Transform &Partial Differential Equtaion Model Q1

MODEL EXAMINATION TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATION Year/Semester & Branch: II / III Common to all Branches Max. Marks: 100 Time: 180 min PART-A Answer ALL Questions (20X2=40) Determine the value of in the Fourier series expansion of . (A/M 08) Define Root mean square of over the range . (Tri-N/D 08) If in the interval (0,4), then find the value of in the Fourier series expansion. (Cbe-N/D 08) State Dirichlet’s condition for Fourier series. (Tnl-N/D 08) (Nov 05) Let be the Fourier cosine transform of . Prove that . (Cbe-N/D 08) State Fourier integral theorem. (A/M 08) (Cbe-N/D 08) If then prove that (Tnl-N/D 08) what is the sine transform of if is the Fourier sine transform of .(Tri-N/D 08) Form the partial differential equation by eliminating arbitrary constants from (Cbe-N/D 08) Find the complete integral of (N/D 08). Eliminate the function ‘f’ from (Tnl-N/D 08). Write the complete integral of (Tnl-N/D 08) A rod 50cm long with insulated sides has its ends A and B kept at and respectively. Find the steady state temperature distribution of the rod. (Cbe-N/D 08) Classify the partial differential equation . (A/M 08) Write the initial conditions of the wave equation if the string has an initial displacement but no initial velocity. (Tnl-N/D 08) List all the possible solutions of the one dimensional wave equation and state the proper solution. (Tri-N/D 08) Form the difference equation from . (A/M 08) Find (A/M 08) (Tnl-N/D 08) If What is ? (Tri-N/D 08) Find the Z- transform of . (Tri-N/D 08) PART-B (Answer ANY 5 questions) (5 X 12 = 60) a) Obtain the Fourier series of of period 2l and defined as follows .Hence deduce (Tnl-N/D 08) b) Find the half range sine series of in (Tnl-N/D 08) Find the Fourier cosine transform of . Hence prove that (Tnl-N/D 08) a) Solve .(Tri-N/D 08) b) Solve .(Tri-N/D 08) If a string of length ‘ l ’ is initially at rest in its equilibrium position and each of its points is given a velocity ‘ v ’ such that . Determine the displacement function at any time t.(Cbe –N/D 08) a) Find the Z-transform of and . Hence find . b) Solve using Z-transform given . a) Obtain the Fourier series upto second harmonic from the data x : 0 : 0.8 0.6 0.4 0.7 0.9 1.1 0.8 (Cbe N/D 08) b) Find the Fourier cosine transform of . Hence deduce the value of .(Tri-N/D 08) The ends A and B of a rod ‘ l ’cm long have their temperatures kept at and , until steady state conditions prevail. The temperature at the end B is suddenly reduced to and that of A is increased to .Find temperature distribution in the rod after time’t’.(Tnl-N/D 08) a) Find using Convolution theorem. (Tnl-N/D 08) b) Find the singular integral of . (Cbe N/D 08)