Question 1: (a) Consider the ordinary differential equation dy = f (t, y). dt (1) By integrating equation (1) over the interval [tn, tn+1] and approximating f(t, y) as a constant on this interval, derive the forward Euler scheme with the step-size h = Yn+1 = yn+hf (tn, Yn), tn+1 — tn (for all n). Define the local truncation error of a numerical scheme, obtain an expression for the local truncation error of the forward Euler scheme (2) and show that it is first order. (b) Verify that y(t) = (1+t) et is the solution to the initial value problem dy dt - y, y(0) 1. Use the forward Euler method (2) with a step-size of h = 1/2 to calculate an approxi- mation of Y at t = 1. Calculate the absolute error from the exact solution. (c) By finding a suitable quadrature formula for integration of equation (1) over the interval [tn, tn+1], derive the two-step Adams-Bashforth scheme Yn+1= Using this scheme, with h = Yn + [3f (tn, Yn) — f (tn−1, Yn−1)]. - 1/2 and the value of y₁ obtained from the forward Euler method (2) in part ((b)), compute the value of y2 for equation (3). Calculate the error from the exact solution.

Question 1: (a) Consider the ordinary differential equation dy = f (t, y). dt (1) By integrating equation (1) over the interval [tn, tn+1] and approximating f(t, y) as a constant on this interval, derive the forward Euler scheme with the step-size h = Yn+1 = yn+hf (tn, Yn), tn+1 — tn (for all n). Define the local truncation error of a numerical scheme, obtain an expression for the local truncation error of the forward Euler scheme (2) and show that it is first order. (b) Verify that y(t) = (1+t) et is the solution to the initial value problem dy dt - y, y(0) 1. Use the forward Euler method (2) with a step-size of h = 1/2 to calculate an approxi- mation of Y at t = 1. Calculate the absolute error from the exact solution. (c) By finding a suitable quadrature formula for integration of equation (1) over the interval [tn, tn+1], derive the two-step Adams-Bashforth scheme Yn+1= Using this scheme, with h = Yn + [3f (tn, Yn) — f (tn−1, Yn−1)]. - 1/2 and the value of y₁ obtained from the forward Euler method (2) in part ((b)), compute the value of y2 for equation (3). Calculate the error from the exact solution.

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter11: Differential Equations

Section11.CR: Chapter 11 Review

Problem 12CR

See similar textbooks