: Transforms and Partial Differential Equations

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Trasforms ad Partial Differetial Equatios 018 SUBJECT NAME : Trasforms ad Partial Differetial Equatios SUBJECT CODE : MA 6351 MATERIAL NAME : Part A questios REGULATION : R013 WEBSITE : wwwharigaeshcom UPDATED ON : April-May 018 TEXT BOOK FOR REFERENCE : Sri Harigaesh Publicatios (Author: C Gaesa) To buy the book visit wwwharigaeshcom/textbook Uit I (Partial Differetial Equatio) x a y b z r 1 Form the PDE from Text Book Page No: 14 Fid the PDE of all spheres whose ceters lie o the x-axis 3 Fid the PDE of the family of spheres havig their ceters o the z axis Text Book Page No: 13 4 Form the partial differetial equatio by elimiatig the costats a ad b from z x a y b Text Book Page No: 16 5 Form the partial differetial equatio by elimiatig the arbitrary costats a ad b from z x a y b Text Book Page No: 15 6 Form the PDE by elimiatig the arbitrary costats a ad b from Text Book Page No: 17 7 Form the PDE by elimiatig the arbitrary costats ab, from the relatio 3 3 z ax by Text Book Page No: 15 z ax by Sri Harigaesh Publicatios (Ph: 9841168917, 8939331876) Page 1

Trasforms ad Partial Differetial Equatios 018 8 Form the partial differetial equatio by elimiatig the arbitrary costats a ad b from log( az 1) x ay b Text Book Page No: 16 9 Form the partial differetial equatio by elimiatig the arbitrary fuctios from f x y, z xy 0 10 Fid the partial differetial equatio by elimiatig the arbitrary fuctio ' f ' from the relatio z f x y y 11 Elimiate the arbitrary fuctio f from z f x ad form the PDE Text Book Page No: 114 1 Form the partial differetial equatio by elimiatig the arbitrary fuctio from z xy f z x Text Book Page No: 115 13 Form the partial differetial equatio by elimiatig arbitrary fuctio f from ay z e f x by 14 Fid the partial differetial equatio of all plaes cuttig equal itercepts from the x ad y axes Text Book Page No: 18 15 Fid the complete solutio of pq 1 (Text Book Page No: 18) 16 Fid the complete itegral of p q 1 17 Fid the complete solutio of the PDE p q 0 3 3 18 Fid the complete itegral of p q pq (Text Book Page No: 19) 19 Fid the complete itegral of z x y pq pq q p 0 Fid the complete solutio of q px () Sri Harigaesh Publicatios (Ph: 9841168917, 8939331876) Page

Trasforms ad Partial Differetial Equatios 018 1 Solve the partial differetial equatio pq x (Text Book Page No: 180) Solve the equatio D D 3 z 0 3 3 Solve (Text Book Page No: 118) D D D z 0 (Text Book Page No: 117) 4 Solve D 7DD D z 0 (Text Book Page No: 113) 3 3 5 Solve 4 4 6 Solve D D D 8DD 1D z 0 D D z 0 (Text Book Page No: 119) 7 Fid the particular itegral of Text Book Page No: 1174 x y D DD D z e 8 Solve z z z 0 x xy x (Text Book Page No: 1179) 9 Solve D 1 D D 1 z 0 (Text Book Page No: 1180) Uit II (Fourier Series) 1 State the Dirichlet s coditios for Fourier series (Text Book Page No: 1) Write the coditios for a fuctio f( x) to satisfy for the existece of a Fourier series Text Book Page No: 1 3 State the sufficiet coditio for a fuctio f( x) to be expressed as a Fourier series Text Book Page No: 1 4 Give the expressio for the Fourier Series co-efficiet b for the fuctio defied i, f ( x) xsi x Text Book Page No: 77 5 Fid the value of a 0 i the Fourier series expasio of f ( x) 0, x e i Sri Harigaesh Publicatios (Ph: 9841168917, 8939331876) Page 3

