MA6351-TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS Question Bank Department of Mathematics FATIMA MICHAEL COLLEGE OF ENGINEERING & TECHNOLOGY MADURAI 625 020, Tamilnadu, India
1. Define PDE and Order of a PDE. Unit-I Partial Differential Equations PART- A (2 marks) 2. Form the PDE by eliminating the arbitrary constants a & b from z = (x 2 +a)(y 2 +b). 3. Obtain the PDE by eliminating the arbitrary constants a & b from (x-a) 2 +(y-b) 2 + z 2 = 1. 4. Form the PDE by eliminating the arbitrary constants a & b from (x-a) 2 +(y-b) 2 = z 2 cot 2 α. 5. Form the PDE of all spheres whose centers lie on the x-axis. 6. Find the PDE of all planes having equal intercepts in the x & y axes. 7. Eliminate the arbitrary function f from z = f (xy/z) and form the PDE. 8. Form the PDE by eliminating the arbitrary function f, from z = f (x 2 + y 2 + z 2 ). 9. Form the PDE by eliminating f, from f (xy + z 2, x + y + z) = 0. 10. Solve 2 z/ x 2 = siny. 11. Solve 2 u/ x t = e t cosx given that u = 0 when t = 0 and u/ t = 0 when x = 0. 12. Define complete integral, Particular integral, Singular integral. 13. Solve pq + p + q = 0. 14. Solve p + q = 1. 15. Find the Complete integral of z = px + qy + c 1+p 2 +q 2. 16. Find the Singular integral of z = px + qy + p 2 q 2. 17. Find the Complete integral of z/pq = x/q + y/p + pq. 18. Solve p (1+q) = qz. 19. Solve p+q = z. 20. Solve p+q = x + y. 21. Find the Complete integral of p + q = x + y. 22. Find the solution of px + qy = z. 23. Find the solution of p tan x + q tan y = tan z. 24. Solve x p+ y q = x. 25. Find the solution of px 2 + qy 2 = z 2. 26. Solve: (D 3 4D 2 D + 4DD 2 ) z = 0. 27. Solve: (D 3 +D 2 D DD 2 - D 3 ) z = 0. 28. Find the Particular integral of (D 2 4DD + 4D 2 ) z = e 2x + y. 29. Find the Particular integral of (D 3 3D 2 D + 4D 3 ) z = e x + 2y. 30. Find the Particular integral of (D 2 2DD + D 2 ) z = cos (x-y). PART B (8 Marks) 1. Form the PDE by eliminating the arbitrary functions f and Φ from z = y f(x) + x Φ(y). 2. Obtain the PDE by eliminating f and g from z = f(y+2x) + g(y-3x). 3. Solve (y-z) p - (2x+y) q = 2x+z. 4. Form the PDE by eliminating the arbitrary functions f and Φ from z = x f(y/x) + y Φ(x). 5. Find the Singular integral of z = px + qy + p 2 + pq + q 2.
6. Solve: (x 2 - y 2 - z 2) p + 2xyq 2xz = 0. 7. Solve x (y 2 + z) p- y(x 2 +z) q = z (x 2 -y 2 ). 8. Solve (3z-4y) p + (4x-2z) q = 2y-3x. 9. Solve (x 2 - yz) p+ (y 2 -zx) q = z 2 -xy. 10. Solve: x (y 2- z 2 ) p+ y(z 2 -x 2 ) q = z (x 2 -y 2 ). 11. Solve z(x p+ y q) = y 2 -x 2. 12. Solve: (D 2-5DD +6D 2 ) z = ysinx. 13. Solve: (D 2 + 2DD +D 2 ) z = x 2 y + e x - y. 14. Solve: (D 2-2DD -20D 2 ) z = e 5x + y + sin (4x-y). 15. Solve: (D 2 + DD -6D 2 ) z = x 2 + y + e 3x + y. 16. Solve: (D 3 7D D 2-6D 3 ) z = sin (4x + y) + e 2x + y. 17. Solve r + s + t = ycosx. 18. Solve: (D 2 + 4DD -5D 2 ) z = 3e 2x - y + sin (x-2y). 19. Solve: (D 2 + 3DD -4D 2 ) z = x + siny. 20. Solve: (D 2-2DD ) z = e 2x + x 3 y. 21. Solve: (D 3 + D 2 D -DD 2 - D 3 ) z = 3sin (x + y). 22. Solve: (D 3 3D 2 D + 4D 3 ) z = e x + 2y. 23. Solve: (D 2 2DD + D 2 ) z = cos (x-y). 24. Solve: (y+z)p+(z+x)q=x+y. 25. Solve: X(y 2+ z 2 )p+y(z 2 +x 2 )q=z(y 2 -x 2 ) 26. Solve: x(y-z)p+y(z-x)q=z(x-y) 27. Solve: (3z-4y)p+(4x-2x)q=2y-3x. 28. Solve: p(1-q 2)= q(1-z) 29. Solve: z 2 =1+p 2 +q 2 30. Solve: Unit- II Fourier series Part A (2 marks) 1. Write down the Dirichlet s condition for a function to be expanded as a Fourier series? 2. State Euler s formula when f(x) is expanded as Fourier series in c x c+2π. 3. If f(x)= is defined in π < x <π. Write the value of a 0 and a n 4. Define root mean square value of f(x) 5. If f(x)=x + x 2 is expressed as a Fourier series in the interval (-2, 2) to which value this series converges at x=2 6. If the Fourier series for the function it is odd or even function. 7. Find the Fourier series for the function f(x)=1, 8. Find the Root mean square value of f(x)=x in π < x <π. 9. What is the constant term a 0 and the coefficient of cos nx, a n in the Fourier series expansion of f(x) = x-x 3 in π < x <π. 10. What do you mean by harmonic Analysis 11. State Parseval s identity for full range expansion of f(x) as Fourier series in (0,2 l )
