2. Fourier Series, Fourier Integrals and Fourier Transforms

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1 Mathematics IV -. Fourier Series, Fourier Itegrals ad Fourier Trasforms The Fourier series are used for the aalysis of the periodic pheomea, which ofte appear i physics ad egieerig. The Fourier itegrals ad Fourier trasforms eted the ideas ad techiques of the Fourier series to the o-periodic pheomeo. The Fourier trasform is commoly used to trasform a problem from the "time domai" ito the "frequecy domai" i which the amplitude ad the phase are give as a fuctio of frequecy.. Fourier Series.. Periodic Fuctios If a fuctio f( ) is defied for all real ad if there is some positive umber p such that f( p) f( ) for all (..) it is called periodic. The umber p is called a period of f( ). From Eq.(..), for ay iteger, f( p) f( ) for all (..) Hece p, 3 p, 4 p, are also period of f ( ). If a periodic fuctio f ( ) has a smallest period p ( ), this is ofte called the primitive period of f ( ). If f ( ) ad g( ) have period p, the the fuctio h ( ) af( ) bg ( ) (, also has the period p. Fig... Periodic fuctio ab costat) (..3).. Fourier Series of a Periodic Fuctio with Period et us assume that f ( ) is a periodic fuctio of period that ca be represeted by a trigoometric series. Fourier series (..4) f ( ) a a cosb si Fourier coefficiets (give by the Euler formulas) a f( ) d a f( )cos d, (,, ) (..5) b f( )si d, (,, ) It ca be obtaied from the orthogoality of trigoometric system o a iterval of legth i Sec..4. ( m) ( m) cos cos m d, si si m d ( m) ( m) (..6) cos si m d, cos d, si d for ay itegers ad m. Fig... Cosie ad sie fuctios havig the period

2 - Mathematics IV If a periodic fuctio f ( ) with period is piecewise cotiuous i the iterval ad has a left-had derivative ad right-had derivative at each poit of that iterval, the the Fourier series Eq.(..4) of f ( ) is coverget. Its sum is f ( ), ecept at a poit at which f ( ) is discotiuous ad the sum of the series is the average of the left- ad right-had limits of f ( ) at. Parseval s Theorem If a fuctio f ( ) is square-itegrable o a iterval, the a a b f( ) d Fig...3 eft- ad right-had limits (..7)..3 Fourier Series of a Fuctio of Ay Period p If a fuctio f ( ) has period p, the Fourier series f ( ) a acos bsi (..8) Fourier coefficiets a f( ) d a f( )cos d, (,, ) (..9) b f( )si d, (,, ) ( et v i Eq.(..4) ad Eq.(..5))..4 Eve ad Odd Fuctios A fuctio g( ) is eve if g( ) g( ) for all. A fuctio h ( ) is odd if h( ) h( ) for all. If g( ) is a eve fuctio, the g( d ) gd ( ). If h ( ) is a odd fuctio, the hd ( ). Eve fuctio Fig...4 Odd fuctio Eve fuctio ad odd fuctio The product of a eve fuctio ad a odd fuctio is odd. The product of a eve fuctio ad a eve fuctio, or that of a odd fuctio ad a odd fuctio is eve. Fourier Series of a Eve Fuctio of Period p f ( ) a a cos ( f ( ) is eve fuctio) Fourier cosie series (..) with coefficiets a f( ) d, a f( )cos d, (,, ) ( b ) (..) Fourier Series of a Odd Fuctio of Period p f ( ) b si ( f ( ) is odd fuctio) Fourier sie series (..) with coefficiets b f( )si d, (,, ) ( a, a ) (..3)

