UNIT I PARTIAL DIFFERENTIAL EQUATIONS PART B. 3) Form the partial differential equation by eliminating the arbitrary functions

Size: px

Start display at page:

Download "UNIT I PARTIAL DIFFERENTIAL EQUATIONS PART B. 3) Form the partial differential equation by eliminating the arbitrary functions"

Ursula George
6 years ago
Views:

1 UNIT I PARTIAL DIFFERENTIAL EQUATIONS PART B 1) Form th artial diffrntial quation b liminating th arbitrar functions f and g in z f ( x ) g( x ) ) Form th artial diffrntial quation b liminating th arbitrar functions f and g in z x f ( ) g( x) ) Form th artial diffrntial quation b liminating th arbitrar functions f and from z f ( ) ( x z) 4) Find th singular solution of z x q q 16 5) Solv z x q 1 q 6) Find th singular intgral of th artial diffrntial quation z x q q 7) Solv z 1 q 8) Solv (1 q ) q(1 z) 9) Solv 9( z q ) 4 10) Solv ( 1 q) qz 11) Solv ( xz) ( z x) q ( x )( x ) 1) Solv ( z) (x ) q x z 1) Find th gnral solution of ( z 4) (4x z) q x 14) Solv x ( z) ( z x) q z( x ) 15) Solv xq x( z ) 16) Solv ( x z) ( zx) q z x 17) Solv ( x ) z ( x ) zq x 18) Solv ( x z) (z ) q x 19) Find th gnral solution of z ( x ) x q 0) Solv x ( z ) ( z x ) q z( x ) 1) Solv ( D DD 0 D ) z 5x sin(4x ) ) Solv ( D 4DD 5D ) z x sin( x ) ) Solv ( D D D DD D ) z x cos( x ) 4) Solv ( D DD 0 D ) z x 6x 5) Solv ( D DD 6D ) z x x 6) Solv z z z x x sinh( x ) x 7) Solv z z z 6 x x cosx 8) Solv ( D 5DD 6D ) z sin x 9) Solv ( D D DD D D 1) z x 0) Solv ( D D D D ) z x 7

2 PART-B 1) Find th Fourir sris of riod for th function hnc find th sum of th sris 1 5 x;(0, ) ) Obtain th Fourir sris for f ( x) x;(, ) 1;(0, ) f (x) and ;(, ) sin x;0 x ) Exand f ( x) as a Fourir sris of riodicit and 0; x hnc valuat ) Dtrmin th Fourir sris for th function f ( x) x of riod in 0 x 5) Obtain th Fourir sris for f ( x) 1 x x in (, ) Dduc that UNIT II FOURIER SERIES 6 6) Exand th function f ( x) xsin x as a Fourir sris in th intrval x 7) Dtrmin th Fourir xansion of f ( x) x in th intrval x 8) Find th Fourir sris for f ( x) cosx in th intrval (, ) 9) Exand f ( x) x x as Fourir sris in (, ) 1 x, x 0 10) Dtrmin th Fourir sris for th function f ( x) 1 x,0 x 1 1 Hnc dduc that ) Find th half rang sin sris of f ( x) xcosx in ( 0, ) 1) Find th half rang cosin sris of f ( x) xsin x in ( 0, ) 1) Obtain th half rang cosin sris for f ( x) x in ( 0, ) 14) Find th half rang sin sris for f ( x) x( x) in th intrval ( 0, ) 15) Find th half rang sin sris of f ( x) x in 0, ) ( Hnc find f (x)

