VALLIAMMAI ENGINEERING COLLEGE SRM Nagar, Kattankulathur DEPARTMENT OF MATHEMATICS QUESTION BANK

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1 SUBJECT VALLIAMMAI ENGINEERING COLLEGE SRM Nagar, Kattankulathur DEPARTMENT OF MATHEMATICS QUESTION BANK : MA6351- TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS SEM / YEAR : III Sem / II year (COMMON TO ALL BRANCHES) UNIT I - PARTIAL DIFFERENTIAL EQUATIONS Formation of partial differential equations Singular integrals -- Solutions of standard types of first order partial differential equations - Lagrange s linear equation -- Linear partial differential equations of second and higher order with constant coefficients of both homogeneous and non-homogeneous types. Bloom s Q.No. Question Taonomy Level Domain PART A 1. Form a partial differential equation by eliminating the. arbitrary constants a and b from z a by Eliminate the arbitrary function from = and form the partial differential equation Construct the partial differential equation of all spheres whose centers lie on the z-ais by the elimination of arbitrary constants Form the partial differential equation by eliminating the arbitrary constants a, b from the relation = + + Form the partial differential equation by eliminating the arbitrary constants a, b from the relation log( az 1) ay b. Form the partial differential equation by eliminating the arbitrary constants a, b from the relation 4(1 a ) z ( ay b). Form the partial differential equation from ( a) ( y b) z cot Form the partial differential equation by eliminating the arbitrary function from ( y, z) BTL- 6 MA6351_TPDE_QBank_ACY 16-17(ODD) COMMON TO ALL BRANCHES OF III SEMESTER B.E., B.TECH. Form the partial differential equation by eliminating the 9. arbitrary function from z y f ( ) z 1. Find the complete solution of p q Find the complete solution of q p

2 1. Find the complete integral of p q pq 13. Solve p qy z 14. Solve ( D 7DD' 6D' ) z Solve ( D 4D D' 4DD' ) z z z z 16. Solve y Solve ( D D' ) z 18. Solve ( D D' 1)( D D' 3) z Solve ( D D') z. Solve ( D 1)( D D' 1) z PART B Form the partial differential equation by eliminating the 1.(a) arbitrary functions f 1, f from z f 1( t) f ( t) 1. (b) Find the singular integral of. (a) z Form the partial differential equation by eliminating arbitrary function from ( y z, a by cz) MA6351_TPDE_QBank_ACY 16-17(ODD) COMMON TO ALL BRANCHES OF III SEMESTER B.E., B.TECH. p qy 1 p q BTL -.(b) Find the singular integral of z p qy p pq q BTL - 3. (a) Form the partial differential equation by eliminating arbitrary functions f and g from z f ( y) y g( ) p q 3.(b) Solve ( ) ( y) 1 4. (a) Solve p y q z( y) Form the partial differential equation by eliminating arbitrary function f from the relation 4.(b) 1 z y f ( log y) 5. (a) Solve ( ) ( ) ( yz p y z q z y ) 5.(b) Solve 9( p z q ) 4 p 6. (a) Find the singular solution of z p qy p q BTL - 6.(b) Solve y z ) p yq z 7. (a) Find the complete solution of z ( p q ) y BTL -4 Analyzing

