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.
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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![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F69cce4ac-4bf6-4e6b-8636-bf160e045b58%2F5a2a3093-b1e0-4072-bbd2-939a97c121f4%2F3y42ivh_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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.
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