12. Define Parseval s theorem on Fourier series? 13. Find a n is expanding e -x as a Fourier series in π < x <π 14. If f(x) = 2x in the interval (0,4) then find the value of a 2 in the Fourier expansion 15. In the Fourier expansion of.find the value of b n the coefficient of sin nx. 16. Find the Fourier constant b n for f(x)=x cosx in ( 17. If is the r.m.s value of f(x) over an interval 2 what is the relation between and the Fourier coefficient a 0, a n, b n. 18. State Euler s formula for Fourier coefficient of a function defined in (c, c+2l) 19. Determine bn in the Fourier series expansion of f(x) = ½ (π-x) in 0<x<2π. 20. If f(x) is an odd function define in (-l,l), what are the values of a 0 and a n? 21. Find the sine series for the function f(x) = 1, 0<x<π. 22. Find the sum of the Fourier series of f(x) = x + x 2 at x = π in π < x < π. 23. If f(x) = 3x 4x 3 defined in the interval (-2,2), then find the value of a 1 in the Fourier series expansion. 24. Find the R.M.S value of f(x) = x in the interval (0, l). 25. State the Parseval s identity for the half range cosine series expansion of f(x) in (0, l). 26. Find the R.M.S Value of f(x) = x 2 in the interval (0,π). 27. If f(x) is discontinuous at x=a, what does its Fourier series represent at the point. 28. Define: Periodic function with an example. 29. Obtain the first term of the Fourier series for the function f(x) = x 2 in (-π, π). PART B (8 marks) 1. Determine the Fourier series for the function f(x)=x 2 in 0 2. Find the complex form of Fourier series for the function f(x)=e -x in -1 x 1 3. Determine the Fourier series for the function deduce that 4. Find the Fourier cosine series for f(x)=x in 0 < x < Show that 5. Obtain the Fourier series of f(x) of period 2l given 6. Find the complex form of Fourier series for the function f(x)=e x in π < x <π. 7. Find the Fourier series for period l for the function hence deduce that the sum of the series 8. Find the Fourier series of in ( 9. Find the Fourier series for the harmonic to represent the function in the following data X 0 1 2 3 4 5 F(x) 9 18 24 28 26 20
10. Find the Fourier series expansion of 11. Find the Half Range sine series of 12. Find the Fourier series upto third harmonic y=f(x) is given by X 0 π/3 2π/3 Π 4π/3 5π/3 π Y 13. Find the Fourier series for 1 1.4 1.9 1.7 1.5 1.2 1 14. Obtain the Fourier series to represent the function in π < x <π and deduce that 15. Find the Fourier series for hence show that 16. Find the Fourier series of f(x)=x+x 2 in ( of periodically 2 Hence deduce that 17. Find the Fourier series of 18. Find the half range cosine series of 19. Find the Fourier series for period for the function 20. Find the first two harmonic Fourier series to represent the function in the following data x 0 0 60 0 120 0 180 0 240 0 300 0 1.98 1.30 1.05 1.30-0.88-0.25 f(x) 21. Find the Fourier series f(x)=2x-x 2 in 0 < x < 2 22. Find the complex form of the Fourier series for π < x <π 23. Expand f(x)=e -x as a Fourier series in (-1, 1) 24. Expand as Fourier series in and hence deduce that the sum 25. If find the Fourier series of period in the interval hence deduce that Unit-III Applications of Partial Differential Equations PART- A (2 Marks) 1. Define initial value problems. 2. Define Boundary value problems. 3. Explain about method of separation of variables.