3 7. (b) Find the general solution of ( D DD' D' ) z cos y sin y BTL - MA6351_TPDE_QBank_ACY 16-17(ODD) COMMON TO ALL BRANCHES OF III SEMESTER B.E., B.TECH. 8. (a) Find the general solution of ( D D' ) z y BTL - 8.(b) Find the complete solution of p y q z BTL - 9. (a) Solve y ( D 3DD' D' ) z ( 4) e 9.(b) Find the general solution of ( z y yz) p ( y z) q ( y z) BTL - 1.(a) Solve ( ) ( ) ( y z p y z q z y ) 1.(b) Solve ( 3 ' ' D DD D ) z sin( 5 y ) 11(a) Solve the Lagrange s equation ( z) p (z y) q y 11(b) Solve 1(a) Solve ( D ( D 4 y DD' D' ) z 3y e DD' 6D' ) z ycos Solve the partial differential equation 1(b) ( z) p (z y) q y y 13(a) Solve ( D D' ) z e sin( 3y) 13(b) Solve 14(a) Solve (D DD' D' 6D 3D' ) z e y 3 y ( D DD') z y e (b) Solve ( D 7DD' 6D' ) z sin( y) UNIT II - FOURIER SERIES Dirichlet s conditions General Fourier series Odd and even functions Half range sine series Half range cosine series Comple form of Fourier series Parseval s identity Harmonic analysis. PART-A Q.No. Question Bloom s Taonomy Level Domain 1. State the sufficient condition for a function to be epanded BTL-1 as a Fourier series State the Dirichlet s condition for Fourier series. BTL-1 3 If = + = < < then deduce that value of =. 4 Does f() = tan posses a Fourier epansion? BTL-

4 Determine the value of in the Fourier series epansion of =,. BTL-5 Evaluating Find the constant term in the epansion of as a Fourier series in the interval (-, BTL- If f() is an odd function defined in (-, ) what are the value of? BTL- If the function f() = in the interval << then find the constant term of the Fourier series epansion of the function. BTL- If the Fourier series of the function f() =, < <. with Period is given by f() = [ + BTL- + ]then find the sum of the series Find the sine series of function f() = Find the RMS value of f() = in (, ). 1 Find the root mean square value of f() = in (,). BTL- BTL- BTL- MA6351_TPDE_QBank_ACY 16-17(ODD) COMMON TO ALL BRANCHES OF III SEMESTER B.E., B.TECH. 13 Find the RMS value of f() = ( -) in BTL- 14 Find the RMS value of f() = in (, ) BTL- 15 Write down Parseval s formula on Fourier coefficients BTL-1 16 Define RMS value of a function f() over the interval (a,b). BTL-1 17 Without finding the value of and the Fourier series, for the function f() = in the Interval (, find the value of { = + } BTL- 18 Find the R.M.S value of f() = 1- in < < BTL- 19 State Parseval s identity for the half-range cosine epansion of f() in (,1). BTL-1 What is meant by Harmonic Analysis? 1.(a) Find the Fourier series epansion of = {. PART-B 1. (b) Find the Fourier series of = < <.Hence deduce the value of. (a).(b) = Obtain the Fourier series to represent the function = < < and Deduce = =. Find the Fourier series of the function = { and Hence Evaluate BTL- BTL- BTL-

5 3. (a) +, < < Epand = { as a full range, < < Fourier series in the interval,.hence deduce that = Find a Fourier series with period 3 to represent 3.(b) =, Determine the Fourier series for the function 4. (a) = < < Obtain the Fourier series for the function f() given by, < < 4.(b) = { Hence deduce that +, < < = Determine the Fourier series for the function 5. (a) = < < BTL- BTL-5 BTL-5 Evaluating Evaluating 5.(b) Obtain the sine series for = { Find the Fourier epansion the following periodic function of period (a) +, = { Hence deduce that, BTL- + = Find the Fourier series of = + in (-, with 6.(b) period. Hence deduce BTL- = = 7. (a) Find the half range sine series of = in the interval (,4).Hence deduce the value of the series + + Find the Fourier series of = < < of 7. (b) periodicity BTL- BTL- 8. (a) 8.(b) Compute the first two harmonics of the Fourier series of f() from the table given / / / / f() Find the comple form of the Fourier series of in, where a is not an integer BTL- 9. (a) Find the Fourier series of =, and hence deduce that = BTL- MA6351_TPDE_QBank_ACY 16-17(ODD) COMMON TO ALL BRANCHES OF III SEMESTER B.E., B.TECH.

