Engineering Mathematics Materials 2013Prepared by C.Ganesan, M.Sc., M.Phil., (Ph: 9841168917) Page 1 SUBJECT NAME : Transforms and Partial Differential Equation SUBJECT CODE : MA 2211 MATERIAL NAME : Part – A questions MATERIAL CODE : JM08AM1011 UPDATED ON : May - June 2013 Name of the Student: Branch: Unit – I (FOURIER SERIES) 1. State the Dirichlet’s conditions for Fourier series. 2. Write the conditions for a function ( ) f x to satisfy for the existence of a Fourier series. 3. State the sufficient condition for a function ( ) f x to be expressed as a Fourier series. 4. Give the expression for the Fourier Series co-efficient n b for the function ( ) sin f x x x = defined in ( ) 2, 2 ÷ . 5. If 2 2 2 1 ( 1) 4 cos 3 n n x nx n t · = ÷ = + ¿ , deduce that 2 2 2 2 1 1 1 ... 1 2 3 6 t + + + = . 6. Obtain the first term of the Fourier series for the function 2 ( ) , f x x x t t = ÷ < < . 7. Find the constant term in the expansion of 2 cos x as a Fourier series in the interval ( ) , t t ÷ . 8. Define Root Mean square value of a function ( ) f x over the interval ( ) , a b . 9. Find the root mean square value of 2 ( ) f x x = in ( ) 0, . 10. Find the root mean square value of the function ( ) f x x = in( ) 0, l . Engineering Mathematics Materials 2013 Prepared by C.Ganesan, M.Sc., M.Phil., (Ph: 9841168917) Page 2 11. Without finding the values of 0 , n a a and n b , the Fourier coefficients of Fourier series, for the function 2 ( ) F x x = in the interval ( ) 0,t find the value of ( ) 2 2 2 0 1 2 n n n a a b · = ( + + ( ¸ ¸ ¿ . 12. What is meant by Harmonic Analysis? Unit – II (FOURIER TRANSFORM) 1. Write the Fourier transform pair. 2. Write the Fourier cosine transform pair. 3. Write the Fourier sine transform pair. 4. Find the Fourier transform of , 0 x e o o ÷ > . 5. Find the Fourier cosine transform of , 0 ax e x ÷ > . 6. Find the Fourier sine transform of ( ) , 0 ax f x e a ÷ = > . 7. Find the Fourier sine transform of 3 x e ÷ . 8. Find the Fourier sine transform of 1 x . 9. If ( ) F s is the Fourier transform of ( ) f x , show that ( ) ( ) ( ) ias F f x a e F s ÷ = . 10. What is the Fourier transform of ( ) f x a ÷ , if the Fourier transform of ( ) f x is ( ) F s ? 11. Find the Fourier transform of , ( ) 0, and ikx e a x b f x x a x b ¦ < < = ´ < > ¹ . 12. State Parseval’s identity on Fourier transform. 13. State convolution theorem in Fourier transform. 14. State and prove the change of scale property of Fourier Transform. Engineering Mathematics Materials 2013 Prepared by C.Ganesan, M.Sc., M.Phil., (Ph: 9841168917) Page 3 15. If { } ( ) ( ) F f x F s = , prove that { } 1 ( ) s F f ax F a a | | = | \ . . 16. If ( ) c F s is the Fourier cosine transform of ( ) f x , prove that the Fourier cosine transform of ( ) f ax is 1 c s F a a | | | \ . . Unit – III (PARTIAL DIFFERENTIAL EQUATION) 1. Form the PDE from ( ) ( ) 2 2 2 2 x a y b z r ÷ + ÷ + = . 2. Find the PDE of the family of spheres having their centers on the z – axis. 3. Form the partial differential equation by eliminating the constants aand bfrom ( ) ( ) 2 2 2 2 z x a y b = + + . 4. Form the partial differential equation by eliminating the arbitrary constants aand bfrom ( ) ( ) 2 2 z x a y b = + + . 5. Eliminate the arbitrary function ‘ f ’ from y z f x | | = | \ . and form the PDE. 6. Form the partial differential equation by eliminating the arbitrary function from 2 x z xy f z | | ÷ = | \ . . 