partial differential equations EXERCISE 3.20 Consider the problem ut =u" + u for u(0,t) u(1, t) 0, u(x,0) f(x). ze(0, 1), t>0, Show that dt and conclude that Use this estimate to bound the difference between two solutions in terms of the difference between the initial functions. Does this problem have a unique solution for each initial function f? EXERCISE 3.20 Consider the problem ut =u" + u for u(0,t) u(1, t) 0, u(x,0) f(x). ze(0, 1), t>0, Show that...
3. Let U-Bt- tB be Brownian bridge on [0, 1], where {BiJosesi is a Brownian process (i) Show E(Ut0 (ii) Show Cov(U,, Ut) s(1- t) for 0 s ts1. (ii) Let Xg(t)B Find functions g and h such that X, has the same covariance as a Brownian bridge. 3. Let U-Bt- tB be Brownian bridge on [0, 1], where {BiJosesi is a Brownian process (i) Show E(Ut0 (ii) Show Cov(U,, Ut) s(1- t) for 0 s ts1. (ii) Let Xg(t)B...
it is Linear Systems Analysis class 1.4-1 Sketch the signals (a) u(t-5)-uſt-7)(b) uſt-5)+u(t-7) (c) lu(t-1)-ut-2)] (d) (t - 4)[u(t - 2) - uſt - 4)]
Consi der the initial value proflem 2) ut (x,0)=u,(x) Find the soltion ,f) for any XER, t2 Sketch the gra pa of u(x,t) as a function cfx r 0, t 2.* AMuene that the improperturtegraes f ( 3) then Prove: If ubtis the sokch on nt: use tu·erden ck integratien dx and ds in eAlemdent's fermla fer ubt) and change , the 2nol ao r Consi der the initial value proflem 2) ut (x,0)=u,(x) Find the soltion ,f) for any...
DIFFERENTIATION: For the signals x(t) in Problems (1-2), (a) Compute the fomula for and (b) sketch the signal's derivative x'(t) = x(t). If necessary, use the Differentiation Product Rule: (f.g)' = fg + fig', or "RUD", e. g u (t) = 8(t). In your plots, label both axes, and indicate key values of time and amplitude. (1) X(t) = 4 rect ). (Hint: express rect(t/10) in terms of the difference of Two unit step functions.) ( 10 points) (2) X(t)...
Let W(s, t) - F(u(s, t), vis, t)), where F, u, and v are differentiable, and the following applies. u(6, -6) - 7 v(6, -6) -9 us(6, -6) - 2 vs(6, -6) -7 (6,-6) --4 V:(6, -6) = 3 Fu(7.-9) - - 1 F (7.-9) - -2 Find W (6, -6) and W.(6, -6). Ws(6, -6) W:(6, -6) =
6. Plot the following functions and then find the Fourier transforms: (a) f (t) - Kt[u(t +a/2) - u(t - a/2)]. What is the value of F(0)? (b) f(t)- A cos (t)[u(t 2) ut -2)] (c) f(t) -Ae-2M-Du(t -1). 2(t-1)
For a particular brand of smart phone, the reliability function is R1(t) = 1 for 0 Rr() expt-3] for t> 3, with time t in years. Here, expla] means e a. Sketch RT(t) versus t. Show proper axis labels, scales, and units. b. Express cumulative distribution function (CDF) Fr(t1) mathematically, and sketch it below Rr() to t 3 and 3. the same horizontal scale. Set Fr(t) 0 for t < 0. Show proper axis labels, scales, and units. Express mathematically...
Let u be the solution to the initial boundary value problem for the Heat Equation 202u(t, ) te (0, o0) (0,3); дли(t, 2) хе _ with boundary conditions ut, 0) 0 u(t, 3) 0 and with initial condition 3 9 u(0, ar) f(x){ 5, | 4' 4 0, Те The solution u of the problem above, with the conventions given in class, has the form ()n "(2)"п (г)"а "," n-1 with the normalization conditions 3 Wn 2n vn (0) 1,...
Compute the Laplace transform of the function on [0, oo). Here, uc (t)ut c where u(t) is the Heaviside function on [0, oo). Give your answer as a function of 8 for 8 〉 0. (f)(s) =