Integral Transforms — DU B.A. (Prog) Mathematics as Major Previous Year Question Papers

BACHELOR OF SCIENCE (PROGRAMME) 1. Let f(x) be piecewise continuous and periodic function with period 2Tt. The function f (x) represented by a Fourier series. Hence, prove the following whereak, bk are Fourier coefficients, for all values of n. Find the Fourier series expansion of the periodic function f(x) = X, -п <x < T and f(x)=f(x+ 2mm), m = 1,2.... Hence, prove 2. Find the maximum value of the Fourier series expansion of the following function, If f(x) = e=*, 0<x<∞. Show that the Fourier cosine integral representation is 3. Find the Fourier transform of e-all and explain the same by plotting a graph. Find the Fourier transform of e-ax and by using this result calculate the value of So x*e-ax ax. 4. If f(t) = Jo (at) is a Bessel function of order zero then show that Obtain the convolutions of : (b) sin at * sin at 5. Using the Fourier transform solve the one dimensional wave equation OR using the Laplace transform solve the matrix differential system de* (t) = Ax (t), x (0) = (*0) 6. A rod with diffusion constant k contains a fuel which produces neutrons by fission. The ends of the rod are perfectly reflecting. If the initial neutron distribution is f(x), find the neutron distribution u(x, t) at any subsequent time t. The problem is governed by Using the Fourier transform solve the PDE OR using the Laplace transform solve the partial differential equation with initial and boundary conditions u(x, 0) = 0 for x > 0, u(0, t) = 0 fort > 0.