3 18 Writ down th aroriat solutions of th two dimnsional hat Equations 19 In two dimnsional hat flow, what is th tmratur along th Normal to th x- lan? 0 If a squar lat has its facs and th dg = 0 insulatd, its dgs x = 0 and x = n ar kt at zro tmratur and its fourth dg is kt at tmratur u, thn what ar th boundar conditions for this roblm? PART B 1 A tightl strtchd string with fixd nd oints x = 0 and x = l is initiall in a osition givn b = 0 sin ( π x / l ) If it is rlasd from rst from this osition, find th dislacmnt ( x, t) A tightl strtchd string of lngth l has its nds fastnd at x = 0, x = l Th mid-oint of th string is thn takn to hight h and thn rlasd from rst in that osition Find th latral dislacmnt of a oint of th string at tim t from th instant of rlas A tightl strtchd string with fixd nd oints x = 0 and x = l At tim t = 0, th string is givn a sha dfind b F(x) = μ x ( l - x ), whr μ is constant, and thn rlasd Find th dislacmnt of an oint x of th string at an tim t >0 4 Th oints of trisction of a string ar ulld asid through th sam distanc on oosit sids of th osition of quilibrium and th string is rlasd from rst Driv an xrssion for th dislacmnt of th string at subsqunt tim and show that th mid-oint of th string alwas rmains at rst 5 A tightl strtchd string of lngth l with fixd nds is initiall in quilibrium osition It is st vibrating b giving ach oint a vlocit v 0 sin ( π x / l ) Find th dislacmnt (x,t) 6 A tightl strtchd string with fixd nd oints x = 0 and x = l is initiall at rst in its quilibrium osition It is st vibrating b giving ach oint a vlocit λ x ( l - x ), find th dislacmnt of th string at an distanc x from on nd at an tim t

4 7 A taut string of lngth 0cmsfastnd at both nds is dislacd from its osition of quilibrium, b imarting to ach of its oints an initial vlocit givn b: v = x in 0 < x < 10 and, x bing th 0 x in 10 < x <0 distanc from on nd Dtrmin th dislacmnt at an subsqunt tim 8 An insulatd rod of lngth l has its nds A and B maintaind at 0 0 C and c rsctivl until stad stat conditions rvail If B is suddnl rducd to 0 0 C and maintaind at 0 0 C, find th tmratur at a distanc x from A at tim t 9 A homognous rod of conducting matrial of lngth 100cm has its nds kt at zro tmratur and th tmratur initiall is u(x,0) = x in 0 < x < x in 50 < x <100 Find th tmratur u(x,t) at an tim 0 An insulatd rod of lngth l has its nds A and B maintaind at 0 0 C and c rsctivl until stad stat conditions rvail If th chang consists of raising th tmratur of A to 0 0 c and rducing that of B to 80 0 c, find th tmratur at a distanc x from A at tim t 1 Th nds A and B of a rod 0cmlong hav th tmratur at 0 0 c and 80 0 c until stad stat rvails Th tmratur of th nds ar changd to 40 0 c and 60 0 c rsctivl Find th tmratur distribution in th rod at tim t Th nds A and B of a rod 10cmlong hav th tmratur at 50 0 c and c until stad stat rvails Th tmratur of th nds ar changd to 90 0 c and 60 0 c rsctivl Find th tmratur distribution in th rod at tim t A squar lat is boundd b th lins x =0, = 0, x =0 and = 0 Its facs ar insulatd Th tmratur along th ur horizontal dg Is givn b u ( x,0) = x ( 0 x) whn 0 < x < 0 whil th othr thr dgs ar kt at 0 0 c Find th stad stat tmratur in th lat 4 Find th stad stat tmratur at an oint of a squar lat whos two adjacnt dgs ar kt at 0 0 c and th othr two dgs ar kt at th constant tmratur c

5 5 Find th stad tmratur distribution at oints in a rctangular lat With insulatd facs th dgs of th lat bing th lins x = 0, x = a, = 0 and = b Whn thr of th dgs ar kt at tmratur zro and th fourth at fixd tmratur a 0 c 6 Solv th BVP U xx + U = 0, 0 < x, < π with u( 0, ) = u(π, ) = u( x, π) = 0 and u(x,0) = sin x 7 A rctangular lat is boundd b th lins x = 0, = 0, x = a, = b Its Surfacs ar insulatd Th tmratur along x = 0 and = 0 ar kt at 0 0 c and th othrs at c Find th stad stat tmratur at an oint of th lat 8 A long rctangular lat has its surfacs insulatd and th two long sids as wll as on of th short sids ar maintaind at 0 0 c Find an xrssion for th stad stat tmratur u(x,) if th short sid = 0 is π cm long and is kt at u 0 o c 9 An infinitl long rctangular lat with insulatd surfac is 10cm wid Th two long dgs and on short dg ar kt at zro tmratur whil th othr short dg x = 0 is kt at tmratur givn b U = 0 for 0 < < 5 0 ( 10 ) for 5 < < 10 Find th stad stat tmratur distribution in th lat 40 An infinitl long lan uniform lat is boundd b two aralll dgs and an nd at right angl to thm Th bradth of this dg x =0 is π, this nd is maintaind at tmratur as u = k (π ) at all oints whil th othr dgs ar at zro tmratur Dtrmin th tmratur u(x,) at an oint of th lat in th stad stat if u satisfis Lalac quation