4. Write down the one dimensional wave equation. 5. Give the examples for one dimensional wave equation. 6. What are the possible solutions of one dimensional wave equation. 7. In one dimensional wave equation what does a 2 stands for? 8. Define temperature gradient. 9. Define steady state temperature distribution. 10. State one dimensional heat equation with the initial and boundary conditions. 11. Give three possible solutions of one dimensional heat equation. 12. State Fourier law of heat conduction. 13. Classify the PDE xf xx + yf yy = 0,x > 0, y > 0. 14. Classify the PDE f xx - 2f xy = 0,x > 0, y > 0. 15. Classify the PDE u xx -2u xy + u yy = 0,x > 0, y > 0. 16. A string is stretched and fastened to two points l aport motion is started by displaying the string into the form y=y 0 sinπx/l from which it is released at time t=0. Formulate this problem as the boundary value problem. 17. State any two laws which are assumed to derive one dimentional heat equation. 18. Write the boundary conditions and initial conditions for solving the vibration of string equation, if the string is subjected to initial displacement f(x) and initial velocity g(x). 19. The ends A and B of a rod of length 10cm long have their temperature kept at 20 0 and 70 0. Find the steady state temperature distribution on the rod. 20. Write down the equation of two dimensional steady state heat conduction. 21. In steady state conditions derive the solution of one dimensional heat flow equation. 22. Write down the three possible solution of laplace equation in two dimensions. 23. Write all variable separable solutions of the one dimensional heat equation u t =α 2 u xx 24. Write any two solutions of the laplace equation u xx +u yy =0 involving exponential terms in x and y. 25. What conditions are assumed in deriving the one dimensional wave equation. PART- B(8 Marks) 1. Using the method of separation of variables solve u/ x = 2 u/ t + u where u (x,0) = 6e -3x. 2. Derive the Solution of wave equation. 3. A string is stretched and fastened to two points x = 0 and x = l apart. Motion is started by displace the string into the form y = k(lx x 2 ) from which it is released at time t = 0.Find the displacement of any point on the string at a distance of x from one end at time t. 4. A string is tightly stretched and its ends are fastened at two points x = 0 and x = l. the midpoint of the string is displaced transversely through a small distance b and the string is released from rest in that position. Find an expression for the transverse displacement of the string at any time during the subsequent motion. 5. If a string of length l is initially at rest in its equilibrium position and each of its points is given the velocity ( y/ t) t=0 = v 0 sin 3 πx /l, 0 < x <l. Determine the displacement function y(x,t).