6 9.(b) The following table gives the variation of a periodic current over a period by harmonic analysis, Show that there is a direct current part of.75amps in the variable current. Also obtain the amplitude of the first harmonic. t secs T/6 T/3 T/ T/3 5T/6 T A amps Find the half range Fourier cosine series of = 1.(a) in the interval,. Hence Find the sum of the series BTL (b) 11.(a) Obtain the Fourier cosine series epansion of = < <.Hence deduce the value of By using Cosine series show that = for = < < Find the Fourier cosine series up to first harmonic to represent the function given by the following data: 11.(b) y Show that the comple form of Fourier series for the function = h < < 1.(a) = + = h + = + Find the comple form of the Fourier series of 1.(b) = < < 13. Calculate the first 3 harmonics of the Fourier of f() from the following data f() Find the comple form of the Fourier series of 14.(a) = < < 14.(b) Find the Fourier series as far as the second harmonic to represent the function f() With period y BTL- BTL- BTL- BTL- MA6351_TPDE_QBank_ACY 16-17(ODD) COMMON TO ALL BRANCHES OF III SEMESTER B.E., B.TECH. UNIT III -APPLICATIONS OF PARTIAL DIFFERENTIAL EQUATIONS Classification of PDE Method of separation of variables - Solutions of one dimensional wave equation - One dimensional equation of heat conduction Steady state solution of two

7 dimensional equation of heat conduction (ecluding insulated edges). Bloom s Q.No. Question Taonomy Level PART - A 1. Classify the PDE = BTL-4 MA6351_TPDE_QBank_ACY 16-17(ODD) COMMON TO ALL BRANCHES OF III SEMESTER B.E., B.TECH. Domain Analyzing. Classify the PDE = BTL-4 Analyzing 3. Solve 4. What are the various solutions of one dimensional wave BTL-1 equation 5. y y In the wave equation c what does C stand t for? BTL- What is the basic difference between the solutions of one 6. dimensional wave equation and one dimensional heat equation with respect to the time? Write down the initial conditions when a taut string of length 7. is fastened on both ends. The midpoint of the string is taken to a height b and released from the rest in that position BTL-1 8. A slightly stretched string of length l has its ends fastened at = and = l is initially in a position given by, =.If it is released from rest from this position, write the boundary conditions BTL- 9. A tightly stretched string with end points = & = is initially at rest in equilibrium position. If it is set vibrating giving each point velocity. Write the initial and boundary conditions BTL- If the ends of a string of length are fied at both sides. The 1. midpoint of the string is drawn aside through a height h and BTL- the string is released from rest, state the initial and boundary conditions 11. State the assumptions in deriving the one dimensional equation BTL-1 1. Write down the various possible solutions of one dimensional heat flow equation? BTL u u In the one dimensional heat equation C t what is C? BTL- The ends A and B of a rod of length cm long have their 14. temperature kept 3 C and 8 C until steady state prevails. Find the steady state temperature on the rod BTL- 15. An insulated rod of length 6 cm has its ends at A and B maintained at C and 8 C respectively. Find the steady BTL-

8 state solution of the rod. An insulated rod of length cm has its ends at A and B maintained at C and 8 C respectively. Find the steady state solution of the rod. Write down the governing equation of two dimensional steady state heat equation. Write down the three possible solutions of Laplace equation in two dimensions A plate is bounded by the lines =, y=, = and y=. Its faces are insulated. The edge coinciding with -ais is kept at 1 C. The edge coinciding with y-ais at 5 C. The other edges are kept at C. write the boundary conditions that are needed for solving two dimensional heat flow equation BTL- BTL-1 BTL-1 BTL-. A rectangular plate with insulated surface is 1cm wide. The two long edges and one short edge are kept at C, while the temperature at short edge = is given by, = { Find the steady state, temperature at any point in the plate BTL PART-B A string is stretched and fastened to two points that are distinct stringl apart. Motion is started by displacing the string into the form = from which it is released at time t=. Find the displacement of any point on the string at a distance of from one end at time t. A tightly stretched string of length l is fastened at both ends. The Midpoint of the string is displaced by a distance transversely and the string is released from rest in this position. Find an epression for the transverse displacement of the string at any time during the subsequent motion. A slightly stretched string of length l has its ends fastened at = and = l is initially in a position given by, =. If it is released from rest from this position, find the displacement at any distance from one end and at any time. A tightly stretched string with fied end points = and = l is initially at rest in its equilibrium position. If it is set vibrating string giving each point a velocity. Find the displacement of the string at any distance from one end at any time. A tightly stretched string with fied end points = and =l is initially at rest in its equilibrium position. If it is vibrating string by giving to each of its points a velocity BTL- BTL- BTL- BTL- MA6351_TPDE_QBank_ACY 16-17(ODD) COMMON TO ALL BRANCHES OF III SEMESTER B.E., B.TECH.

