![. Consider the function v(r) r(r 1/2) (r-1) for r e (0, 1]. Determine the transformed function u(E) introduced in the previou](http://img.homeworklib.com/images/490a747b-d199-4737-8e40-2b303abf7165.png?x-oss-process=image/resize,w_560)
Homework Answers
Add Answer to:
. Consider the function v(r) r(r 1/2) (r-1) for r e (0, 1]. Determine the transformed...
3. Consider the function()x(x-1/2)(1) for z E 0,1]. Determine the transformed function u() introduced in the previous q...
3. Consider the function()x(x-1/2)(1) for z E 0,1]. Determine the transformed function u() introduced in the previous question. Show that | -1 u(E)dE-: 0. (Hint: you can do this without evaluating the function.) Determine the values of the midpoint rule,the simple trapezoidal rule (with two points) and of the Gaussian rule with 2 quadrature points. What do you observe about the accuracy of these rules? [10pts]
3. Consider the function()x(x-1/2)(1) for z E 0,1]. Determine the transformed function u() introduced...
Question 1 (Quadrature) [50 pts I. Recall the formula for a (composite) trapezoidal rule T, (u) for 1 = u(a)dr whic...
Question 1 (Quadrature) [50 pts I. Recall the formula for a (composite) trapezoidal rule T, (u) for 1 = u(a)dr which requires n function evaluations at equidistant quadrature points and where the first and the last quadrature points coincide with the integration bounds a and b, respectively. 10pts 2. For a given v(r) with r E [0,1] do a variable transformation g() af + β such that g(-1)-0 and g(1)-1. Use this to transform the integral に1, u(z)dz to an...
3. (15p.) Approximate the following integral using the two-point Gaussian quadrature rule | (2 + a)*e¢8–1)-+de...
3. (15p.) Approximate the following integral using the two-point Gaussian quadrature rule | (2 + a)*e¢8–1)-+de 2 B=1 ju a=8 0
1. The two-point forward difference quotient with error term is given by where ξ e ll, l + hl. In...
1. The two-point forward difference quotient with error term is given by where ξ e ll, l + hl. In class we showed an additional error term appears to due to computer rounding error, e(r). Denoting (z) f(x) +e(x) as what the com- puter stores, and supposing f"(x)M and e() e where e, M are constants, we obtained an upper bound for the error between f(r) and the computed forward difference quotient 2c h Find the minimum value of the...
5. Let f a, b R be a 4 times continuously differentiable function. For n even, consider < tn = b, a to < t< an...
5. Let f a, b R be a 4 times continuously differentiable function. For n even, consider < tn = b, a to < t< an uniform partition of [a, b] with b- a , i = 0,1,.. , n - 1 h t Let T denote the composite Trapezoidal rule associated with the above partition which approx imates eliminate the term containing h2 in the asymptotic expansion. Interprete the result which you obtain as an appropriate numerical quadrature rule...
1. Solve the boundary value problem ut =-3uzzzz + 5uzz, u(z, 0) = r(z) (-00 < z < oo, t > 0), usi...
1. Solve the boundary value problem ut =-3uzzzz + 5uzz, u(z, 0) = r(z) (-00 < z < oo, t > 0), using direct and inverse Fourier transforms U(w,t)-홅启u(z, t) ei r dr, u(z,t)-二U( ,t) e ur d . You need to explain where you use linearity of Fourier transform and how you transform derivatives in z and in t 2. Find the Fourier transform F() of the following function f(x) and determine whether F() is a continuous function (a)...
(40 pts) 2a. Show that u(z) is the solution to the problem where k(x)-1 for x < 1/2 and k = 2 for...
(40 pts) 2a. Show that u(z) is the solution to the problem where k(x)-1 for x < 1/2 and k = 2 for x > 1 /2. 2b. Set up the weak form for the differential equation above and the resulting element stiffness and element load vector and calculate the element stiffness matrix and load vector for 4 quadratic elements by using the Gaussian quadrature that is going to exactly calculate the integrals Then set up the global K and...
Matrix operations 22. Suppose you are given a matrix of the form cos(() - sin(0) R(0)...
Matrix operations 22. Suppose you are given a matrix of the form cos(() - sin(0) R(0) = sin() cos(0) Consider now the unit vector v = [1,0)" in a two dimensional plane. Compute R(O)v. Repeat your computations this time using w = [0, 1]". What do you observe? Try thinking in terms of pictures, look at the pair of vectors before and after the action of R(O). 23. You may have recognised the two vectors in the previous question to...
