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1 SUBJECT NAME : Trasforms ad Partial Diff Eq SUBJECT CODE : MA MATERIAL NAME : Problem Material MATERIAL CODE : JM8AM6 REGULATION : R8 UPDATED ON : April-May 4 (Sca the above QR code for the direct dowload of this material) Name of the Studet: Brach: Uit I (Fourier Series) ) Develop a Fourier series for the fuctio f ( ) ( ) i the iterval(, ) ) Fid the Fourier series for the fuctio f ( ) ( ) i the iterval (, ) ad deduce sum of series as ad also fid ) Fid the Fourier series of periodic π for 5 8 4) Obtai the Fourier epasio of periodicity π for f ( ) i (, ) ad deduce that i (, ) whe Deduce that ad also fid the value of ) Fid the Fourier series of f ( ) i the iterval(, ) also fid the value of Hece deduce that 6) Fid the Fourier series for the fuctio f ( ) i the iterval (, ) ad also deduce the value of 6 si si4 si6 7) Show that whe, / 8) Develop a sie series of the fuctio / 9) Fid the Fourier series of f ( ) cos i the iterval(, ) Prepared by CGaesa, MSc, MPhil, (Ph: ) Page

2 ) Fid the Fourier series of f ( ) cos i the iterval [-π, π] ) Obtai the Fourier series of f( ) period i) ii) ) Fid the Fourier series of f ( ) ad defied as follows ad hece deduce the followig i the iterval(, ) Hece deduce the sum of series ) Fid the Fourier series of f ( ) ( ) i Deduce the sum of the series 4 4) Fid the comple form of the Fourier series of the fuctio f ( ) e ad f ( ) f ( ) a whe 5) Fid the comple form of the Fourier series of the periodic fuctio f ( ) si whe 6) Fid the Fourier Series up to three harmoic for y = f()i (,π) for the followig data π/ π/ π 4π/ 5π/ π y ) Fid the Fourier Series up to two harmoic for y = f()i (,π) for the followig data y ) Fid the Fourier Series up to two harmoic for y = f()i (,π) for the followig data 4 5 y ) The values of ad the correspodig values of f() over a period T are give below Show that 75 7 cos 4 si where T Prepared by CGaesa, MSc, MPhil, (Ph: ) Page

3 T/6 T/ T/ T/ 5T/6 T f() Uit II (Fourier Trasforms) Part I ) Show that the Fourier trasform of a, a is, a si as as cos as s Hece deduce that si t t cos t dt Usig t 4 Parseval s idetity show that si t t cos t dt t 5, ) Fid the Fourier trasform of Hece prove that, si s scos s s cos ds s 6 ) Fid the Fourier trasform of f( ) if i) si t dt ii) t si t t, a Hece deduce that, a dt 4) Fid the Fourier trasform of f( ) if Part II si t t dt ad 4 si t dt t Note: The same problem they may ask deductio, Hece deduce that, a, a, a with same a 5) Fid the Fourier cosie ad sie trasform of the fuctio f ( ) e, a Prepared by CGaesa, MSc, MPhil, (Ph: ) Page

4 6) Evaluate 7) Evaluate 8) Evaluate 9) Evaluate ) Evaluate Part III d a a a b d a b d 4, if a usig Parseval s idetity d, a usig Parseval s idetity, if a, b usig Parseval s idetity d, if a, b usig Parseval s idetity usig trasforms method, ) Fid the Fourier sie trasform of f ( ),, si, a ) Fid the Fourier sie ad cosie trasform of, a ) Fid the Fourier itegral represetatio of f( ) defied as for f ( ) for e for Part IV ) Fid the Fourier cosie ad sie trasform of f ( ) e a e ) Fid the Fourier cosie trasform of e ) Fid the Fourier sie trasform of 4) Fid the Fourier cosie trasform of f ( ) e 5) Fid the Fourier cosie trasform of e of a e a 6) Show that the Fourier sie trasform of e Prepared by CGaesa, MSc, MPhil, (Ph: ) Page 4 a a, a ad hece deduce that sie trasform is self reciprocal

5 7) Fid the Fourier sie ad cosie trasform of e ad Fourier cosie trasform of 8) State ad Prove Covolutio theorem i Fourier trasform Uit III (Partial Differetial Equatios) also fid the Fourier sie trasform of Higher Order Homogeeous ad No Homogeeous PDE ) Solve D 6DD 5D z e sih y y As: 4 y y y z f( y ) f( y 5 ) e e ) Solve 4 cos D DD D z y y ) Solve D 4D D 4DD z 6si( 6 y) As: z f( y) f( y ) f( y ) cos( 6 y) 8 4) Solve D DD z si si y As: z f( y) f( y ) cos( y) cos( y) 6 z z z 5) Solve 7 6 si( ) z y e y y y y As: z f( y ) f( y ) f( y ) cos( y) e 75 6) Solve D DD 6D z y cos As: z f( y ) f( y ) ycos si 7) Solve D DD D D D z si( y) y 8) Solve D D D D z e y 7 9) Solve D DD D D D z cosh( y) ) Solve D DD D D D z cos cos y y ) Solve D DD D 6D D z e Stadard Types ) Form the PDE by elimiatig the arbitrary fuctio from y z, a by cz Prepared by CGaesa, MSc, MPhil, (Ph: ) Page 5