6. A tightly stetched string with fixed end points x = 0 and x = l is initially at rest in its equilibrium position. If it is set vibrating giving each point a velocity λx (l - x), the show that 7. A rod of length l has its ends A and B kept at 0 o C and 100 o C until steady state Conditions prevail. If the temperature at B is reduced suddenly to 0 o C and kept so while that of A is maintained. Find the temperature u(x,t) at a distance x from A and at time t. 8. A string of length 2l is fastend at both ends. The mid point of the string is taken to a height b and then released from rest in that position. Show that the displacement is 9. A string of length l has its ends x=0,x=l fixed. The point where x=l/3 is drawn aside a small distance h, the displacement y(x, t) satisfies find y(x, t) at any time t. 10. A tightly stretched string with fixed end points x=0 and x=l is initially at rest in its equilibrium position. If it is set vibrating string giving each points a velocity. show that displacement 11. If a string of length l is initially at rest in its equilibrium position and each of its points is given the velocity from one end at time t. determine the displacement of a point distant x 12. A string of length l is initially at rest in its equilibrium position and motion is started by given each of its points a velocity given by find the displacements function y(x, t). 13. A rod 30cm long has its ends A and B kept at 20 0 and 80 0 respectively until steady state conditions prevail the temperature at each end is then suddenly reduced to 0 0 c and kept so. Find the resulting temperature function u(x, t) taking x=0 at A. 14. The ends A and B of a rod l cm long have their temperature kept at 30 0 C and 80 0 C, until steady state conditions prevail. The temperature of the end B is suddenly reduced to 60 0 C and that of A is increased to 40 0 C. Find the temperature distribution in the rod after time t. 15. A bar 10cm long with insulated sides, has its ends A and B kept at 20 0 C and 40 0 C respectively until steady state condition prevail. The temperature at A is then suddenly
raised to 50 0 C and at the same instant that at B is lowered to 10 0 C and maintained there after. Find the subsequent temperature distribution in the bar. 16. Find the steady state temperature distribution in a rectangular plate of sides a and b insulated at the lateral surface and satisfying the boundary conditions 17. A rectangular plate is bounded by the lines x=0,y=0,x=a and y=b. its surface are insulated and the temperature along two adjacent edges are kept at 100 0 C, while the temperature along the other two edges are at 0 0 C. find the steady state temperature at any point in the plate. Also find the steady state temperature at any point of a square plate of side a if two adjacent edges are kept at 100 0 C and the other at 0 0 C 18. An infinitely long rectangular plate with insulated surface is 10cm wide. The two long edges and one short edge are kept at zero temperature, while the other short edge x=0 is kept at temperature given by find the steady state temperature distribution in the plate. 19. A rectangular plate with insulated surface is 10cm wide and so long compared to its width that may be considered infinite in length without introducing appreciable error. The temperature at short edge y=0 is given by steady state temperature at any point in the plate. and the 20. A square plate is bounded by the lines x=0,y=0,x=20 and y=20. Its faces are insulated, the temperature along the edge y=20 is given by x(20-x) while the other three edges are kept at 0 0 C. Find the steady state temperature distribution on the plate. UNIT IV FOURIER TRANSFORMS PART-A (2 marks) 1. Write the Fourier transform pair 2. If F c [s] is the Fourier cosine transform of f(x) prove that the Fourier cosine transform of f(x) is 1/a F c [s/a] 3. If F(s) is the Fourier transform of f(x) write the formula for the Fourier transform of f(x)cos ax in terms of F. 4. State the convolution theorem for Fourier transform 5. If F(s) is the Fourier transform of f(x) find the Fourier transform of f(x-a) 6. Find the Fourier transform of f(x) if 7. Find the Fourier cosine transform of e -x 8. Find (i) F[x n f(x)] (ii) F[d n f(x)/dx n ] in terms of Fourier transform of f(x) 9. Find the Fourier sine transform of 1/x 10. State the Fourier integral theorem 11. State modulation property on Fourier transform
