Question

I really need help with Part B of this question

Problem 2: a) If F(a) is the Fourier transform (FT) of a function qx), show that the inverse FT of ewb F(a) is q -b), with b a constant. This is the shift theorem for Fourier transforms. Hint: Y ou will need the orthogonality relation: where y-y) is the Dirac delta function] [ Joeo(y-y')dus2πδ(y-y'), b) Solve the diffusion equation with convection: vetneuzkat.aax au(x,t) аги, ди with-c < 鱸8: and ux,0)-far). by the convolution theorem. Fourier transform the differential equation and use the convolutior theorem together with the shift theorem to solve for u,t

Answer 1

Given Partialdifferential u/Partialdifferential t = k Partialdifferential^2 u/Partialdifferential x^2 + alpha Partialdifferential u/Partialdifferential x (1) Define u (x, t) = 1/squareroot 2 pi integral^infinity_-infinity u^~ (p, t) e^i px dp (2) Sub: (2) in (1) =. Partialdifferential u^~/Partialdifferential t = [k (i)^2 + alpha (ip)] u^~ Partialdifferential u^~/Partialdifferential t = (-kp^1 + alpha ip) u^~ u^~ = u^~ (p, 0) e^(-kp^2 + alpha ip) t u (x, t) = 1/squareroot 2 pi integral^infinity_-infinity u^~ (p, 0) e^ipx e^(ip alpha - kp^2) t dp (3) given u (x, 0) = f (x) From (2), inverse FT is, u^~ (p, t) = 1/squareroot 2 pi integral^infinity_-infinity u (x, t) e^-ipx dx At t = 0, u^~ (p, 0) = 1/squareroot 2 pi integral^infinity_-infinity u (x, 0) e^-ipx dx u^~ (p, 0) = 1/squareroot 2 pi integral^infinity_-infinity f (x) e^-i px dx (4) Sub: (4) in (3) = > u (x, t) = 1/2 pi integral^infinity_-infinity integral^infinity_-infinity f (y) e^-ipy