3 Mathematics IV Comple Fourier Series The Fourier series of a periodic fuctio f ( ) of period (..4) f ( ) a a cosb si a f ( ) d, a f ( )cos d, b f ( )si d, (,, ) (..5) ca be writte i comple form. Usig the Euler formula, we get i e cos isi, i e cos isi (..6) i i i i i i i cos e e, si e e e e (..7) i With this, Eq.(..4) becomes i i i i f( ) a ae e be e i (..8) i i a a ibe a ibe If we itroduce the otatios, c a f( ) d i c a ib f ( ) cos i si d f ( ) e d (..9) i c a ib f( ) cos isi d f( ) e d we obtai i i f ( ) ce, c f( ) e d, (,,, ) (..)..6 Comple Fourier Series of Fuctio f ( ) of Period p i f( ) ce i, c f( ) e d, (,,, ) (..) Parseval s Theorem If a fuctio f ( ) is square-itegrable o a iterval, the c f( ) d (..) Summary Fourier series of a periodic fuctio f ( ) of period p f( ) a a cos bsi ce Fourier coefficiets a f( ) d a f( )cos d, (,, ) b f( )si d, (,, ) i c f( ) e d, (,,, ) i (..3) (..4)

4 -4 Mathematics IV Eample Fid the Fourier series of the followig periodic fuctio of period p, ad graph their partial sums. if f( ) if Solutio Sice the period p, we obtai by the Euler formulas a f( ) d d 4 a f( )cos d cos d si cos for odd for eve f () S S 3 Fig...5 Partial sums of the Fourier series of f ( ) S 5 f () (..5) for odd cos si b f( )si d si d (..6) for eve Hece, the Fourier series of f ( ) is f ( ) cos cos3 cos5 si si si (..7) Figure..5 shows the partial sum m m ( ) ( ) S cos m si m, (,, ) (..8) 4 m m m m This figure shows oscillatios ear the poits of discotiuity of f ( ). We might epect that these oscillatios disappear as approaches ifiity, but this is ot true; with icreasig, they are shifted closer to the poits of discotiuity. This uepected behavior is kow as the Gibbs pheomeo. Net, we obtai the comple Fourier coefficiets c a 4 i i i c f( ) e d e d e ( i ) (,, ) Thus, we should ofte separately calculate oly c. (..9) Problem Fid the Fourier series of the followig fuctios, which are assumed to be periodic of the period p, ad graph their partial sums. () f( ) ( ) () f( ) ( ) (3) f( ) ( ) (4) f( ) e ( ) (5) f ( ) si (6) f ( ) 4cos cos4 (7) if f( ) if (9) if f( ) if if () f( ) si if if (8) f( ) if if () f( ) if () f( ) si ( )

5 Mathematics IV -5. Fourier Itegrals ad Fourier Trasforms I the previous sectio, we foud that a periodic fuctio ca be represeted by a Fourier series. We wat to eted the method of Fourier series to operiodic fuctios. We see what happes to the periodic fuctio of period p if we let... Fourier Itegral We cosider ay periodic fuctio f ( ) of period p that ca be represeted by a Fourier series f( ) a acosbsi (..) f() v dv cos f()cos v vdv si f()si v vdv where We ow set ( ) (..) The, ad we may write Eq.(..) i the form f ( ) ( ) (cos ) ( )cos (si ) ( )si f v dv f v vdv f v vdv This represetatio is valid for ay fied, arbitrarily large, but fiite. (..3) We ow let ad assume that the resultig operiodic fuctio f ( ) lim f ( ) is absolutely itegrable o the -ais; that is, f ( ) deists ad it is fiite. The, ad the value of the first term o the right side of Eq.(..3) approaches zero. Also ad it seems plausible that ifiite series i Eq.(..3) becomes a itegral from to, which represets f ( ), f ( ) cos f( v)cosvdvsi f( v)sivdvd (..4) If we itroduce the otatios A( ) f( v)cos vdv, B( ) f( v)sivdv (..5) we ca write this i the form f ( ) A( )cos B( )sid (..6) This is a represetatio of f ( ) by a Fourier itegral. This aive approach merely suggests the represetatio Eq.(..6) but by o meas establishes it. Sufficiet Coditios for the Fourier Itegral If f ( ) is piecewise cotiuous i every fiite iterval ad has a right-had derivative ad a left-had derivative at every poit ad if it is absolutely itegrable o the -ais, the f ( ) ca be represeted by a Fourier itegral Eq.(..6). At a poit where f ( ) is discotiuous the value of the Fourier itegral equals the average of the left- ad right-had limits of f ( ) at that poit. Comple Fourier Itegral Similarly, we ca derive the followig comple Fourier itegral from the comple Fourier series. i ( v) f ( ) f( v) e dvd (..7)