Trasforms ad Partial Differetial Equatios 018 6 If the Fourier series of the fuctio f ( x) x, x with period is give by 7 If x si x si 3x si4x f ( x) si x 3 4, the fid the sum of the series 1 1 1 1 Text Book Page No: 77 3 5 7 ( 1) 1 1 1 4 cosx, deduce that 3 1 3 6 1 8 If x cosx 1 4 i 0 x, the deduce the value of 3 1 1 9 Obtai the first term of the Fourier series for the fuctio f ( x) x, x Text Book Page No: 77 10 Fid the value of b i the Fourier series expasio of x i (,0) f( x) x i ( o, ) 11 Fid the costat term i the expasio of, Text Book Page No: 77 cos x as a Fourier series i the iterval 1 Expad f( x) 1 as a half rage sie series i the iterval 0, 13 Fid the half rage sie series expasio of f( x) 1 i 0, Text Book Page No: 108 14 Defie Root Mea square value of a fuctio f( x) over the iterval ab, Text Book Page No: 93 15 Fid the root mea square value of f ( x) 0, x i Sri Harigaesh Publicatios (Ph: 9841168917, 8939331876) Page 4

Trasforms ad Partial Differetial Equatios 018 Text Book Page No: 95 16 Fid the root mea square value of the fuctio f ( x) Text Book Page No: 108 x i0,l 17 Without fidig the values of a, 0 a ad b, the Fourier coefficiets of Fourier series, for the fuctio a 108 F( x) 0 of a b x i the iterval 0, fid the value Text Book Page No: 1 18 Write the complex form of Fourier series for a fuctio f( x ) defied i x 19 What is meat by Harmoic Aalysis? Text Book Page No: 14 Uit III (Applicatio of Partial Differetial Equatios) 1 Classify the partial differetial 1 x zxx xyz xy 1 y z yy xzx 3x yz y z 0 equatio Text Book Page No: 35 u u Classify the partial differetial equatio 4 x t Text Book Page No: 3 3 Classify the partial differetial equatio u u f ( x, y) 4 Write dow all possible solutios of oe dimesioal wave equatio Text Book Page No: 318 5 Write dow the three possible solutios of oe dimesioal heat equatio Text Book Page No: 350 6 What is the basic differece betwee the solutios of oe dimesioal wave equatio ad oe dimesioal heat equatio with respect to the time? xx xy Sri Harigaesh Publicatios (Ph: 9841168917, 8939331876) Page 5

Trasforms ad Partial Differetial Equatios 018 y 7 I the wave equatio c t Text Book Page No: 315 y, what does x c stad for? 8 I the oe dimesioal heat equatio u t c uxx, what is c? Text Book Page No: 346 9 A tightly stretched strig with fixed ed poits x 0 ad x is iitially i a positio give by y( x,0) si boudary coditios Text Book Page No: 34 3 y0 10 Defie steady state coditio o heat flow Text Book Page No: 355 x If it is released from rest i this positio, write the 11 State the assumptios i derivig the oe-dimesios heat flow equatio 1 A isulated rod of legth cm has its eds A ad B maitaied at 0⁰C ad 80⁰C respectively Fid the steady state solutio of the rod Text Book Page No: 356 13 A rod 40 cm log with isulated sides has its eds A ad B kept at 0⁰C ad 60⁰C respectively Fid the steady state temperature at a locatio 15 cm from A Text Book Page No: 369 14 A rod 30 cm log has its eds A ad B kept at 0⁰C ad 80⁰C respectively util steady state coditio prevail Fid this steady state temperature i the rod Text Book Page No: 369 15 Give three possible solutios of two dimesioal steady state heat flow equatio Text Book Page No: 378 16 Write all three possible solutios of steady state two dimesioal heat equatio Text Book Page No: 378 Sri Harigaesh Publicatios (Ph: 9841168917, 8939331876) Page 6

