1. (15 points) For any parameter t, show that the R-K method (ks = /(z,-+ (1-1)h,y,, + (1-1) ). has the local truncation error O(h3) 1. (15 points) For any parameter t, show that the R-K...
2. Show that the local truncation error of of the midpoint method is O(k2)
2. Use Taylor expansion to show that the local truncation error for backward Euler's method applied to Y' (2) = f(2,Y (2)) is Tn+1 = O(h).
3. (15 points) Derive the Adams-Moulton Two-Step method and its local truncation error by using an appropriate form of an interpolating polynomial.
3. (15 points) Derive the Adams-Moulton Two-Step method and its local truncation error by using an appropriate form of an interpolating polynomial.
ASAP PLEASE
e) Explain the idea of the Gauss integration formula. f Show on a figure the local and global truncation error for the first two iterations of a ODE solver g) Solve graphically the ODE h) Explain how numerical adaptive ODE solvers works i) When is a numerical method for solving differential equations considered to be dy dx unstable ? Which parameter(s) is (are) influencing this stability (or instability)? j In general the total error done by any numerical...
please show all steps and equations used, please write
neatly.
Problem 16. Given the Runge-Kutta method for the initial value problem y' = f(t,y) for a
Problem 1: Let y()- r(t+2)-r(t+1)+r(t)-r(t-1)-u(t-1)-r(t-2)+r(t-3), where r(t) is the ramp function. a) plot y(t) b) plot y'() c) Plot y(2t-3) d) calculate the energy of y(t) note: r(t) = t for t 0 and 0 for t < 0 Problem 2: Let x(t)s u(t)-u(t-2) and y(t) = t[u(t)-u(t-1)] a) b) plot x(t) and y(t) evaluate graphically and plot z(t) = x(t) * y(t) Problem 3: An LTI system has the impulse response h(t) = 5e-tu(t)-16e-2tu(t) + 13e-3t u(t) The input...
2. Chose a and k such that the system in unknowns r, y, z has a (a) no solution, (b) a unique solution, and (c) many solutions. Give separate answers for each part. (15 points) kx+y+z=1 1+y+z=2 x+y + kz = a
(k)) In the power method, let r,-φ(z(k+1))/φ(z(k)). We know that limk-oork Show that the relative errors obey We know Ai. (는) Ck where the numbers ck form a convergent (and hence bounded) sequence.
(k)) In the power method, let r,-φ(z(k+1))/φ(z(k)). We know that limk-oork Show that the relative errors obey We know Ai. (는) Ck where the numbers ck form a convergent (and hence bounded) sequence.
Mechanics. Need help with c) and d)
1. A particle of mass m moves in three dimensions, and has position r(t)-(x(t), y(t), z(t)) at time t. The particle has potential energy V(x, y, 2) so that its Lagrangian is given by where i d/dt, dy/dt, dz/dt (a) Writing q(q2.93)-(r, y, z) and denoting by p (p,P2, ps) their associated canonical momenta, show that the Hamiltonian is given by (show it from first principles rather than using the energy) H(q,p)H(g1, 92,9q3,...
2. Consider the following transformations of R2 Tİ (z, y) (-r, y), T3(x, y) (z, _y), T,(zw) (y, x). Show that, for any j 1,2,3, a subset A C R2 is a Jordan region if and only if T,(A) is a Jordan region. What is the relation between the volumes of A and T, (A)?
2. Consider the following transformations of R2 Tİ (z, y) (-r, y), T3(x, y) (z, _y), T,(zw) (y, x). Show that, for any j 1,2,3,...