6 -6 Mathematics IV.. Fourier Trasform Writig the epoetial fuctio i Eq.(..7) as a product of epoetial fuctios, we have iv i f ( ) f( v) e dv e d (..8) The epressio i brackets is a fuctio of, is deoted by f ˆ( ), ad is called the Fourier trasform of f ( ); writig v, we have Fourier trasform ˆ( ) ( ) ( ) i f F f f e d (..9) Iverse Fourier trasform ˆ ( ) ( ) ˆ i f F f f( ) e d (..) Sufficiet Coditios for Eistece of the Fourier Trasform If f ( ) is piecewise cotiuous o every fiite iterval ad it is absolutely itegrable o the -ais, the the Fourier trasform of a fuctio f ( ) eists. Summary Fourier itegral of a fuctio f ( ) i f ( ) A( )cos B( )si d C( ) e d (..) where A( ) f( )cos d, B( ) f( )sid (..) i C( ) f( ) e d Fourier trasform ˆ( ) ( ) ( ) i f F f f e d (..3) Iverse Fourier trasform ˆ ( ) ( ) ˆ i f F f f( ) e d (..4) (a) Fourier series (b) arge T (c) T (Fourier trasform) Fig... Fourier series ad Fourier trasform (period p T )

7 Mathematics IV -7 Table.. Formulas of the Fourier trasform ( a ad b are costats) () iearity af ( ) bg( ) afˆ ( ) bgˆ ( ) () Derivative of fuctio ( f ) ( ) i fˆ( ) ia (3) Shiftig o the -ais e f( ) f ˆ( a) i a (4) Shiftig o the -ais f ( a) e fˆ( ) (5) Differetiatio of trasform f( ) if ˆ ( ) (6) Scalig ( a ) f ( a ) (7) Covolutio ( f g)( ) f( p) g( p) dp f ( p) g( p) dp ˆ( ) a f a fˆ ( ) gˆ ( ) (8) Duality f ˆ( ) f ( )..3 Formulas of the Fourier Trasform () iearity ( a ad b are costats) F af ( ) bg( ) i af ( ) bg( ) e d a i i f( e ) db ge ( ) d af f( ) bf g( ) () Fourier trasform of the derivative of a fuctio et f ( ) be cotiuous o the -ais ad f( ) as. Furthermore, let f ( ) be absolutely itegrable o the -ais. The i i i f ( ) f( ) e d f( ) e ( i ) f( ) e d (..5) F (..6) i i f( ) e d= iff( ) Two successive applicatios of the above equatio give F f ( ) if f( ) ( i) F f( ) F f( ) (..7) (3) Shiftig o the -ais ˆ( ) ˆ i F f a f( a) e d et a v, the va, d dv. We obtai ˆ ˆ ( ) iva ia ( ) () ˆ iv F f a fve dv e fve () dv ia ˆ( ) ia e F f e f ( ) (4) Shiftig o the -ais i F f ( a) f( a) e d et a v, the v a, d dv. We obtai F ( ) i va ia iv f ( a) f( v) e dv e f( v) e dv e ia F f( ) (..8) (..9)

