| APPLIED MATHEMATICS-3 |
| PAPER NO. 2 |
 |
 |
 |
|
| PUT ON: 16th Aug.,2002 |
|
AM-201 APPLIED MATHEMATICS-3 (B.Tech 3rd Semester,2122) Time : 3 Hours Maximum Marks : 60 NOTE:- This paper consist of Three Sections. Section A is compulsory. Do any Four questions from Section B and any two questions from Section C Section-A Marks : 20 1(a) Write Fourier half range sine series for the function: f(x) = 1, 0 < x < π/2, = 0, π/2 < x < π (b) Obtain Laplace Transform of: f(t) = cost, 0 < t< 2π = 0 t < 2π (c) Define Impulse Function. What is the Laplace Transform ? (d) Use Frobenius method to obtain series solution for y'' = y. (e) Use Rodrigues formula to obtain polynomial expression for P2 (x). (f) What does Jn (x) represent ? What is the value of J0 (x) ? (g) Write two dimensional Laplace equation in Cartesial coordinates. Is this PDE elliptic, parabola or hyperbolic ? (h) Obtain D' Alembert's solution for the one dimensional wave equation. (i) Is the function w = z2 + 1 analytic ? Justify your answer. (j) Evaluate  , where C is thje circle |z| = 1.Section-B Marks:5 Each 2. Obtain Fourier series expansion for the function: f(x) = x2, -π < x< π and use it to deduce the result  3. (a) Evaluate  (b)Prove that with usual notation J1 + J3(x) = (4/x)J2(x). 4. Use method of separation of variables to obtain temperature distribution u (x,t) in a laterally insulated copper bar 80 cm long, if th einitial temperature inside the bar is 100 sin(πx/80) C, and the ends of the bar are kept at 0 C throughout. How long will it take for the maximum temperature inside the bar to drop to 50 C ? 5. (a) Is the function w = z + (1/z) analytic ? Justify your answer. (b) State Cauchy's integral formula and use it to evaluate  where z is the circle | z - 1 | = 1. 6. (a) Find the laplace transform of t2 sin at. Find the invers Laplace transform of 1/s(s2 + a2). Section-C Marks : 10 Each 7. (a) Write one dimensional wave equation and obtain its solution in terms of Fourier series expansion using method of seperation of variables. (b) Obtain steady state temperature distribution inside a circular plate of radius a whose both surfaces are perfectly insulated and whose circumference is being maintained at temperature u(θ) = 10 θ (π - θ), 0 < θ < π = 0, π < θ < 2π 8. (a) Define bessel function and establish the orthogonal property of Bessel functions. (b) Use method of laplace transform to (D3 - 2 D  9. (a) Is mapping w = z2 conformal ? What is the image of the circle |z| = 1 under this mapping? (b) Use the method of residues to evaluate the real integral  |
| <<PREVIOUS 1 2 3 NEXT>> |
| SIMILAR PAPERS:- MATHEMATICS 3 (CSE) |