9 k l if l v. Find the displacement of the string k( l ) l if l l at any distance from one end at any time t. A tightly stretched string of length is initially at rest in this equilibrium position and each of its points is given the BTL- l 3 velocity v u u Solve C t subject to the conditions (i) u(,t)= for all t (ii), = l if (iii) u (,). l l if l A rod 3 cm long has its ends A and B kept at and 8 respectively until steady state conditions prevail. The temperature at each end is then suddenly reduced to C and BTL- kept so. Find the resulting temperature function u(, t) taking = at A. A bar 1 cm long with insulated sides has its ends A and B maintained at temperature 5 C and 1 C respectively. Until steady state conditions prevails. The temperature at A is suddenly raised to 9 C and at the same time lowered to 6 C at B. Find the temperature distributed in the bar at time t. A square plate is bounded by the lines =, =, = =. Its faces are insulated. The temperature along the upper horizontal edge is given by, = when < < while the other three edges are kept at C. Find the steady state temperature in the plate. A square metal plate is bounded by the lines =, =a, y=, y=a.the edges =a, y=, y=a are kept at zero degree temperature while the temperature at the edge = is. Find the steady state temperature distribution at in the plate. A rectangular plate with insulated surface is 1 cm wide and so long compared to its width that it may be considered infinite in length without introducing appreciable error. The temperature at short edge y= is given by, = { and all the other three, edges are kept at C. Find the steady state temperature at BTL- BTL- BTL- BTL- MA6351_TPDE_QBank_ACY 16-17(ODD) COMMON TO ALL BRANCHES OF III SEMESTER B.E., B.TECH.

10 any point in the plate. 13. An infinitely long rectangular plate with insulated surface is 1cm wide. The two long edges and one short edge are kept at C, while the other short edge = is kept at temperature, = {. Find the steady state, temperature distribution in the plate. BTL- 14. A long rectangular plate with insulated surface is cm. If the temperature along one short edge y= is u(,)=k(l- ) degrees, for <<l, while the other edges = and =l as well as the other short edge are kept at C, find the steady state temperature function u(,y). BTL- UNIT -IV FOURIER TRANSFORM Statement of Fourier integral theorem Fourier transform pair Fourier sine and cosine transforms Properties Transforms of simple functions Convolution theorem Parseval s identity. Q. No. Question Bloom s Domain Taonomy Level PART-A 1. State Fourier integral Theorem BTL -1. Write Fourier Transform in Pairs BTL If F s F denote the Fourier Transform of f, Prove that 1 a s a f a F, a. is f then showthat F f a If the Fourier Transform of f F s F ias e F( )., s MA6351_TPDE_QBank_ACY 16-17(ODD) COMMON TO ALL BRANCHES OF III SEMESTER B.E., B.TECH. a 5. Find the Fourier Transform of e. BTL - 6. Find the Fourier Transform of i k e, if a b f, if a & b. BTL - 7. State and Prove Modulation theorem on Fourier Transforms BTL - 8. ia If F s F f, then find F e f. BTL - 9. Define self-reciprocal with respect to Fourier Transform BTL Find the Fourier sine Transform of. BTL -