8 -8 Mathematics IV (5) Differetiatio of the Fourier trasform ˆ( ) i ˆ df d ( ) ( ) i de ( ) ( ) i f f e d f d if e d d d d if f( ) Thus, ˆ F f( ) if( ) (..) (6) Scalig o the -ais If a the i F f ( a) f ( a) e d et a v, we get v i a f( a) f( v) e dv f ˆ F ( ) a a a We ca similarly derive it if a. Hece, ( ) ˆ F f a f( ) (..) a a (7) Covolutio i F ( f g)( ) f( pg ) ( pe ) dpd (..) et p q, we get i ( pq) F ( f g)( ) f ( p) g( q) e dqdp (..3) ip iq f ( pe ) dp gqe ( ) dq f( ) g ( ) F F Also, F f ( g ) ( ) ( fˆ* gˆ)( ) ˆ f( vg ) ˆ( ) dv (..4) F f ( g ) ( ) F f( ) F g ( ) Note: I geeral, (8) Duality et ad simultaeously i the iverse Fourier trasform. We get ( ) ˆ i f f( ) e d fˆ( ) F (..5) Eample Fid the real Fourier itegral ad the Fourier trasform of the followig fuctio. if f( ) (..6) otherwise Solutio Substitutig Eq.(..6) ito Eq.(..5), we obtai cos si A( ) f ( )cos d cos d (..7) si cos B( ) f( )sid sid Hece, we obtai the Fourier itegral cos si si cos f ( ) cos si d (..8) Substitutig Eq.(..6) ito Eq.(..9), we obtai the Fourier trasform i ( ) ˆ( ) ( ) i i e i f f e d e d (..9)

9 Mathematics IV -9 Figure.. shows A( ), B( ) ad f ( ) fˆ ( ), which is a frequecy domai represetatio of the fuctio f ( ). Figure..3 shows the itegral a cos si si cos Sa ( ) cos si d (..3) which approimates the itegral i Eq.(..8). Although Sa ( ) approaches f ( ) as a icreases, the oscillatio kow as Gibbs pheomeo occurs ear the discotiuity poit of f ( ).. f (). f ~ () S 6 S 3 f ().. B() A() Fig... Amplitude spectrums of f ( ) Fig...3 Itegral Sa ( ) ad f ( ) S 4..4 Fourier Trasforms of Some Fuctios Gaussia fuctio f ( ) e a ( a ) We use the defiitio of the Fourier trasform. i i i a a a i a a 4a a F f ( ) e d e d e e d We deote the itegral by I ad we use ai a v as a ew variable of itegratio. The d dv a, so that i a a v I e d a e dv We square the itegral, covert it to a double itegral, ad use polar coordiates r u v ad. Sice dudv rdrd, we get I e du e dv e dudv u v ( u v ) a a r r e rdrd e a a a Fig...4 Gaussia fuctio ad its Fourier trasform Hece I a. From this ad the first equatio i this solutio, a a i 4a 4a F e e d e e (..3) a a The Fourier trasform of the Gaussia fuctio is a Gaussia fuctio. Square wave if fa ( ) a if Area: f ( ) d a a a (..3) Fig...5 Square wave ad its Fourier trasform

10 - Mathematics IV Takig the Fourier trasform, we get F a i i i fa( ) fa( ) e d e d e a a a i ia ia e e ai sia si i a ai a a a (..33) Dirac delta fuctio (uit impulse fuctio) if lim fa ( ) ( ) (..34) a if Properties: ( ) d f ( ) ( a) d f( a) (..35) f ( ) ( ) d f() Takig the Fourier trasform, we get i F ( ) ( e ) d (..36) i e Fig...6 Dirac delta fuctio ( ) ad its Fourier trasform Usig the formula of shiftig o the -ais, we get i a F ( a) e F ( ) ia e (..37) cos a isi a Fig...7 ( a) ad its Fourier trasform From the duality, we obtai F ( ) ( ) (..38) ad ia F e ( a) ( a) F ia e ( a) (..39) Fig...8 Costat ad its Fourier trasform Trigoometric fuctio f( ) acos bsi ( aib ) e ( aib ) e where is costat. Takig the Fourier trasform, we get i i Fig...9 Trigoometric fuctio ad its Fourier trasform F acosbsi ( aib) ( ) ( aib) ( ) (..4)

