

1 TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS UNIT-I PARTIAL DIFFERENTIAL EQUATIONS PART-A. Elimit th ritrry ott & from = ( + )(y + ) Awr: = ( + )(y + ) Diff prtilly w.r.to & y hr p & q y p = (y + ) ; q = ( + ) y (y + ) = p/ ; ( + ) = q/y = (p/)(q/y) 4y = pq. Form th PDE y limitig th ritrry futio from = f(y) Awr: = f(y), Diff prtilly w.r.to & y hr p & q y p = f ( y). y q = f ( y). p/q = y/ p qy = 3. Form th PDE y limitig th ott d from = + y Awr: = + y, Diff. w.r. t. d y hr p & q y p = - ; q = y - p ; q y p q y y p qy 4. Fid th omplt itgrl of p + q =pq Awr: Put p =, q =, p + q =pq += = - Th omplt itgrl i = + 5. Fid th olutio of p q y + Awr: = +y+ ----() i th rquird olutio giv p q -----() put p=, q = i () ( ) ( ) y

2 6. Fid th Grl olutio of p t + q ty = t. Awr: d dy d t t y t ot d ot y dy ot d tk ot d ot y dy ot y dy ot d log i log i y log log i y log i log i i y i y i i i y, i y i 7. Fid th qutio of th pl who tr li o th -i Awr: Grl form of th phr qutio i Whr r i ott. From () +(-) p= () y +(-) q = (3) From () d (3) y p q 8. Fid th igulr itgrl of p qy pq Awr: Th omplt olutio i y ; y ; y y r (), Tht i py -q = whih i rquird PDE. ( y) ( ) y ( y. ) y y y y y 9. Fid th grl olutio of p+qy= Awr: Th uiliry qutio i From d dy y dy d y y y Thrfor, y d dy d y Itgrtig w gt log = log y + log, o implifyig. i grl olutio. y

3 . Fid th grl olutio of p +qy = Awr: d dy d y Th uiliry qutio i d dy From y dy d Alo y Thrfor. Solv Awr: Auiliry qutio i. Solv Awr:, Itgrtig w gt y Itgrtig w gt y, y y D DD 3D i grl olutio. m m 3, 3 Th olutio i f y f y 3 y D 4DD 3D Auiliry qutio i m m, m, m 3 m m, m, m 3 m 4m 3, 3 Th CF i CF f y f y 3 PI y D 4DD 3D Put D, D Domitor =. PI y y D 4D Z=CF + PI y f y f y 3 3. Fid P.I D 4DD 4D y Awr: PI D 4DD 4D Put D, D PART-B. Solv. Solv y PI D D y p y q y m y l ly m y 3. Solv 3 4y p 4 q y 3 4. Solv y p yq y y 6 y 5. Solv y p yq 6. Solv y p q y 7. Solv y p q y

4 8. Solv 9. Solv. Solv. Solv. Solv 3. Solv 4. Solv 3 y D DD D y 3 i(3 ) o o y y D DD 6D y o D DD 3D y 6 y D 6DD 5D ih y y D 4DD 4D y 3 3 y D D D DD D y o( ) 5. Solv 6. Solv p qy p q p qy p q 7. Solv p q 8. Solv 9. Solv (i) ( p q ) ( p q ) y (ii) ( p q ) y UNIT-II FOURIER SERIES PART-A. Dfi priodi futio with mpl. If futio f() tifi th oditio tht f( + T) = f(), th w y f() i priodi futio with th priod T. Empl:- i) Si, o r priodi futio with priod ii) t i r priodi futio with priod.. Stt Dirihlt oditio. (i) f() i igl vlud priodi d wll dfid pt poily t Fiit umr of poit. (ii) f () h t mot fiit umr of fiit diotiuou d o ifiit Diotiuou. (iii) f () h t mot fiit umr of mim d miim. 3. Stt Eulr formul. Awr: I (, l) o f o i l whr f ( ) d o l f ( )o d l f ( )i d

5 4. Writ Fourir ott formul for f() i th itrvl (, ) Awr: o f ( ) d f ( )o d f ( )i d 5. I th Fourir pio of, f() = i (-π, π ), fid th vlu of, Awr: Si f(-)=f() th f() i v futio. H. = 6. If f() = 3 i π < < π, fid th ott trm of it Fourir ri. Awr: Giv f() = 3 f(-) = (- ) 3 = - 3 = - f() H f() i odd futio Th rquird ott trm of th Fourir ri = o = 7. Wht r th ott trm d th offiit of o i th Fourir Epio f() = 3 i π < < π Awr: Giv f() = 3 f(-) = - - (- ) 3 = - [ - 3 ] = - f() H f() i odd futio Th rquird ott trm of th Fourir ri = = 8. Fid th vlu of for f() = ++ i (, ) Awr: f ( ) d o 3 ( ) d Fid i th pio of Fourir ri i (, ). Awr: Giv f() = f(-) = = f() H f() i v futio I th Fourir ri =. Fid i th pio of i Fourir ri i (, ) Awr: Giv f() = i f(-) = (-)i(-) = i = f() H f() i v futio I th Fourir ri =

