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( x Question 19 1 pts Problem 19: Numerical solution of Ordinary differential equations Consider the...
Question 19 1 pts Problem 19: Numerical solution of Ordinary differential equations Consider the following initial...
Question 19 1 pts Problem 19: Numerical solution of Ordinary differential equations Consider the following initial value problem GE: + 15y = + 1.C: y(0) 0.5 Using Euler's method, and a time step of At = 0.2. do you expect the numerical solution not to oscillate and to be stable? No, because the time step far exceeds the critical value At stable < 0.067 for this problem. None of the above Yes, because Euler's method is explicit and there is...
Question 22 1 pts Problem 22: Numerical solution of Ordinary differential equations Consider the following initial...
Question 22 1 pts Problem 22: Numerical solution of Ordinary differential equations Consider the following initial value problem GE:+15y = 1.C:y(0) -0.5 Carry out two-steps of the modified Euler (trapezoidal) method solution from the initial condition with a time step of At = 0.1. and the predicted solutions is y(0.2)-0.20 None of the above. y(0.2) - -0.75 y(0.2)-1.27 y(0.2)=0.25
Question 21 1 pts Problem 21: Numerical solution of Ordinary differential equations Consider the following initial...
Question 21 1 pts Problem 21: Numerical solution of Ordinary differential equations Consider the following initial value problem G.EE +15y = 1.C:y(0) - 0.5 Carry out a single step of the modified Euler (trapezoidal) method solution from the initial condition with a time step of At = 0.2, and the predicted solutions is Y(0.2)-0.20 None of the above y(0.2)-1.27 Y(0.2)-0.25 (0.2)--0.75
Question 20 1 pts Problem 20: Numerical solution of Ordinary differential equations Consider the following initial...
Question 20 1 pts Problem 20: Numerical solution of Ordinary differential equations Consider the following initial value problem GE: H+ 15y =t 1.C:y(0) = 0.5 Carry out two consecutive steps of the Euler solution from the initial condition with a time step of At = 0.2. and the predicted solutions are None of the above. y(0.2)--0.25 and y(0.4)-0.13 (0.2)-0.05 and y(0.4)-0.03 y(0.2) -- 1.00 and y(0.4)-2.04 y(0.2)-0.13 and y(0.4)-0.20
Question 23 1 pts Problem 23: Numerical solution of Ordinary differential equations Consider the following initial...
Question 23 1 pts Problem 23: Numerical solution of Ordinary differential equations Consider the following initial value problem GE: + 15y = 1.C:y(0) = 0.5 Using the results in question 21 and 22, the computed absolute value of the error estimate e for the modified Euler predicted solution using a time step of At = 0.2.is None of the above. Ec-0.12 Ec-0.42 Ec-1.42 Ec-15.42 21 Y(0.2) = 0.5 +0.77=1.27 k, = 0.2 [0.15 (0.5))=-1.5 K2=0.2 [02-151-1)] = 3.04 k=kitky =...
Question is as follows: NOTE: • Subject: Numerical Methods for Ordinary Differential Equations: Initial Value Problems....
Question is as follows:
NOTE:
• Subject: Numerical Methods for Ordinary Differential
Equations: Initial Value Problems.
______
Useful Information
Implicit Nystrom Method: (aka Milne-Simpson)
Explicit Nystrom Method: (for reference)
Adams-Bashforth Method (AB): (for reference)
Zero-Stability:
Root Condition: (for reference)
3. Consider the 2-step Implicit Nyström Method (aka Simpson's Method): Un+2 – Un = (F(UM) + 4f(Un+1) + f(Un+2)]. (a) Determine the two characteristic polynomials, p() and o(5), for this method. Use these to show that this method is consistent. (b)...
NOTE: • Subject: Numerical Methods for Ordinary Differential Equations: Initial Value Problems. 1. Consider the family...
NOTE:
• Subject: Numerical Methods for Ordinary Differential
Equations: Initial Value Problems.
1. Consider the family of linear multistep methods Un+1 = qUN+ (2(1 – a) f (Un+1) + 3a f (UN) – af (Un-1)). Ppt" (a) Determine the order of accuracy as a function of the parameter a. Find the optimal a to give the highest order of accuracy. Let's call the optimal value Qopt. (b) Is the method with Qopt zero-stable? Is the method with Qopt convergent? Explain...
please show excel formulas for both explicit and implicut methods please solve using excel and show...
please show excel formulas for both explicit and implicut
methods
please solve using excel and show formulas You are given the following system of differential equations: 99x, +2999x 2000x1 - 3000x2 If x1(0)=x2(0)-1, obtain a solution from t=0 to 0.3 using a step size of 0.03 with a. The explicit Euler's method b. The implicit Euler's method (note that this problem can be solved via a set of simultaneous linear equations for each time step) C. Plot all results on...