6 ) Solve z p qy pq As: z y ) Solve z p qy p q As: 4) Solve z p qy p q As: 5) Solve z p qy p pq q 6) Solve z y pq As: pq q p Hit: Multiply pq o both side, we get z p qy pq / 7) Solve p y q Hit: Rearrage the above equatio, we get This is of the form f (, p) f ( y, q) p q 4z y y z k (says) y 8) Solve 9 p z q 4 As: z a ay b Hit: This is of the form f ( z, p, q) 9) Solve z p q z ay b a As: cosh Hit: This is of the form f ( z, p, q) ) Solve pq qz As: ay log c az Hit: This is of the form f ( z, p, q) ) Solve y z p yz q z y As: y z yz, ) Solve y z p y z q z y As: y z yz, ) Solve the partial differetial equatio y z p y z q z y 4) Solvemz y p zq y m As: y z y z, m 5) Solvez 4 y p 4 zq y As: y z, y 4 z 6) Solve yz p y z q z y y As:, y yz z y z Prepared by CGaesa, MSc, MPhil, (Ph: ) Page 6

7 Uit IV (Applicatio of PDE) Oe dimesioal Wave equatio ) Problems o ero iitial velocity (Pre work: ) For the followig f( ) a) f ( ) a si (aother ame Siusoid of legth a) b) f ( ) y si c) f ( ) ( ) d) The midpoit of the strig is displaced to a small height h or b is give e) Ay problems with strig of legth l ) Problems o No ero iitial velocity (Pre work: ) For the followig f( ) f) f ( ) a si (aother ame Siusoid of legth a) g) f ( ) v si h) f ( ) ( ) i) c, / v c( ), / j) Ay problems with strig of legth l Oe dimesioal Heat flow equatio ) Problems o zero boudary coditios (Pre work: ) 4) Problems o Oe ed zero ad aother ed o zero boudary coditio adreduced to C (Pre work: 4) 5) Problems o both eds o zero boudary coditio ad reduced to C 6) Problems o both eds o zero boudary coditio ad reduced to o zero temperature Uit V ( Trasform) Problems o trasform ) Fid the trasform of the fuctiocos, si, a cos, a si, cos /, si /, a cos /, a si /, a si, a cos ) State ad Prove fial value theorem i trasform ) State ad Prove Iitial value theorem i trasform Prepared by CGaesa, MSc, MPhil, (Ph: ) Page 7

8 4) Fid the trasform of the followig fuctios (i) a (ii) a! (v) a (vi)! (vii) (viii) 5) Fid the trasform of the fuctio f( ) (i) () (iii) (iv) Problems o Iverse trasform usig Partial fractio method ) Fid the iverse trasform of z z z 5 method ) Fid z z z z z z z by usig partial fractio method ) Fid the iverse trasform of z z 4 4) Fid z z 7z by usig partial fractio method z usig partial fractio usig partial fractio method z z 5) Fid the iverse trasform of usig partial fractio method z z 4 6) Fid z z z by usig residues method Problems o Covolutio Theorem z ) State Covolutio theorem ad use it to evaluate z a( z b) ) Fid ) Fid z usig covolutio theorem ( z a) 8z usig covolutio theorem (z)(4z) Prepared by CGaesa, MSc, MPhil, (Ph: ) Page 8

9 z 4) Usig Covolutio theorem, fid z z z 5) Fid the iverse trasform of, usig Covolutio theorem z a 6) Use Covolutio theorem to fid the iverse trasform Solvig Differece Equatio ) Form the differece equatio from the relatio y a b z 4z ) Usig trasform, solve y 6 9 k k yk yk give y y ) Solve u u u give u, u, usig trasform 4) Solve y( ) y( ) y( ) give that y() 4, y() ad y() 8 5) Solve the system y 5y 6 y u, with y, y ad u for,,, by trasform method Hit: Substitute u i give equatio, we get y 5y 6y ad do as usual method 6) Usig trasform solve y( ) y( ) 4 y( ), give that y() ad y() ----All the Best---- Prepared by CGaesa, MSc, MPhil, (Ph: ) Page 9