6 Unit IV FOURIER TRANSFORMS PART B 1 A function f(x) is dfind as f(x) = 1 if x < 1 0, othrwis Using Fourir intgral rrsntation of f(x) Hnc valuat [sin s/ s] cos sx ds in 0 < x< A function f(x) is dfind as f(x) = 1 if x < 1 0, othrwis Using Fourir cosin intgral rrsntation of f(x) Hnc valuat [sin s/ s] cos sx ds in 0 < x < Find th FT of f(x) is dfind as f(x) = 1 if x < 1 0, othrwis Hnc valuat (i) [sin / ] d (ii) [sin / ] d in (0, ) 4 Find th FT of f(x) is dfind as f(x) = a - x if x < a 0, othrwis Hnc valuat (i) [sin / ] d in (0, ) (ii) [sin / ] 4 d in (0, ) 5 Find th FT of f(x) is dfind as f(x) = 1 x if x < 1 0, othrwis Hnc valuat (i) [sin t t cos t/ t ] dt in (0, ) (ii) [x cos x - sin x / x ] cos (x /)dx in (0, ) 6 Find th FT of f(x) is dfind as f(x) = -ax,a >0 Hnc ST -x / is slf rcirocal undr FT 7 Find th FT of - x and hnc find th FT of - x cos x 8 Obtain th FST of f(x) = x if 0 <x < 1 x if 1<x< 0, othrwis 9 Find th FCT of f(x) is dfind as f(x) = cos x if 0 <x < a 0, othrwis 10 Stat and Prov Parsval s Idntit 11 Find th FST and FCT of x n-1, whr 0 < n< 1, x >0 Dduc that 1/ x is slf-

7 rcirocal undr both FST and FCT 1 Find th FST of -ax / x Hnc find FST of 1 / x 1 Evaluat [dx / (a + x ) (b + x ) ] dx in (0, ) 14 Find F c {f (x)} 15 Solv th intgral quation [f(x) cos x] dx in (0, ) and also [cos x / ( 1 + )] d in (0, )

8 Z TRANSFORM PART B 1 Find Z [ a n cos nθ ] and Z [a n sin nθ ] Find Z [a n n ] Find Z [ cos nπ/ ] and Z [ sin nπ/ ] 4 Find th Z transforms of th following (i) an (ii) n an 5 Find th Z transform of (i) cosh nθ (ii) a n cosh nθ 6 Find Z [ cos ( nπ/ + π/4 ) ] 7 Find th Z transform of (i) n c (ii) n+ c 8 Find th Z transform of unit imuls squnc and unit st squnc 9 Find th Z transform of (i) sinh nθ (ii) a n sinh nθ 10 Find Z [ t sint ] and Z [ -t sint ] 11 Find th invrs Z transform of z / ( z + 1 ) b division mthod 1 Find th invrs Z transform of { z + z } / ( z + ) ( z 4 ) b artial fractions mthod 1 Find th invrs Z transform of ( z 0 z ) / ( z ) ( z 4 ) b artial fraction mthod 14 Find th invrs Z transform of 10 z / ( z-1) ( z-) b invrsion intgral mthod 15 Find th invrs Z transform of z / ( z -1 ) ( z - i ) ( z + I ) b invrsi on intrgral mthod 16 Using convolution thorm, valuat th invrs Z transform of z / ( z -a ) ( z - b ) 17 Using convolution thorm, valuat th invrs Z transform of z / ( z a) 18 Show that ( 1/ n! ) * (1/ n! ) = n / n! 19 Solv n+ + 6 n n = n with 0 = 1 = 0, using Z transform 0 Solv n+ - n+1 + n = n + 5

10. The Discrete-Time Fourier Transform (DTFT)

10. The Discrete-Time Fourier Transform (DTFT) Th Discrt-Tim Fourir Transform (DTFT Dfinition of th discrt-tim Fourir transform Th Fourir rprsntation of signals plays an important rol in both continuous and discrt signal procssing In this sction w