3. Consider the following stiff system of autonomous ordinary differential equations du f(u, u) =-3u +3,...
3. Consider the following stiff system of autonomous ordinary differential equations du f(u, u) =-3u +3, u(0)2 = ' dt de g(u, v) -2000u - 1000, v(0)-3 Note that 1 u<2 and -4 <v < 3 for all t. (a) Find the Jacobian matrix for the system of equa tions (b) Find the eigenvalues of the Jacobian matrix. (c) In the figure the shaded region shows the region of absolute stability, in the complex h plane, for third order explicit...
this is numerical analysis. Please do a and b 4. Consider the ordinary differential equation 1'(x)...
this is numerical analysis. Please do a and b
4. Consider the ordinary differential equation 1'(x) = f(x, y(x)), y(ro) = Yo. (1) (a) Use numerical integration to derive the trapezoidal method for the above with uniform step size h. (You don't have to give the truncation error.) (b) Given below is a multistep method for solving (1) (with uniform step size h): bo +1 = 34 – 2n=1 + h (362. Yn) = f(n=1, 4n-1)) What is the truncation...
Question 19 1 pts Problem 19: Numerical solution of Ordinary differential equations Consider the following initial value problem GE: + 15y = + 1.C: y(0) 0.5 Using Euler's method, and a time step of At = 0.2. do you expect the numerical solution not to oscillate and to be stable? No, because the time step far exceeds the critical value At stable < 0.067 for this problem. None of the above Yes, because Euler's method is explicit and there is...
Question 22 1 pts Problem 22: Numerical solution of Ordinary differential equations Consider the following initial value problem GE:+15y = 1.C:y(0) -0.5 Carry out two-steps of the modified Euler (trapezoidal) method solution from the initial condition with a time step of At = 0.1. and the predicted solutions is y(0.2)-0.20 None of the above. y(0.2) - -0.75 y(0.2)-1.27 y(0.2)=0.25
Question 21 1 pts Problem 21: Numerical solution of Ordinary differential equations Consider the following initial value problem G.EE +15y = 1.C:y(0) - 0.5 Carry out a single step of the modified Euler (trapezoidal) method solution from the initial condition with a time step of At = 0.2, and the predicted solutions is Y(0.2)-0.20 None of the above y(0.2)-1.27 Y(0.2)-0.25 (0.2)--0.75
Question 20 1 pts Problem 20: Numerical solution of Ordinary differential equations Consider the following initial value problem GE: H+ 15y =t 1.C:y(0) = 0.5 Carry out two consecutive steps of the Euler solution from the initial condition with a time step of At = 0.2. and the predicted solutions are None of the above. y(0.2)--0.25 and y(0.4)-0.13 (0.2)-0.05 and y(0.4)-0.03 y(0.2) -- 1.00 and y(0.4)-2.04 y(0.2)-0.13 and y(0.4)-0.20
Question 23 1 pts Problem 23: Numerical solution of Ordinary differential equations Consider the following initial value problem GE: + 15y = 1.C:y(0) = 0.5 Using the results in question 21 and 22, the computed absolute value of the error estimate e for the modified Euler predicted solution using a time step of At = 0.2.is None of the above. Ec-0.12 Ec-0.42 Ec-1.42 Ec-15.42 21 Y(0.2) = 0.5 +0.77=1.27 k, = 0.2 [0.15 (0.5))=-1.5 K2=0.2 [02-151-1)] = 3.04 k=kitky =...
Question is as follows:
NOTE:
• Subject: Numerical Methods for Ordinary Differential
Equations: Initial Value Problems.
______
Useful Information
Implicit Nystrom Method: (aka Milne-Simpson)
Explicit Nystrom Method: (for reference)
Adams-Bashforth Method (AB): (for reference)
Zero-Stability:
Root Condition: (for reference)
3. Consider the 2-step Implicit Nyström Method (aka Simpson's Method): Un+2 – Un = (F(UM) + 4f(Un+1) + f(Un+2)]. (a) Determine the two characteristic polynomials, p() and o(5), for this method. Use these to show that this method is consistent. (b)...
NOTE:
• Subject: Numerical Methods for Ordinary Differential
Equations: Initial Value Problems.
1. Consider the family of linear multistep methods Un+1 = qUN+ (2(1 – a) f (Un+1) + 3a f (UN) – af (Un-1)). Ppt" (a) Determine the order of accuracy as a function of the parameter a. Find the optimal a to give the highest order of accuracy. Let's call the optimal value Qopt. (b) Is the method with Qopt zero-stable? Is the method with Qopt convergent? Explain...
please show excel formulas for both explicit and implicut
methods
please solve using excel and show formulas You are given the following system of differential equations: 99x, +2999x 2000x1 - 3000x2 If x1(0)=x2(0)-1, obtain a solution from t=0 to 0.3 using a step size of 0.03 with a. The explicit Euler's method b. The implicit Euler's method (note that this problem can be solved via a set of simultaneous linear equations for each time step) C. Plot all results on...
3. Consider the following stiff system of autonomous ordinary differential equations du f(u, u) =-3u +3, u(0)2 = ' dt de g(u, v) -2000u - 1000, v(0)-3 Note that 1 u<2 and -4 <v < 3 for all t. (a) Find the Jacobian matrix for the system of equa tions (b) Find the eigenvalues of the Jacobian matrix. (c) In the figure the shaded region shows the region of absolute stability, in the complex h plane, for third order explicit...
this is numerical analysis. Please do a and b
4. Consider the ordinary differential equation 1'(x) = f(x, y(x)), y(ro) = Yo. (1) (a) Use numerical integration to derive the trapezoidal method for the above with uniform step size h. (You don't have to give the truncation error.) (b) Given below is a multistep method for solving (1) (with uniform step size h): bo +1 = 34 – 2n=1 + h (362. Yn) = f(n=1, 4n-1)) What is the truncation...
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