11 Mathematics IV - Impulse trai (Dirac comb) T ( ) ( T) (..4) Because it is a periodic fuctio of period p T, it has a Fourier series. The comple Fourier coefficiets are T i c T( ) e d T T T where T. Hece we ca epress the impulse trai as i Fig... Impulse trai ad its Fourier trasform T ( ) e (..4) T Takig the Fourier trasform, we get F T ( ) ( ) T ( ) (..43) ( ) The Fourier trasform of the impulse trai is a impulse trai. Fig... Property of impulse trai Periodic fuctio f ( ) of period p T Takig the Fourier trasform, we get ( ) T ˆ( ) ( ) i ( ) i f f e d f T e d ( ) T et t T, the T ( ) T ˆ( ) () i tt it () it f f t e dt e f t e dt T T Where, it if e T otherwise it From the impulse trai, e ( ) T T Furthermore, usig the comple Fourier coefficiets c, T it f () t e dt Tc at T T Hece we get fˆ( ) Tc ( ) c ( ) where T (..44) The Fourier trasform of the periodic fuctio f ( ) of period p T has values oly at discrete poits, ad these values are represeted by the comple Fourier coefficiets of f ( ).

12 - Mathematics IV..5 Applicatios of Fourier Series ad Fourier Trasforms Approimatio et f ( ) be a fuctio o the iterval that ca be represeted o this iterval by a Fourier series. The a trigoometric polyomial of degree N N F( ) A AcosBsi (N fied) (..45) is a approimatio of the give f ( ). The square error of F i Eq.(..45) (with fied N ) related to the fuctio f o the iterval E ( f F) d (..46) is miimum if ad oly if the coefficiets of F i Eq.(..45) are the Fourier coefficiets of f, that is, A a, A a,, B b. This miimum value E * is give by N * E f d a ( a b) (..47) * From Eq.(..47) we see that E caot icrease as N icreases, but may decrease. Hece, with icreasig N, the partial sums of the Fourier series of f N f ( ) a a cosb si yield better ad better approimatios to f, cosidered from the viewpoit of the square error. Frequecy aalysis The ature of presetatio Eq.(..) of f ( ) becomes clear if we thik of it as a superpositio of siusoidal oscillatios of all possible frequecies, call a spectral represetatio. I Eq.(..), the spectral desity f ˆ( ) measures the itesity of f( ) i the frequecy iterval betwee ad ( small, fied). The itegral ˆ f ( ) d ca be iterpreted as the total eergy of the physical system; hece a itegral of fˆ ( ) from a to b gives the cotributio of the frequecies betwee from a to b to the total eergy. If a fuctio f( ) is square-itegrable, the we have Parseval s theorem ˆ f( ) d f( ) d (..48) If the system has a periodic solutio y f( ) that ca be represeted by a Fourier series, the we get a series of squares c of Fourier coefficiets c give by Eq.(..). I this case we have a discrete spectrum (or poit spectrum) cosistig of coutably may isolated frequecies (ifiitely may, i geeral), the correspodig c beig the cotributios to the total eergy. A system whose solutio ca be represeted by a Fourier itegral Eq.(..) leads to the above itegral for the eergy. Fig... Frequecy aalysis Solvig method of differetial equatios I the et sectio, we will see that the partial differetial equatios ca be solved by the Fourier series or the

13 Mathematics IV -3 Fourier trasforms methods. Problem- Fid the real Fourier itegral ad the Fourier trasform of the followig fuctios. () f( ) ( ) () f( ) ( ) (3) f( ) ( ) (4) f( ) e ( ) (5) f ( ) e (6) f ( ) e si (7) if if f( ) (8) f( ) if otherwise otherwise (9) if if f( ) () f( ) if otherwise otherwise si if si if () f( ) () f( ) otherwise otherwise Problem- Fid the Fourier trasform of the followig fuctios. () f ( ) si () f ( ) 4cos cos4