7. Find the partial differential equation of all planes cutting equal intercepts from the x and y axes. 8. Find the complete integral of p q pq + = . 9. Solve the partial differential equation pq x = . 10. Solve the equation( ) 3 0 D D z ' ÷ = . 11. Solve ( ) 3 2 2 0 D D D z ' ÷ = . 12. Solve ( ) 2 2 7 0 D DD D z ' ' ÷ + = . Engineering Mathematics Materials 2013 Prepared by C.Ganesan, M.Sc., M.Phil., (Ph: 9841168917) Page 4 13. Find the particular integral of ( ) 2 2 2 x y D DD D z e ÷ ' ' ÷ + = . 14. Solve the equation( ) 3 0 D D z ' ÷ = . 15. Solve( ) ( ) 1 1 0 D D D z ' ÷ ÷ + = . Unit – IV (APPLICATION OF PARTIAL DIFF. EQN.) 1. Classify the partial differential equation 2 2 4 u u x t c c = c c . 2. Write down all possible solutions of one dimensional wave equation. 3. Write down the three possible solutions of one dimensional heat equation. 4. What is the basic difference between the solutions of one dimensional wave equation and one dimensional heat equation with respect to the time? 5. In the wave equation 2 2 2 2 2 y y c t x c c = c c , what does 2 c stand for? 6. In the one dimensional heat equation 2 t xx u c u = , what is 2 c ? 7. A tightly stretched string with fixed end points 0 x = and x = is initially in a position given by 3 0 ( , 0) sin x y x y t | | = | \ . . If it is released from rest in this position, write the boundary conditions. 8. A rod 40 cm long with insulated sides has its ends Aand Bkept at 20⁰C and 60⁰C respectively. Find the steady state temperature at a location 15 cm from A. 9. Give three possible solutions of two dimensional steady state heat flow equation. 10. Write all three possible solutions of steady state two – dimensional heat equation. 11. Write down the partial differential equation that represents steady state heat flow in two dimensions and name the variables involved. 12. Write down the three possible solutions of Laplace equation in two dimensions. Engineering Mathematics Materials 2013 Prepared by C.Ganesan, M.Sc., M.Phil., (Ph: 9841168917) Page 5 13. Write down the two dimensional heat equation both in transient and steady states. 14. A plate is bounded by the lines 0, 0, x y x l = = = and y l = . Its faces are insulated. The edge coinciding with x – axis is kept at100 C . The edge coinciding with y – axis is kept at 50 C . The other two edges are kept at 0 C . Write the boundary conditions that are needed for solving two dimensional heat flow equation. Unit – V (Z – TRANSFORM) 1. Define the unit step sequence. Write its Z – transform. 2. Find the Z – transform of sin 2 nt . 3. If 2 ( ) 1 1 3 2 4 4 z F z z z z = | | | | | | ÷ ÷ ÷ | | | \ . \ . \ . , find (0) f . 4. Find the Z – transform of for 0 ( ) ! 0 otherwise n a n x n n ¦ > ¦ = ´ ¦ ¹ . 5. Find the Z – transform of n a . 6. Find the Z – transform of n. 7. Find the Z – transform of 1 ! n . 8. Obtain ( ) ( ) 1 z 1 2 z Z z ÷ ( ( + + ¸ ¸ . 9. What advantage is gained when Z – transform is used to solve difference equation? 10. Form a difference equation by eliminating arbitrary constants from 1 2 n n U A + = . 11. Form a difference equation by eliminating the arbitrary constant Afrom .3 n n y A = . Engineering Mathematics Materials 2013 Prepared by C.Ganesan, M.Sc., M.Phil., (Ph: 9841168917) Page 6 12. Find the difference equation generated by 2 n n n y a b = + . 13. Solve 1 2 0 n n y y + ÷ = given 0 3 y = . ----All the Best----
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