4. (a) Consider the ordinary differential equation dy dt = f (t, y). (3) By integrating equation (3) over the interval [tn, tn+1] and approximating f(t, y) as a constant on this interval, derive the forward Euler scheme Yn+1 = Yn + hf (tn, Yn), (4) with the step-size h = 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 (4) and show that it is first order. (b) Taking f(t, y) = −λy and the initial condition yo = 1, obtain the exact solution to the ordinary differential equation and determine its behaviour in the limit t→ ∞. (c) Show that the value of the approximation at the nth step can be written as Уп Yo (1 - Ah)". (5) = Using 5 and step-sizes of h € 0.25 and h = = 0.5, calculate the approximation = of y at t = 1. Determine the absolute errors using the exact solution in 4.b. Do the numerical solutions behave in the same way as the exact solution? (d) Using equation (5), determine the condition on h that ensures stability of the Euler scheme. Does this stability criterion explain the behaviour of solutions in 4.c?
4. (a) Consider the ordinary differential equation dy dt = f (t, y). (3) By integrating equation (3) over the interval [tn, tn+1] and approximating f(t, y) as a constant on this interval, derive the forward Euler scheme Yn+1 = Yn + hf (tn, Yn), (4) with the step-size h = 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 (4) and show that it is first order. (b) Taking f(t, y) = −λy and the initial condition yo = 1, obtain the exact solution to the ordinary differential equation and determine its behaviour in the limit t→ ∞. (c) Show that the value of the approximation at the nth step can be written as Уп Yo (1 - Ah)". (5) = Using 5 and step-sizes of h € 0.25 and h = = 0.5, calculate the approximation = of y at t = 1. Determine the absolute errors using the exact solution in 4.b. Do the numerical solutions behave in the same way as the exact solution? (d) Using equation (5), determine the condition on h that ensures stability of the Euler scheme. Does this stability criterion explain the behaviour of solutions in 4.c?
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
![4. (a) Consider the ordinary differential equation
dy
dt
= f (t, y).
(3)
By integrating equation (3) over the interval [tn, tn+1] and approximating f(t, y)
as a constant on this interval, derive the forward Euler scheme
Yn+1 = Yn + hf (tn, Yn),
(4)
with the step-size h
=
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 (4) and show that it is first order.
(b) Taking f(t, y) = −λy and the initial condition yo
=
1, obtain the exact solution to
the ordinary differential equation and determine its behaviour in the limit t→ ∞.
(c) Show that the value of the approximation at the nth step can be written as
Уп
Yo (1 - Ah)".
(5)
=
Using 5 and step-sizes of h
€ 0.25 and h
=
= 0.5, calculate the approximation
=
of y at t = 1. Determine the absolute errors using the exact solution in 4.b. Do
the numerical solutions behave in the same way as the exact solution?
(d) Using equation (5), determine the condition on h that ensures stability of the Euler
scheme. Does this stability criterion explain the behaviour of solutions in 4.c?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F69cce4ac-4bf6-4e6b-8636-bf160e045b58%2F8250ab50-cb79-4a14-b60f-203868b873ba%2Frkn56j4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4. (a) Consider the ordinary differential equation
dy
dt
= f (t, y).
(3)
By integrating equation (3) over the interval [tn, tn+1] and approximating f(t, y)
as a constant on this interval, derive the forward Euler scheme
Yn+1 = Yn + hf (tn, Yn),
(4)
with the step-size h
=
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 (4) and show that it is first order.
(b) Taking f(t, y) = −λy and the initial condition yo
=
1, obtain the exact solution to
the ordinary differential equation and determine its behaviour in the limit t→ ∞.
(c) Show that the value of the approximation at the nth step can be written as
Уп
Yo (1 - Ah)".
(5)
=
Using 5 and step-sizes of h
€ 0.25 and h
=
= 0.5, calculate the approximation
=
of y at t = 1. Determine the absolute errors using the exact solution in 4.b. Do
the numerical solutions behave in the same way as the exact solution?
(d) Using equation (5), determine the condition on h that ensures stability of the Euler
scheme. Does this stability criterion explain the behaviour of solutions in 4.c?
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