14 -4 Mathematics IV.3 iear Partial Differetial Equatios I this sectio, we see that the partial differetial equatios ca be solved by the Fourier series or the Fourier trasforms methods..3. Eamples of iear Partial Differetial Equatios of the Secod Order Heat equatio u u c t Wave equatio u u c t aplace equatio u u y iear Homogeeous Partial Differetial Equatios The superpositio or liearity priciple holds. (.3.) (.3.) (.3.3) Eample- Fid the temperature ut (, ) i a bar of legth govered by the followig heat equatio ad the coditios. u u c t (, t ) (.3.4) Boudary coditios: u(, t), u(, t) ( t ) (.3.5) Iitial coditio: if u (,) f( ) if (.3.6) Solutio by method of separatig variables (review) We determie solutio of the Eq.(.3.4) of the form ut (, ) FGt ( ) ( ) (.3.7) By differetiatig ad substitutig it ito Eq.(.3.4), we obtai dg() t d F( ) F( ) c G( t) dt d (.3.8) Dividig by cfgt ( ) ( ), we fid dg( t) d F( ) k cgt () dt F( ) d (.3.9) This yields two ordiary differetial equatios, d F( ) kf( ) d (.3.) dg() t kc G() t dt (.3.) From Eq.(.3.) ad the boudary coditios, F(), F( ), () For k, F( ) Ae k k Be. From Boudary coditios, A B. Hece F( ) () For k, F( ) a b. From Boudary coditios, a b. Hece F( ) (3) For k, F( ) Acos p Bsi p, where p k From the boudary coditios, F() A, F ( ) Bsi p We must take B. Hece si p, so p (,, ) Thus, we obtai ( ) si F B (,, ) (.3.) For p k, a solutio of Eq.(.3.) is Gt () e c p t (.3.3) The followig is omitted.

15 Mathematics IV -5 Solutio by Fourier series We solve the problem usig the method of separatig variables ad the Fourier series. From the boudary coditios, the temperature ut (, ) ca be epressed i the form of the Fourier sie series. ut (, ) G ( t)si (.3.4) Substitutig Eq.(.3.4) ito Eq.(.3.4), we obtai G ()si t c G() t si (.3.5) Hece, G () t s must be satisfy the followig equatio. G () t c G() t (.3.6) The solutio of this equatio is c G() t B ep t (.3.7) From this ad Eq.(.3.4), we obtai c ut (, ) B si ep t (.3.8) From the iitial coditio, we have u (,) f( ) B si (.3.9) This equatio represets the Fourier sie epasio of f ( ). Hece, B s are the coefficiets of the Fourier sie series. B f( )si d (.3.) Hece the formal solutio is c v ut (, ) si ep t f( v)si dv (.3.) From Eq.(.3.), we get 4 (,5,9, ) 4 B si d ( )si d si 4 (.3.) ( 3,7,, ) (, 4,6, ) Hece the solutio is 4 c 3 9c ut (, ) si ep t si ep t (.3.3) 9 Figure.3. shows the temperature for c, ad various values of time. u (,t) t = u (,t) t = t =. t =..5 t =.5 t = Fig..3. Temperature i the bar of egth.5 t =. t =.5 t = t = 4 4 Fig..3. Temperature i the ifiite bar

