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Advanced Engineering Mathematics
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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You want to find a numerical solution to dy = -0.21 (1+y)? dx with y(0)=1. You want to use the implicit Simpson scheme (Milne's method) n- to estimate the solution at x=0.5 using a step length h= 0.25. To do this you need an estimate of y, · Use Euler's explicit method to estimate y, and as a predictor for the above scheme before taking three iterations of the implicit scheme to find and estimate of y2. Give all your answers to 5 decimal places (no more and no less). The estimate of y, is The first estimate of y, using the predictor is Using Simpson's method, the estimate of y, after 3 iterations of the scheme is y, = Find the true solution of the differential equation and calculate the correct value at x=0.5: y(0.5) = The size of the error of this first estimate = In order to improve the accuracy of your numerical estimate you are to use a power series expansion of y(x) to estimate y,. Find the expansion up to the x term, filling in the coefficients below: y(x) = + x2 + + The estimate of y, using the power series expansion up to x* is The second estimate of y, , again using the Euler forward difference scheme as a predictor and three iterations of the Simpson scheme, is y, = The size of the error of your second estimate =
Transcribed Image Text:You want to find a numerical solution to dy = -0.21 (1+y)? dx with y(0)=1. You want to use the implicit Simpson scheme (Milne's method) n- to estimate the solution at x=0.5 using a step length h= 0.25. To do this you need an estimate of y, · Use Euler's explicit method to estimate y, and as a predictor for the above scheme before taking three iterations of the implicit scheme to find and estimate of y2. Give all your answers to 5 decimal places (no more and no less). The estimate of y, is The first estimate of y, using the predictor is Using Simpson's method, the estimate of y, after 3 iterations of the scheme is y, = Find the true solution of the differential equation and calculate the correct value at x=0.5: y(0.5) = The size of the error of this first estimate = In order to improve the accuracy of your numerical estimate you are to use a power series expansion of y(x) to estimate y,. Find the expansion up to the x term, filling in the coefficients below: y(x) = + x2 + + The estimate of y, using the power series expansion up to x* is The second estimate of y, , again using the Euler forward difference scheme as a predictor and three iterations of the Simpson scheme, is y, = The size of the error of your second estimate =
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