u'(t) = 20u(t) u(t) 10+0.1 sin(2лt), Exercise 3.3.3 Consider the logistic ODE (3.23) with initial condition u(0) = 0.5. To ten significant figures, the solution at t = 1 is u(1) = 9.971345698. (a) Estimate u(1) by using Euler's method with steps of size h = 0.1, and compute the error. (b) Estimate u(1) by using the improved Euler's method with steps of size h = 0.1, and compute the error. (c) Estimate u(1) by using the RK4 method with steps of size h = 0.1, and compute the error. (d) Estimate u(1) by using whatever numerical ODE solver you have available, without specifying the method, then compute the error. Most software, e.g., Mathematica, Maple, Matlab, and Sage, will use a good ODE solver with adaptive stepsizing for error control. (e) Explore the error tolerances in the solver you are using. For example, the Maple dsolve command, when solving numerically using the usual RKF45 method, accepts arguments abserr (default value 1.0 × 107) and relerr (default value 1.0 × 10-6) that can be used to control error. In Matlab the ode45 command accepts arguments AbsTol (default value 1.0 × 10-6) and RelTol (default value 1.0 × 10-³). In Mathematica the NDSolve command has arguments AccuracyGoal and PrecisionGoal.

u'(t) = 20u(t) u(t) 10+0.1 sin(2лt), Exercise 3.3.3 Consider the logistic ODE (3.23) with initial condition u(0) = 0.5. To ten significant figures, the solution at t = 1 is u(1) = 9.971345698. (a) Estimate u(1) by using Euler's method with steps of size h = 0.1, and compute the error. (b) Estimate u(1) by using the improved Euler's method with steps of size h = 0.1, and compute the error. (c) Estimate u(1) by using the RK4 method with steps of size h = 0.1, and compute the error. (d) Estimate u(1) by using whatever numerical ODE solver you have available, without specifying the method, then compute the error. Most software, e.g., Mathematica, Maple, Matlab, and Sage, will use a good ODE solver with adaptive stepsizing for error control. (e) Explore the error tolerances in the solver you are using. For example, the Maple dsolve command, when solving numerically using the usual RKF45 method, accepts arguments abserr (default value 1.0 × 107) and relerr (default value 1.0 × 10-6) that can be used to control error. In Matlab the ode45 command accepts arguments AbsTol (default value 1.0 × 10-6) and RelTol (default value 1.0 × 10-³). In Mathematica the NDSolve command has arguments AccuracyGoal and PrecisionGoal.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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