11 11. Find the Fourier sine Transform of f ( ) e. BTL - 1. Give an eample of a function which is self- reciprocal under Fourier Sine & Cosine Transform 13. Write down the Fourier cosine Transform pair of formulae BTL If F(s) is the Fourier Transform of f. Show that the Fourier Transform of e f ( ) is F( s a). Show that the Fourier Transform of the derivatives of a n 15. d n function F f ( ) is F( s). n d n 16. If F s F f, then find F f. BTL Find the Fourier cosine Transform of e. BTL - Let s F c be the Fourier cosine Transform of f. Prove that Fc f ( )cos a Fc s a Fc s a. 19. State Convolution theorem in Fourier Transform BTL -1. State Parseval s Identity on Fourier Transform BTL -1 PART-B Find the Fourier Transform of 1, a f and, a 1.(a) sin t hence evaluate dt. Also using Parseval s t Identity Prove that sin t dt t Find the Fourier Cosine Transform of the function a b 1. (b) e e f, Find the Fourier Transform of the function 1, if 1 f() =. (a) Hence deduce that, if 1 ( i ) sin t t dt ( ii) 4 sin t dt. t 3.(b) Show that the function e is self-reciprocal under the Fourier Transform by finding the Fourier Transform of e a, a BTL - BTL - BTL - MA6351_TPDE_QBank_ACY 16-17(ODD) COMMON TO ALL BRANCHES OF III SEMESTER B.E., B.TECH.

12 3.(a) Show that the Fourier Transform of a, if a 1 cos as f is. Hence, if a s sin t deduce that dt. t Prove that 3.(b) d d Fc f ( ) Fs f and Fs f ( ) Fc f ds ds Find the Fourier Transform of 1, if 1 f ( ), if 1 4. (a) Hence Show that (i) sin s s cos s 3 s ds. 15 sin s s cos s s 3 cos 3 ds, (ii) s 16 BTL -1, 1 4.(b) Find the Fourier Sine Transform of f, 1 BTL -, Show that the Fourier transform of a, a 5. (a) f ( ) is sin as as cos as. Hence 3, a s 5.(b) deduce that sin t t cos t. 3 dt t 4 a Find the Fourier sine transform of e (a>). Hence find a F s e a e and Fs hence deduce the value of sin s d s 6. (a) Find the Fourier Transform of. 6.(b) 7. (a) 1 f BTL - MA6351_TPDE_QBank_ACY 16-17(ODD) COMMON TO ALL BRANCHES OF III SEMESTER B.E., B.TECH. Using Parseval s Identity evaluate the following integrals. d d i, ii. where a BTL -5 Evaluating a a n 1 Find the Fourier Cosine and Sine Transforms of. and 1 BTL - hence show that is self reciprocal with respect to both. 7. (b) Evaluate a b d, using Fourier Transform BTL -5 Evaluating

13 8. (a) Find the function whose Fourier Sine Transform is e a s BTL -1, a s 8.(b) State and Prove Convolution Theorem on Fourier Transform 9. (a) Find the Fourier Transform of e and hence find the BTL - Fourier Transform of f e cos. Find the Fourier Cosine transform of e 4. Deduce 9.(b) cos 8 sin 8 that BTL - d e and d e 16 Find the Fourier Transform of and hence deduce that cos t a ( i) 1.(a) dt a e as ( ii) F e = a t a i, a s here F stands for Fourier Transform. a e BTL -4 Analyzing 1.( b) Using Fourier Sine Transform prove that ( d a )( b ) ( a b) Find the Fourier cosine & sine Transform of e. Hence evaluate 11.(a) 1 i d and ii d (b) Find a F e and F e a. s c UNIT -V Z - TRANSFORMS AND DIFFERENCE EQUATIONS MA6351_TPDE_QBank_ACY 16-17(ODD) COMMON TO ALL BRANCHES OF III SEMESTER B.E., B.TECH. BTL - BTL - Find the Fourier Sine Transform of the function 1.(a) sin, a BTL - f, a Find the Fourier Cosine Transform of f e e a and 1.(b) hence find the Fourier Cosine Transform of and the Fourier Sine Transform of e. 13.(a) Find the Fourier sine transform of and + Fourier cosine transform of. BTL (b) Using Parseval s Identity evaluate d 5 9. Verify the Convolution Theorem for Fourier Transform if 14.(a) f g( ) e BTL - BTL -5 BTL (b) Derive the Parseval s Identity for Fourier Transform Evaluating Evaluating

