1. For a special case, show that the Improved Euler’s Method may be referred to as...

1. For a special case, show that the Improved Euler’s Method may be referred to as the Trapezoid Rule. Support this claim by writing and solving an example problem using both methods.

2. The error associated with Euler's Method is proportional to step size. Euler's method will yield error-free prediction for a certain condition. What is the condition? Support the claim.

3. Use the Taylor Series to show that the Improved Euler’s Method and Midpoint Method have the same order of accuracy, which is greater than the order of accuracy for Euler’s Method.

4. Use the forward finite difference approximation to derive Euler’s formula.

Expert Solution

Colby Messinger answered 2 years ago

plz short respsonse Describe the basic idea behind Euler’s Method. Compare this with the Improved Euler’s...

plz short respsonse Describe the basic idea behind Euler’s Method. Compare this with the Improved Euler’s Method - in what way is it an improvement? Finally, compare both these methods with the Runge-Kutta method - what is the difference, and why does Runge-Kutta give more accurate results? Why would we choose to use one of the numerical methods, instead of solving the differential equation to obtain an exact answer? What is the characteristic equation? What is the connection between the...

2. (Improved Euler’s Method ) Second, work out the first three steps by hand. Then approximate...

2. (Improved Euler’s Method ) Second, work out the first three steps by hand. Then approximate y(2) for each of the initial value problems using Improved Euler’s method, first with a step size of h = .1 and then with a step size of h = .05 using the Excel spreadsheet. (a) dy dx = 2xy, y(0) = 1 (b) dy dx = x − y x + 2y , y(0) = 1 (c) dy dx = y + x,...

Justify jackknife method is a special case of bootstrap method.

Euler’s Method Let’s get our hands dirty and actually use Euler’s method to estimate the value...

Euler’s Method Let’s get our hands dirty and actually use Euler’s method to estimate the value of y(2) where y is the solution to the initial value problem y′=y−2x y(0) = 1 Recall that Euler’s method says: Approximate values for the solution of the initial value problem y′=F(x, y),y(x0) =y0 with step size h, at xn=xn−1+h, are yn=yn−1+hF(xn−1, yn−1) Fill in the table for steps of size h= 0.2. n xn yn=yn-1+0.2F(xn-1,Yn-1 y'=F(xn,yn) 0 0 1 1 .2 2 .4...

Plot the Euler’s Method approximate solution on [0,1] for the differential equation y* = 1 +...

Plot the Euler’s Method approximate solution on [0,1] for the differential equation y* = 1 + y^2 and initial condition (a) y0 = 0 (b) y0 = 1, along with the exact solution (see Exercise 7). Use step sizes h = 0.1 and 0.05. The exact solution is y = tan(t + c)

Euler’s method Consider the initial-value problem y′ = −2y, y(0) = 1. The analytic solution is...

Euler’s method Consider the initial-value problem y′ = −2y, y(0) = 1. The analytic solution is y(x) = e−2x . (a) Approximate y(0.1) using one step of Euler’s method. (b) Find a bound for the local truncation error in y1 . (c) Compare the error in y1 with your error bound. (d) Approximate y(0.1) using two steps of Euler’s method. (e) Verify that the global truncation error for Euler’s method is O(h) by comparing the errors in parts (a) and...

1. (Euler’s method) First, work out the first three steps by hand. Then approximate y(2) for...

1. (Euler’s method) First, work out the first three steps by hand. Then approximate y(2) for each of the initial value problems using Euler’s method, first with a step size of h = .1 and then with a step size of h = .05 using the Excel spreadsheet. (a) dy dx = 2xy, y(0) = 1 (b) dy dx = x − y x + 2y , y(0) = 1 (c) dy dx = y + x, y(0) = 1...

1. The thermal efficiency of a Rankine power cycle may be improved by i. Superheating the...

1. The thermal efficiency of a Rankine power cycle may be improved by i. Superheating the steam ii. Reheating the steam between high and lower pressure sections of the turbine iii. Regenerative Feedwater Heating iv. Insulating the turbine and decreasing the entropy production during the expansion process v. Incorporating a Rankine cycle power system as part of a cogeneration system a. Items i), ii), and iv) only b. Items i), iii), and v) only c. Items ii), iv) and v)...

1. The thermal efficiency of a Rankine power cycle may be improved by i. Superheating the...

1. The thermal efficiency of a Rankine power cycle may be improved by i. Superheating the steam ii. Reheating the steam between high and lower pressure sections of the turbine iii. Regenerative Feedwater Heating iv. Insulating the turbine and decreasing the entropy production during the expansion process v. Incorporating a Rankine cycle power system as part of a cogeneration system a. Items i), ii), and iv) only b. Items i), iii), and v) only c. Items ii), iv) and v)...

Show that 2D potential flows satisfy Euler’s equations even if μ ≠ 0.

Question

1. For a special case, show that the Improved Euler’s Method may be referred to as...

Solutions

Expert Solution

Related Solutions

plz short respsonse Describe the basic idea behind Euler’s Method. Compare this with the Improved Euler’s...

2. (Improved Euler’s Method ) Second, work out the first three steps by hand. Then approximate...

Justify jackknife method is a special case of bootstrap method.

Euler’s Method Let’s get our hands dirty and actually use Euler’s method to estimate the value...

Plot the Euler’s Method approximate solution on [0,1] for the differential equation y* = 1 +...

Euler’s method Consider the initial-value problem y′ = −2y, y(0) = 1. The analytic solution is...

1. (Euler’s method) First, work out the first three steps by hand. Then approximate y(2) for...

1. The thermal efficiency of a Rankine power cycle may be improved by i. Superheating the...

1. The thermal efficiency of a Rankine power cycle may be improved by i. Superheating the...

Show that 2D potential flows satisfy Euler’s equations even if μ ≠ 0.