12. Find the Fourier sine transform of f(x)=e -x 13. Prove that F[f(ax)]=1/a F(s/a), a>0 14. Find the Fourier transform of 15. State Parseval s identity theorem in Fourier transform 16. Find the Fourier sine transform of f(x)=1 in (0,l) 17. What is the Fourier transform of f(x-a) if the Fourier transform of f(x) is f(s) 18. State the Fourier transform of the derivation of a function. 19. Write down the FCT pair of formulae. 20. Write down the FST pair of formulae. 1. Find the Fourier cosine transform of PART B (8 marks) 2. Evaluate using Fourier transforms 3. Find the Fourier transform of f(x) is deduce that 4. Using Parseval s Identity calculate 5. If F c [f(x)]=f c (s). Prove that F c [F c (x)]=f(s) 6. Using Parseval s identity calculate 7. Find the Fourier sine and cosine transform of 8. Find the Fourier cosine transform of and Fourier sine transform of 9. Find the Fourier sine transform of and evaluate 10. Show that is a self reciprocal under Fourier cosine transform. 11. If F(s) is the Fourier transform of f(x) then 12. State and prove Parseval s identity in Fourier transform 13. Find the Fourier cosine transform of 14. Find the Fourier transform of f(x) is hence find 15. Find the Fourier Cosine transform of PART B (16 marks) 1. Find the Fourier transform of hence P.T. 2. Find the Fourier transform of, hence show that and 3. Find the Fourier transform of, hence show that and
4. Express the function as a Fourier integral. Hence evaluate and find the value of 5. Find the Fourier transform of hence prove that using Parseval s identity show that 6. Find the Fourier transform of Hence prove that and Unit-V Z-Transforms and Difference Equations PART- A (2 Marks) 1. Define Z-Transforms. 2. Prove that Z(1) = z / z-1.find also the region of convergence. 3. Find Z(a n /n ). 4. Find Z(1/n). 5. Find the value of Z f(n) when f(n) = na n. 6. Find Z( e -iat ). 7. Find Z (cos n /2) and Z (sin n /2). 8. Find Z (cos at) and Z (sin at). 9. Find the Z transform of (n + 1) (n + 2). 10. Find the Z transform of (n + 2). 11. State and prove First Shifting property on Z-transform. 12. Define Unit Step function. 13. Define Unit impulse function. 14. State and prove initial value theorem. 15. State final value theorem. 16. Evaluate Z -1 [z / z 2 +7z + 10]. 17. Define convolution of two functions. 18. State Convolution theorem on Z-Transforms. 19. Find Z -1 [z 2 / (z -a) 2 ] using convolution theorem. 20. Find Z -1 [z 2 / (z -1) (z-2)] using convolution theorem. PART- B (8 Marks) 1. Find Z transform of 1/(n + 1) (n + 2).
2. State and prove initial value theorem. 3. Find Z -1 [z 3 / (z -1) 2 (z-2)] by partial fraction method. 4. Find Z -1 [z (z 2 -z + 2) / (z +1) (z-1) 2 ] by partial fraction method. 5. Find Z -1 [z 2-3z / (z -5) (z+2)] by partial fraction method. 6. Find Z -1 [4z 2-2z / z 3-5z 2 +8z-4] by partial fraction method 7. Find Z -1 [z (z +1) / (z-1) 3 ] using residues. 8. Find Z -1 [z / z 2-2z + 2] using residues. 9. Find Z -1 [z / (z -1) (z-2)] using residues. 10. State and Prove convolution theorem. 11. Find the inverse Z- transform of 8z 2 / (2z -1) (4z +1) using convolution theorem. 12. Solve the difference equation y (n+2) 3 y (n+1) + 2y (n) = 2 n given that y (0) = 0 and y (1) = 0. 13. Solve the difference equation y (n) + 3 y (n-1) - 4y (n-2) = 0 n 2 given that y (0) = 3 and y (1) = -2. 14. Solve the difference equation y (n+2) + 6 y (n+1) + 9y (n) = 2 n given that y (0) = 0 and y (1) = 0. 15. Solve the difference equation y (k+2) -4y (k+1) + 4y (k) = 0 given that y (0) = 1 and y (1) = 0. 16. Solve the difference equation y n+2 + y n = 2 given that y 0 = y 1 = 0. 17. Solve the difference equation y (n+3) -3 y (n+1) + 2y (n) = 0 given that y (0) = 4 and y (1) = 0 y (2) = 8. 18. Find Z(cos n ) and Z(n cos n ). 19. Sol ve using z-transform y n+2-3y n+1-10y n =0 given y 0=1 and y 1 =0. 20. Solve the equation (using z-transform) y(n+1)-5y(n+1)+6y(n)=36 given that y(0)=y(1)=0. 21. Find the z-transform of {a n } and {na n } 22. Find the inverse z-transform of 8z 2 /(2z-1)(4z-1) 23. Find the z-transform fo {1/n} and {cos nπ/2} 24. Solve the equation y(n+3)-3y(n+1)+2y(n)=0 given that y(0)=4, y(1)=0 and y(2)=8 by the method of z-transform 25.State and prove the convolution theorem of z-transform.