6 . If f() i odd futio i ll,. Wht r th vlu of & Awr: If f() i odd futio, o =, =. I th Epio f() = Fourir ri i (-. ) fid th vlu of Awr: Giv f() = f(-) = - = = f() H f() i v futio d o 3. Dfi R.M.S vlu. If lt f() futio dfid i th itrvl (, ), th th R.M.S vlu of f() i dfid y y f ( ) d 4. Stt Prvl Thorm. Lt f() priodi futio with priod l dfid i th itrvl (, +l). l o f ( ) d l 4 Whr, & r Fourir ott o 5. Fid th RMS vlu of f() =, Awr: Giv f() = R.M.S vlu PART-B. Epd l y f ( ) d d l f( ) (, ) Fourir ri d h ddu tht (, ). Fid th Fourir ri for f() = i (-. ) d lo prov tht (i)... (ii) Epd f() = o Fourir ri i (-. ). 4. Epd f() = i Fourir ri i (, ) 5. Epd f() = Fourir ri i (-. ) d ddu to 6. If f( ), (,) i, (, ) Fid th Fourir pio of Fid th Fourir ri d h ddu tht f ( ) ( ) i (, ) d H ddu tht

7 8. Fid Fourir ri to rprt 9. Fid th Fourir ri of f i (, ).. Fid th Fourir ri for f f ( ) with priod 3 i th rg (,3) i (, ) i (, ) d h.t Fid th th hlf rg i ri for f i th itrvl (, ) d ddu tht Oti th hlf rg oi ri for f i (,) d lo ddu tht 3. Fid th Fourir ri for f() = i (-. ) d lo prov tht 4. Fid th Fourir ri up to od hrmoi X Y Fid th Fourir ri up to third hrmoi X π/3 π/3 π 4π/3 5π/3 π F() UNIT-III APPLICATIONS OF PARTIAL DIFFERENTIAL EQUATIONS PART-A u u. Clify th Prtil Diffrtil Equtio i) y Awr: u u hr A=,B=,&C=- y B - 4AC=-4()(-)=4>. Th Prtil Diffrtil Equtio i hyproli. u u u. Clify th Prtil Diffrtil Equtio y y y Awr: u u u y y y B -4AC=-4()()=>. hr A=,B=,&C= Th Prtil Diffrtil Equtio i hyproli. 3. Clify th followig od ordr Prtil Diffrtil qutio Awr: u u u u y y hr A=,B=,&C= B -4AC=-4()()=-4<. Th Prtil Diffrtil Equtio i Ellipti. 4. Clify th followig od ordr Prtil Diffrtil qutio u u u u u y y y Awr: 4 u u u 4 6 u 8 u y y y hr A= 4,B =4, & C = B -4AC =6-4(4)() =. Th Prtil Diffrtil Equtio i Proli. 4 u u u u y y

8 5. I th wv qutio y y t wht do td for? Awr: y y T hr t m T-Tio d m- M 6. I o dimiol ht qutio ut = α u wht do α td for? Awr:- u u t = k i lld diffuivity of th ut Whr k Thrml odutivity - Dity Spifi ht 7. Stt y two lw whih r umd to driv o dimiol ht qutio Awr: i) Ht flow from highr to lowr tmp ii) Th rt t whih ht flow ro y r i proportiol to th r d to th tmprtur grdit orml to th urv. Thi ott of proportiolity i kow th odutivity of th mtril. It i kow Fourir lw of ht odutio 8. A tightly trthd trig of lgth i ftd t oth d. Th midpoit of th trig i dipld to dit d rld from rt i thi poitio. Writ th iitil oditio. Awr: (i) y(, t) = (ii) y(,t) = y (iii) t t (iv) y(, ) = ( ) 9. Wht r th poil olutio of o dimiol Wv qutio? Awr: Th poil olutio r y(,t) = (A + B - ) (C t + D - t ) y(,t) = (A o + B i )( C o t + D i t) y(,t) = (A + B) ( Ct + D). Wht r th poil olutio of o dimiol hd flow qutio? Awr: Th poil olutio r u(, t) ( A B ) C u(, t) ( Ao Bi ) C u(, t) ( A B) C t. Stt Fourir lw of ht odutio Awr: Q u ka t (th rt t whih ht flow ro r A t dit from o d of r i proportiol to tmprtur grdit) Q=Qutity of ht flowig k Thrml odutivity, A=r of ro tio ; u =Tmprtur grdit