Trasforms ad Partial Differetial Equatios 018 17 Write dow the partial differetial equatio that represets steady state heat flow i two dimesios ad ame the variables ivolved 18 Write dow the three possible solutios of Laplace equatio i two dimesios Text Book Page No: 378 19 Write dow the two dimesioal heat equatio both i trasiet ad steady states 0 A plate is bouded by the lies x 0, y 0, x l ad y l Its faces are isulated The edge coicidig with x axis is kept at100 C The edge coicidig with y axis is kept at 50 C The other two edges are kept at 0 C Write the boudary coditios that are eeded for solvig two dimesioal heat flow equatio Uit IV (Fourier Trasform) 1 State Fourier itegral theorem (Text Book Page No: 41) Write the Fourier trasform pair (Text Book Page No: 45) 3 Write the Fourier cosie trasform pair (Text Book Page No: 415) 4 Write the Fourier sie trasform pair (Text Book Page No: 415) 5 Fid the Fourier trasform of a derivative of the fuctio f( x ) if f( x) 0 as x x 6 Fid the Fourier trasform of e, 0 (Text Book Page No: 418) ax 7 Fid the Fourier cosie trasform of e, x 0 (Text Book Page No: 446) ax 8 Fid the Fourier sie trasform of f ( x) e, a 0 (Text Book Page No: 446) 3 x 9 Fid the Fourier sie trasform of e (Text Book Page No: 467) 10 Fid the Fourier sie trasform of 1 (Text Book Page No: 457) x Sri Harigaesh Publicatios (Ph: 9841168917, 8939331876) Page 7

Trasforms ad Partial Differetial Equatios 018 11 Fid the Fourier trasform of ikx e, a x b f( x) 0, x a ad x b Text Book Page No: 417 1 Defie self-reciprocal with respect to Fourier Trasform Text Book Page No: 415 13 State Parseval s idetity o Fourier trasform (Text Book Page No: 413) 14 State covolutio theorem i Fourier trasform (Text Book Page No: 41) 15 State ad prove the chage of scale property of Fourier trasform Text Book Page No: 47 16 State ad prove modulatio theorem o Fourier trasform Text Book Page No: 48 17 If Fs () is the Fourier trasform of ( ) Text Book Page No: 46 ias F f ( x a) e F( s) f x, show that 18 What is the Fourier trasform of f ( x a), if the Fourier trasform of f( x) is Fs ()? Text Book Page No: 46 19 If F f ( x) F( s), prove that ( ) Text Book Page No: 47 F f ( x) F( s) 0 If 1 s F f ax F a a iax, the fid ( ) F e f x (Text Book Page No: 48) 1 If Fc () s is the Fourier cosie trasform of f( x ), prove that the Fourier cosie trasform of f ( ax) is 1 F a c s a (Text Book Page No: 443) Sri Harigaesh Publicatios (Ph: 9841168917, 8939331876) Page 8

Trasforms ad Partial Differetial Equatios 018 Uit V (Z Trasform) 1 Defie the uit step sequece Write its Z trasform Text Book Page No: 5; 59 Z f ( ) F( z) If z Z a f F a, the show that ( ) Text Book Page No: 54 3 Fid the Z trasform of si Text Book Page No: 519 z 4 If F( z) 1 1 3 z z z 4 4 Text Book Page No: 534, fid f (0) a for 0 5 Fid the Z trasform of x ( )! 0 otherwise Text Book Page No: 517 6 Fid the Z trasform of a (Text Book Page No: 51) 7 Fid the Z trasform of (Text Book Page No: 513) 8 Fid the Z trasform of (Text Book Page No: 514) 9 Fid the Z trasform of 1 (Text Book Page No: 514) 10 Fid the Z trasform of 1 (Text Book Page No: 514)! 11 Fid Z cos isi 1 State iitial value theorem o Z trasforms (Text Book Page No: 53) 13 State the covolutio theorem o Z trasforms (Text Book Page No: 573) Sri Harigaesh Publicatios (Ph: 9841168917, 8939331876) Page 9

Trasforms ad Partial Differetial Equatios 018 14 Fid the iverse Z-trasform of z z 1 (Text Book Page No: 560) 1 z 15 Obtai Z (Text Book Page No: 559) z1 z 16 What advatage is gaied whe Z trasform is used to solve differece equatio? Text Book Page No: --- 1 17 Form a differece equatio by elimiatig arbitrary costats from U A Text Book Page No: 5111 18 Form a differece equatio by elimiatig the arbitrary costat A from y A3 Text Book Page No: 58 19 Form the differece equatio by elimiatig arbitrary costat ' a ' form y a 0 Fid the differece equatio geerated by y a b Text Book Page No: 5111 1 Solve y 1 y 0 give y0 3 (Text Book Page No: 5111) Textbook for Referece: TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS Editio: 3 rd Editio Publicatio : Harigaesh Publicatios Author : C Gaesa To buy the book visit wwwharigaeshcom/textbook ----All the Best---- Sri Harigaesh Publicatios (Ph: 9841168917, 8939331876) Page 10

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