Integral Transforms — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

2. (a) Show that the Fourier series of the function cos 3x cos 5x Hence, prove that (b) State and prove the pointwise convergence theorem. (c) Find the Fourier series of f(x) = x -П <X <П and hence deduce (a) Define the Fourier transform of a function f(t) and show that (b) Let f(x) and its first derivative vanish as x → 0o. If Fc(k) is the Fourier cosine transform, then show that (c) If F(k) and G(k) are Fourier transform of f(x) and g (x) respectively, then show that the Fourier transform of the convolution (f * g) is the product (a) Show that: (ii) cleat cosct] = (s-a) +w (b) Let f be piecewise continuous in the interval [O, T] for every positive T, and let f be of exponential order, that is, f(t) = O(eat) as t → oo for some a > O. Then, show that the Laplace transform of f(t) exist for Re s > a. (c) Let f(s) be the Laplace transform of f(t) and let f be of exponential order as t → ∞o, then show that f (s) Solve the following ordinary differential equation: -U"+U= f(x), -00<x<∞, where u (X) vanishes as (x] → 0o, using Fourier transform method. (b) Find the solution of the Laplace equation in the half-plane, with the boundary conditions u(x, 0) = f(x), -0<x<0, u(x, y) → 0 as (x) → ∞, y → 00. (c) Find the solution of the problem of an infinite medium which slows neutrons, in which a source of neutrons is located, и (x, 0) =б (x), -∞<X ∞0, u(x,t) → 0 as (x) → ∞0, where u (x, t) represents the number of neutrons per unit volume per unit time and 8(x) S(t) represents the source function. 7.5, 7.5, 7.5 (a) Using Laplace transform, solve the third order ordinary differential equation D = de t > O, with the initial condition x (0) = 1, x'(0) = O and x"(0) = 5. (b) Use Laplace transform method to solve the first order partial differential equation xUx+ Ux =x, * > 0, t > 0, with the initial and boundary conditions u(x, 0) = 0 for x > 0 and u (0, t) = 0 (c) Find the displacement of a semi-infinite string which is initially at rest in its equilibrium position by solving the one-dimensional wave equation Ut=cUzx 0≤x<00, t > 0, with the initial and boundary conditions u (0, t) = A f(t) for t ≥ 0, u(x, t) → 0 as x → ∞ for t ≥ 0, and u (x, 0) = 0 = u (x, 0) for 0 < x < 0, where A is a constant.