14 Z- Transforms - Elementary properties Inverse Z - transform (using partial fraction and residues) Convolution theorem - Formation of difference equations Solution of difference equations using Z - transform. Q.No. Question Bloom s Domain Taonomy Level PART-A f n. BTL Define Z Transform of the sequence. Find the Z Transform of BTL - n a 3. Find Z n! BTL Find Z n! BTL - 5. Find 1 Z BTL - n 1 6. Define the unit step sequence.write its z-transform BTL State initial value theorem and final value theorem BTL Find Z n. BTL - 9. Find inverse Z transform of BTL - 1. Find 1 Z BTL - n 1! 11. Find Z e t sin t. BTL - z 1. Prove that Z a ( ) f n f ( ) a BTL -5 Evaluating z 13. Prove that Z a z a BTL -5 Evaluating 14. Find [ ] BTL - Analyzing 15. Find Z transform of, BTL Find [ + ] BTL Solve y n+1 + y n = given that y()= 18. State Convolution theorem in Z Transforms BTL Form the difference equation by eliminating arbitrary constants from = + MA6351_TPDE_QBank_ACY 16-17(ODD) COMMON TO ALL BRANCHES OF III SEMESTER B.E., B.TECH.

15 . Form the difference equation by eliminating arbitrary constants from = 1.(a) Find the z transform of PART-B 1 f n BTL - n 1 n 1 1. (b) Find the 1z Z BTL - z 3z. (a) Find the z-transform of (n+1) and sin(3n+5) BTL -.(b) Find the inverse Z Transform of z z z z 1z 1 3. (a) Find i Z r n cos n, ii Z r n sin n iii) Z( e at cos bt) 3.(b). BTL - BTL- Using convolution theorem find inverse Z transform of z z a z b z Z BTL - 4. (a) Find the 1 z 1 z 4.(b) Using convolution theorem find 1 Z z 1 z 3 5. (a) Find inverse Z -Transform of z z 4 5.(b) z z by the method of Partial fraction Using convolution theorem find inverse Z transform of + 6. (a) Using residue find [ 6.(b) Using convolution theorem find 7.(a) Using Z transform Solve y y y 7.(b) given that y()=,y(1)=1 Find inverse Z transform of BTL - Analyzing ] 1 z Z z 4 z 5 n 3 n 1 n z z z 1 3. BTL - Analyzing 8.(a) Using Z transform solve = given that y()=,y(1)=1 Using the inversion method (Residue theorem )find the 8.(b) z inverse Z transform of U(z)= z z (a) Using Residue method find z Z z z MA6351_TPDE_QBank_ACY 16-17(ODD) COMMON TO ALL BRANCHES OF III SEMESTER B.E., B.TECH.

16 1 9.(b) Using convolution theorem evaluate z 1 z 3 Using convolution theorem evaluate 1.(a) [ ] z Z 1.(b) Using Z transform solve + + = with y()=,y(1)=1 11.(a) Find the Z transform { } and { } 11.(b) Using Z transform solve = given that y()=,y(1)= 1.(a) 1.(b) Form the difference equation = h =, = = State and prove final value theorem and their inverse transformation 13.(a) Find the Z transform of { } and{ } BTL - Solve the difference equation y 13.(b) n 3 y n 1 yn given that y()=4,y(1)=,y()=8 Solve the equation using Z Transform 14.(a) y y 6 y 36 given that y() = y(1) = n 5 n 1 n 1 z 14.(b) Find Z by convolution theorem BTL - z 7z 1 ****ALL THE BEST**** MA6351_TPDE_QBank_ACY 16-17(ODD) COMMON TO ALL BRANCHES OF III SEMESTER B.E., B.TECH.