1. Let U C IRt be open, UR be a function, a U and 0 A v E R" such that Dof(a) exists. Show that DAvf(a) exists f...
1. Let U C IRt be open, UR be a function, a U and 0 A v E R" such that Dof(a) exists. Show that DAvf(a) exists for every 0 λ E R, and DAwf(a-λDuf(a). 3 marks
1. Let U C IRt be open, UR be a function, a U and 0 A v E R" such that Dof(a) exists. Show that DAvf(a) exists for every 0 λ E R, and DAwf(a-λDuf(a). 3 marks
3. Consider the function()x(x-1/2)(1) for z E 0,1]. Determine the transformed function u() introduced in the previous question. Show that | -1 u(E)dE-: 0. (Hint: you can do this without evaluating the function.) Determine the values of the midpoint rule,the simple trapezoidal rule (with two points) and of the Gaussian rule with 2 quadrature points. What do you observe about the accuracy of these rules? [10pts]
3. Consider the function()x(x-1/2)(1) for z E 0,1]. Determine the transformed function u() introduced...
Question 1 (Quadrature) [50 pts I. Recall the formula for a (composite) trapezoidal rule T, (u) for 1 = u(a)dr which requires n function evaluations at equidistant quadrature points and where the first and the last quadrature points coincide with the integration bounds a and b, respectively. 10pts 2. For a given v(r) with r E [0,1] do a variable transformation g() af + β such that g(-1)-0 and g(1)-1. Use this to transform the integral に1, u(z)dz to an...
3. (15p.) Approximate the following integral using the two-point Gaussian quadrature rule | (2 + a)*e¢8–1)-+de 2 B=1 ju a=8 0
1. The two-point forward difference quotient with error term is given by where ξ e ll, l + hl. In class we showed an additional error term appears to due to computer rounding error, e(r). Denoting (z) f(x) +e(x) as what the com- puter stores, and supposing f"(x)M and e() e where e, M are constants, we obtained an upper bound for the error between f(r) and the computed forward difference quotient 2c h Find the minimum value of the...
5. Let f a, b R be a 4 times continuously differentiable function. For n even, consider < tn = b, a to < t< an uniform partition of [a, b] with b- a , i = 0,1,.. , n - 1 h t Let T denote the composite Trapezoidal rule associated with the above partition which approx imates eliminate the term containing h2 in the asymptotic expansion. Interprete the result which you obtain as an appropriate numerical quadrature rule...
1. Solve the boundary value problem ut =-3uzzzz + 5uzz, u(z, 0) = r(z) (-00 < z < oo, t > 0), using direct and inverse Fourier transforms U(w,t)-홅启u(z, t) ei r dr, u(z,t)-二U( ,t) e ur d . You need to explain where you use linearity of Fourier transform and how you transform derivatives in z and in t 2. Find the Fourier transform F() of the following function f(x) and determine whether F() is a continuous function (a)...
(40 pts) 2a. Show that u(z) is the solution to the problem where k(x)-1 for x < 1/2 and k = 2 for x > 1 /2. 2b. Set up the weak form for the differential equation above and the resulting element stiffness and element load vector and calculate the element stiffness matrix and load vector for 4 quadratic elements by using the Gaussian quadrature that is going to exactly calculate the integrals Then set up the global K and...
Matrix operations 22. Suppose you are given a matrix of the form cos(() - sin(0) R(0) = sin() cos(0) Consider now the unit vector v = [1,0)" in a two dimensional plane. Compute R(O)v. Repeat your computations this time using w = [0, 1]". What do you observe? Try thinking in terms of pictures, look at the pair of vectors before and after the action of R(O). 23. You may have recognised the two vectors in the previous question to...
1. Let U C IRt be open, UR be a function, a U and 0 A v E R" such that Dof(a) exists. Show that DAvf(a) exists for every 0 λ E R, and DAwf(a-λDuf(a). 3 marks
1. Let U C IRt be open, UR be a function, a U and 0 A v E R" such that Dof(a) exists. Show that DAvf(a) exists for every 0 λ E R, and DAwf(a-λDuf(a). 3 marks
Most questions answered within 3 hours.
-
Which of the following statements is true regarding the nature
of culture?
A.We are born with...
asked 8 minutes ago -
(a) 2
NO2(g) à 2
NO(g) +
O2(g)
The rate law for this reaction
is 2nd...
asked 11 minutes ago -
Write the overall reaction for the RedOx
reaction (below) when it occurs in an acidic environment....
asked 18 minutes ago -
Two forces are acting on an object. Force 1 on the object is 6.0
N and...
asked 19 minutes ago -
You are planning for retirement. You plan to work for 20 years.
Starting a year from...
asked 32 minutes ago -
The African Adventures exhibit water system holds approximately
850,000 gallons (3.2x109 g) of water. The temperature...
asked 23 minutes ago -
which of the following is capital?
the barn a farmer uses to hold the chickens he...
asked 34 minutes ago -
Attribute
Sample Value
Sample Value
Sample Value
Sample Value
pirate_num
11
22
43
54
pirate_lname
James...
asked 27 minutes ago -
10. Two forces act on a 3 kg object, the gravitational force and
a second, constant...
asked 44 minutes ago -
Dividing Partnership Net Loss
Leigh Meadows and Byron Leef formed a partnership in which the
partnership...
asked 41 minutes ago -
Discuss how Entrepreneurial leadership style differs
from traditional leadership?
asked 1 hour ago -
A futuristic design for a car is to have a large solid
disk-shaped flywheel within the...
asked 1 hour ago