7. For fixed x', the Gaussian kernel function f(x ) = - is the solution to...
3. Let the function f be a real valued bounded continuous function on R. Prove that there is a solution of the equation f(x) = x, xER. Now choose a number a with f(a) > a and define the sequence (an) recursively by defining al = a and a叶1 = f(an), where n E N. If f is strictly increasing on R, show that (an) converges to a solution of the equation (0.1). This method for approximating the solution is...
2. Consider the following 1-D wave equation with initial condition u (x, 0)- F (x) where F(x) is a given function. a) Show that u (x, t)-F (x - t) is a solution to the given PDE. b) If the function F is given as 1; x< 10 x > 10 u(x, 0) = F(x) = use part (a) to write the solution u(x, t) c) Sketch u(x,0) and u(x,1) on the same u-versus-x graph d) Explain in your own...
Determine if the given function y- f(x) is a solution of the accompanying differential equation Differential equation: 9xy' + 9y-cos x Initial condition: y()0 Solution candidate: y- x O a. No b. Yes
Determine if the given function y- f(x) is a solution of the accompanying differential equation Differential equation: 9xy' + 9y-cos x Initial condition: y()0 Solution candidate: y- x O a. No b. Yes
6. Solve the heat equation (5.17) with initial condition u(x, 0) = H(x)e-x. Write the solution of the Cauchy problem for the heat equation u = kuyx - < x <®, t> 0, (5.17) with initial condition u(t,0) = {(H(x + 1) - H (1 - x)) in terms of the error function Erf () = * e ** dy.
Please answer all questions
Q2 2015
a) show that the function f(x) = pi/2-x-sin(x)
has at least one root x* in the interval [0,pi/2]
b)in a fixed-point formulation of the root-finding problem, the
equation f(x) = 0 is rewritten in the equivalent form x = g(x).
thus the root x* satisfies the equation x* = g(x*), and then the
numerical iteration scheme takes the form x(n+1) = g(x(n))
prove that the iterations converge to the root, provided that
the starting...
Solve the DE for x(t) given the following DE and volume
solution of V(t)
then answer the case1 and case 2 questions
V(t)=180-100e-0.01t+20e-0.05t
Case 1 Let i(t) = e-0.01t and r(t) =
e-0.05t
Solve for x(t) and plot a graph for x(t) and the function V(t)
What is the limiting value of x(t) that is what is x(t) as t goes
to infinity.
How does the solution vary as a function given the initial
conditions of X0=0,...
Let u be the solution to the initial boundary value problem for the Heat Equation, dụı(t, x)-20 11(t, x), IE(0, oo), XE(0,3); with initial condition u(0,x)-f (x), where f(0) 0 andf'(3)0, and with boundary conditions Using separation of variables, the solution of this problem is with the normalization conditions 3 a. (5/10) Find the functions wn, with index n 1. wn(x) = 1 . b. (5/10) Find the functions vn, with index n
Let u be the solution to the...
1. Solve the Cauchy problem (2.1)-(2.2) for the following initial condition a) $(x) = 1 if |2<1 and $(x) = 0 if |z| > 1. b) p(x) = e-x, x > 0; $(x) = 0, x < 0. with the heat, or diffusion, equation on the real line. That is, we We begin with the hea sider the initial value problem Ut = kuxx, XER, t > 0, u(x,0) = 0(2), XER. (2.1) (2.2)
(4) (a) Compute the Fourier series for the function f(x) interval [-π, π]. 1-z on the (b) Compute the solution u(t, z) for the partial differential equation on the interval [0, T): 16ut = uzz with u(t, 0)-u(t, 1) 0 for t>0 (boundary conditions) (0,) 3 sin(2a) 5 sin(5x) +sin(6x). for 0 K <1 (initial conditions) (20 points) Remember to show your work. Good luck.
(4) (a) Compute the Fourier series for the function f(x) interval [-π, π]. 1-z on...
Let u be the solution to the initial boundary value problem for the Heat Equation, au(t,z 382u(t,z), tE (0,oo), E (0,3); with initial condition u(0,x)-f(x)- and with boundary conditions Find the solution u using the expansion u(t,x) n (t) wn(x), with the normalization conditions vn (0)1, Wn (2n -1) a. (3/10) Find the functionswn with index n 1. b. (3/10) Find the functions vn, with index n 1 C. (4/10) Find the coefficients cn , with index n 1.
Let...