16 -6 Mathematics IV Eample- Fid the temperature ut (, ) i the ifiite bar ( ) govered by the followig heat equatio ad the iitial temperature. u u c (, t ) (.3.4) t if u (,) f( ) (.3.5) if Solutio et the Fourier trasforms of ut (, ) ad f ( ) with respect to be ˆ i i uˆ(, t) u(, t) e d, f ( ) f ( ) e d (.3.6) The Fourier trasforms of Eq.(.3.4) ad Eq.(.3.5) become duˆ(, t) c ( i ) uˆ(, t), uˆ(,) fˆ ( ) (.3.7) dt The geeral solutio of this differetial equatio is c t uˆ(, t) Ce (.3.8) From the iitial coditio, we get uˆ(,) C fˆ ( ) (.3.9) Hece the solutio of this iitial value problem is ˆ(, ) ˆ c t u t f( ) e (.3.3) Takig the iverse Fourier trasform, we obtai the formal solutio i ˆ c t i ut (, ) uˆ (, te ) d f( ) e e d (.3.3) c t i ( v) ( v) f () v e d dv f()ep v dv c t 4ct The last equatio ca be obtaied by a similar method to Eq.(..3). Hece the solutio is ( v) ( v) ut (, ) ( v)ep dv ( v)ep dv c t 4ct (.3.3) 4ct Figure.3. shows the temperature for c ad various values of time. Takig z ( v) ( c t) as a variable of itegratio i Eq.(.3.3), we get the alterative form z ut (, ) f( cz t) e dz (.3.33) Eample-3 Fid the deflectio i the strig of legth of govered by the followig wave equatio ad the coditios. u u c (, t ) (.3.34) t u(, t), u(, t) ( t ) (.3.35) if u u (,) f( ), g ( ) (.3.36) if t t Solutio Similar to Eample, from the boudary coditios Eq.(.3.35), ut (, ) G ( t)si (.3.37) Substitutig it ito the wave equatio Eq.(.3.34), we obtai dg() t c G () t (.3.38) dt

17 Mathematics IV -7 Hece, c t ct ut (, ) B cos Csi si (.3.39) From the iitial coditio Eq.(.3.36), u (,) f( ) Fourier sie series B f( )si d (.3.4) u g ( ) Fourier sie series C g( )si d t t c (.3.4) Hece the formal solutio is c t ut (, ) B cos si, B f( )si d (.3.4) Eample-4 Fid the deflectio i the ifiite strig ( ) govered by the followig wave equatio ad the coditios. u u c (, t ) (.3.43) t if u u (,) f( ), (.3.44) if t t D Alembert s solutio of the wave equatio (review) By itroducig the ew idepedet variables, ct, ct, (.3.45) a geeral solutio of the wave equatio Eq.(.3.43) is ut (, ) ( ct) ( ct) (.3.46) The we fid the solutio satisfyig the iitial coditios. Solutio by Fourier trasform The Fourier trasforms with respect to of the wave equatio Eq.(.3.43) ad the coditios Eq.(.3.44) become duˆ(, t) c uˆ(, t) (.3.47) dt ˆ duˆ(, t) uˆ(,) f( ), (.3.48) dt t We obtai a geeral solutio of Eq.(.3.47), cit cit uˆ(, t) Ae Be (.3.49) From the iitial coditios Eq.(.3.48), the solutio is ˆ cit cit uˆ(, t) f( ) e e (.3.5) Takig the iverse Fourier trasform of Eq.(.3.5), we obtai the formal solutio ut (, ) f( ct) f( ct) (.3.5) Eample-5 Fid the solutio of the followig aplace equatio. u u (, y K) (.3.5) y u(, y), u(, y) ( y K) (.3.53) if u (,) f( ), uk (, ) ( ) (.3.54) if

18 -8 Mathematics IV Solutio From the boudary coditios, Eq.(.3.53) uy (, ) G ( y)si (.3.55) Substitutig it ito the aplace equatio Eq.(.3.5), we obtai dg( y) G ( ) y (.3.56) dy Hece, y y G( y) a ep bep (.3.57) From the boudary coditio, uk (, ) ( ), K K G( K) aep bep (.3.58) We obtai K b a ep (.3.59) Hece, y K y G( y) a ep ep K ( K y) ( K y) a ep ep ep (.3.6) K ( K y) a ep sih K et aep Cad from the above equatio, ( K y) uy (, ) C sih si (.3.6) From the boudary coditio, u (,) f( ), K f( ) C sih si, K Csih f( )si d B (.3.6) Hece the formal solutio is ( K y) sih uy (, ) B si, B f( )si d K (.3.63) sih Problem Fid the temperature i a bar of legth govered by the followig heat equatio ad the coditios. u u c (, t ) (.3.64) t u u, ( t ) (.3.65) if u (,) f( ) if (.3.66) Hit From the boudary coditios, the temperature ut (, ) ca be epressed i the form of the Fourier cosie series. ut (, ) G ( t)cos (.3.67)

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