Exam 8: Numerical Methods

Consider the initial value problem This question is related to using the fourth-order backward differentiation formula to estimate the solution y(0.2) using a step size of h = 0.05 . For the following problem, you will need these values to carry out the computation: Which of the following expressions represents the error incurred in using this method to estimate y(0.2)?

Consider the initial value problem This question is related to using the predictor-corrector method to estimate the solution y(0.2) using a step size of h = 0.05. Which of the following is the correct formula for

For the following differential equation, use the fourth-order Adams-Moulton formula to estimate the solution at t = 0.4 using a step size of h = 0.1. To start the process, you are given some Y_i values. Calculate f₄ = ________

For the following differential equation, use the fourth-order Adams-Moulton formula to estimate the solution at t = 0.4 using a step size of h = 0.1. To start the process, you are given some Y_i values. Using the fourth-order Adams-Moulton formula, approximate Y₅. Express your answer accurate to 7 decimal places. Y₅ = ________

Consider the following initial value problem The following question pertains to the various computational steps and identifications involved in applying Runge-Kutta's 4th order method to evaluate y(0.55). x₀ = ________

Consider the initial value problem This question relates to using the Euler method to approximate the solution of this problem at t = 0.5 using a step size of h = What are the correct values of

Consider the initial value problem This question is related to using the Runge-Kutta method for approximating the solution y at with a step size of h = 0.05. To find Y₂ , first calculate the following in order to apply the Runge-Kutta method. Express your answers accurate to seven decimal places. (i) K₁ = ________ (ii) K₂ = ________ (iii) K₃ = ________ (iv) K₄= ________ (v) So, using the Runge-Kutta method, Y₂ = ________

For the following differential equation, use the fourth-order Adams-Moulton formula to estimate the solution at t = 0.4 using a step size of h = 0.1. To start the process, you are given some Y_i values. Calculate the following approximations. Express your answers accurate to 7 decimal places. (i) f₀ = ________ (ii) f₁ = ________ (iii) f₂ = ________ (iv) f₃ = ________

Consider the initial value problem This question is related to using the predictor-corrector method to estimate the solution y(0.2) using a step size of h = 0.05. For this problem, you will need these values to carry out the computations: Which of the following is the correct formula for

Consider the initial value problem Use the backward Euler method with step size h = 0.01 to find the following approximation. Express your answer to 5 decimal places. Y₁________

Consider the initial value problem This question is related to using the fourth-order backward differentiation formula to estimate the solution y(0.2) using a step size of h =0.05 . For the following problem, you will need these values to carry out the computation: Which of the following is the value of

Consider the system of initial value problems given by This problem can be expressed using matrix notation as where When using the improved Euler method with h = 0.05, ________

For the following differential equation, use the fourth-order Adams-Moulton formula to estimate the solution at t = 0.4 using a step size of h = 0.1. To start the process, you are given some Y_i values. Using the fourth-order Adams-Moulton formula, approximate Y₄ . Express your answer accurate to 7 decimal places. Y₄ = ________

Consider the initial value problem (Note: The exact solution is Using a step size of h = 0.25, compute the following approximations. Give your answers accurate to 6 decimal places. (i) = ________ (ii) = ________ (iii) = ________ (iv) = ________

Consider the initial value problem This question is related to using the predictor-corrector method to estimate the solution y(0.2) using a step size of h = 0.05. For this problem, you will need these values to carry out the computations: Which of the following is the predicted value for

Consider the initial value problem Use the backward Euler method with step size h = 0.01 to find the following approximation. Express your answer to 5 decimal places. Y₂________

Consider the following initial value problem on the interval [0, 1] Approximate the solution y(t) at t = 2.5 using the fourth-order backward differentiation formula with step size h = 0.05. Using the Runge-Kutta method you are provided with some initial Y_n values to start the fourth-order backward differentiation formula. Y₁ = 2.157777 Y₂= 2.3322791 Y₃ = 2.5253944 Use the fourth-order backward Euler differentiation formula to compute the following approximation. Express your answer accurate to 6 decimal places. Y₄ = ________

Consider the system of initial value problems given by This problem can be expressed using matrix notation as When using the Euler method with h = 0.05, what are the values of when using the Euler method?

Consider the following initial value problem The following question pertains to the various computational steps and identifications involved in applying Runge-Kutta's 4th order method to evaluate y(0.6). To compute identify the parameter below. In what follows, K₃ = ________

Consider the initial value problem Use the backward Euler method with step size h = 0.01 to find the following approximation. Express your answer to 5 decimal places Y₃________