9 . Wht r th poil olutio of two dimiol hd flow qutio? Awr: Th poil olutio r u(, y) ( A B )( C o y Di y) y y u(, y) ( Ao Bi )( C D ) u(, y) ( A B)( Cy D) 3. Th tdy tt tmprtur ditriutio i oidrd i qur plt with id =,y =, = d y =. Th dg y = i kpt t ott tmprtur T d th thr dg r iultd. Th m tt i otiud uqutly. Epr th prolm mthmtilly. Awr: U(,y) =, U(,y) =,U(,) =, U(,) = T. PART-B. A tightly trthd trig with fid d poit = d = l i iitilly t rt i it quilirium poitio. If it i t virtig givig h poit vloity 3 (l-). Fid th diplmt.. A trig i trthd d ftd to two poit d prt. Motio i trtd y diplig th trig ito th form y = K(l- ) from whih it i rld t tim t =. Fid th diplmt t y poit of th trig. 3. A trig of lgth l i ftd t oth d. Th midpoit of trig i tk to hight d th rld from rt i tht poitio. Fid th diplmt of th trig. 4. A tightly trthd trig with fid d poit = d = l i iitilly t rt i it poitio giv 3 y y(, ) = y i. If it i rld from rt fid th diplmt. l 5. A trig i trthd tw two fid poit t dit l prt d poit of th trig r < < l giv iitil vloiti whr V. Fid th diplmt. ( l ) < < l 6. Driv ll poil olutio of o dimiol wv qutio. Driv ll poil olutio of o dimiol ht qutio. Driv ll poil olutio of two dimiol ht qutio. 7. A rod 3 m log h it d A d B kpt t o C d 8 o C, rptivly util tdy tt oditio prvil. Th tmprtur t h d i th rdud to o C d kpt o. Fid th rultig tmprtur u(, t) tkig =. 8. A rtgulr plt with iultd urf i 8 m wid o log omprd to it width tht ita r m log, with iultd id h it d A & B kpt t o C d 4 o C rptivly util th tdy tt oditio prvil. Th tmprtur t A i uddly rid to 5 o C d B i lowrd to o C. Fid th uqut tmprtur futio u(, t). 9. A rtgulr plt with iultd urf i 8 m wid d o log omprd to it width tht it my oidrd ifiit plt. If th tmprtur log hort dg y = i u (,) = i 8, <<8 Whil two dg = d = 8 wll th othr hort dg r kpt t o C. Fid th tdy tt tmprtur.. A rtgulr plt with iultd urf i m wid o log omprd to it width tht it my oidrd ifiit plt. If th tmprtur log hort dg y = i giv y 5 u d ll othr thr dg r kpt t o C. Fid th tdy tt ( ) 5 tmprtur t y poit of th plt.

10 UNIT-IV FOURIER TRANSFORMS PART-A. Stt Fourir Itgrl Thorm. Awr: If f( ) i pi wi otiuouly diffrtil d olutly o, th, i( t) f ( ) f t dt d.. Stt d prov Modultio thorm. F f o F F Proof: i F f o f o d i i i f d i( ) i( ) f d f d F F F f o F F 3. Stt Prvl Idtity. Awr: If F i Fourir trform of f, th F d f d 4. Stt Covolutio thorm. Awr: Th Fourir trform of Covolutio of f d g i th produt of thir Fourir trform. F f g F G 5. Stt d prov Chg of l of proprty. Awr: If F F f, th F f F i F f f d i t dt f t ; whr t, F f F

11 d 6. Prov tht if F[f()] = F() th F f ( ) ( i) F( ) d Awr: i F f d, Diff w.r.t tim d i F f i d d i f ( i) ( ) d d i ( ) F f d () i d d i ( i) F ( ) f d d d d F f i F 7. Solv for f() from th itgrl qutio Awr: f ( )o d f ( )o d F f f o d F f f ( ) F f o d o d o d o d, 8. Fid th ompl Fourir Trform of f( )

12 Awr: i F f f d ; i F f d (o ii ) d i (o ) d i [U v d odd proprty od trm om ro] 9. Fid th ompl Fourir Trform of f( ) Awr: i F f f d i d ; (o i i ) d i o i ( ( ii ) d () i o i [U v d odd proprty firt trm om ro]. Writ Fourir Trform pir. Awr: If f( ) i dfid i,, th it Fourir trform i dfid i F f d, If F i Fourir trform of f, th t vry poit of Cotiuity of f, w hv i f F d.

13 . Fid th Fourir oi Trform of f() = - Awr: F f f o d F o d F. Fid th Fourir Trform of Awr: i F f f d f( ) im,, im i i m d d othrwi i m i m i m 3. Fid th Fourir i Trform of. o d i m i m Awr: F f f i d i d F 4. Fid th Fourir i trform of F f f i d F i d F i d 5. Fid th Fourir oi trform of Awr: F f f o d

14 F o d o d o d 4 4 PART-B. Fid th Fourir Trform of f( ) if if d h o i 3 ddu tht (i) o 3 d 6. Fid th Fourir Trform of 3.Fid th Fourir Trform of i) i d ii) i f( ) f( ) d i o (ii) 3 d 5 i o. h S.T 3 d if if d h vlut if 4. Fid Fourir Trform of f( ) d h vlut if 4 i) i d ii) i 4 d 5. Evlut i) d d ii) d 6. Evlut () 4 () t dt t 4 t 9 7. Fid th Fourir i trform of f( ) i ; wh o ; wh 8. Fid th Fourir oi trform of f( ) o ; wh o ; wh 9. Show tht Fourir trform i.oti Fourir oi Trform of. Solv for f() from th itgrl qutio d h fid Fourir i Trform f ( )o d

15 .Solv for f() from th itgrl qutio 3. Fid Fourir i Trform of, t f ( )i t d, t, t, > d h ddu tht i d UNIT-V Z- TRANSFORMS PART-A. Dfi Z trform Awr: Lt {f()} qu dfid for =,, d f() = for < th it Z Trform i dfid Z f ( ) F f ( ) (Two idd trform) Z f ( ) F f ( ) (O idd trform). Fid th Z Trform of Awr: Z f f Z () Fid th Z Trform of Awr: Z Z f f Z 4. Fid th Z Trform of. Awr: d Z Z Z d, y th proprty, d d ( ) ( ) 4 3

16 5. Stt Iitil & Fil vlu thorm o Z Trform Iitil Vlu Thorm If Z [f ()] = F () th f () = lim F ( ) Fil Vlu Thorm If Z [f ()] = F () th lim f ( ) lim( ) F( ) 6. Stt ovolutio thorm of Z- Trform. Awr: Z[f()] = F() d Z[g()] = G() th Z{f()*g()} = F() G() 7. Fid Z Trform of Awr: Z f f Z Fid Z Trform of o d i Awr: Z W kow tht o Z f f o o =>> Z o Z Similrly Z o i 9. Fid Z Trform of o i o i i =>> o Awr: Z f f

17 Z log log log. Fid Z Trform of Awr:! Z f f Z!! 3...!!! 3!. Fid Z Trform of Awr: Z f f Z ( ) log log

18 . Stt d prov Firt hiftig thorm t Sttmt: If Z f t F, th Z f () t F Proof: Z f ( t) f ( T ) t T A f(t) i futio dfid for dirt vlu of t, whr t = T, th th Z-trform i T Z f ( t) f ( T ) F( ). t T T Z f ( t) f ( T ) F( ) 3. Dfi uit impul futio d uit tp futio. Awr: Th uit mpl qu i dfid follow: for ( ) for Th uit tp qu i dfid follow: u ( ) for for 4. Fid Z Trform of Awr: t Z t T T T Z 5. Fid Z Trform of Awr: Z t t T Z t Z t T t T T [Uig Firt hiftig thorm] T T T [Uig Firt hiftig thorm] t 6. Fid Z Trform of Z o t o Awr: Z o t Z o t T o T T ot t T T T ot [Uig Firt hiftig thorm]

19 7. Fid Z Trform of Z i t T Awr: Lt f (t) = it, y od iftig thorm Z i( t T) Z f ( t T) F( ) f () i t i t t t o o 8. Fid Z trform of Awr: Z f f Z Z Z 3 3 PART-B. Fid (i) Z. Fid (i) Z 3. Fid (i) Z 8 ( )(4 ) 3 3 (ii) Z (ii) Z ( )( ) ( ) (ii) Z ( ) ( )(4 ) 8 y ovolutio thorm. y ovolutio thorm ( )( 3) y ovolutio thorm 4. Stt d prov Iitil & Fil vlu thorm. 5. Stt d prov Sod hiftig thorm 6. Fid th Z trform of (i) 3 (ii) ( )( ) ( )( ) 7. Fid Z ( 4) y ridu. 8. Fid th ivr Z trform of 9. Fid (i) Z. Fid th Z trform of. Fid (ii) Z f( )! ( ) 7 H fid y prtil frtio. Z d ( )! Z d lo fid th vlu of i( ) d o( ).!. Solv y 6y 9y with y & y 3. Solv y 4y 4y y() =,y() = 4. Solv y 3y 4y, giv y() 3& y () 5. Solv y 3 3y y, y 4, y & y 8, Z. ( )!