a) Consider the ordinary differential equation dy = f (t, y). dt (1) By integrating equation (1) over the interval [tn, tn+1] and approximating ƒ(t, y) as a constant on this interval, derive the forward Euler scheme Yn+1 = Yn + hf (tn, Yn), (2) 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 (2) and show that it is first order. [5]
a) Consider the ordinary differential equation dy = f (t, y). dt (1) By integrating equation (1) over the interval [tn, tn+1] and approximating ƒ(t, y) as a constant on this interval, derive the forward Euler scheme Yn+1 = Yn + hf (tn, Yn), (2) 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 (2) and show that it is first order. [5]
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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![a) Consider the ordinary differential equation
dy
= f (t, y).
dt
(1)
By integrating equation (1) over the interval [tn, tn+1] and approximating ƒ(t, y) as a
constant on this interval, derive the forward Euler scheme
Yn+1 = Yn + hf (tn, Yn),
(2)
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 (2) and show that it is first order. [5]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7560de44-7bbe-4fed-b63a-532eb75ed369%2F519d8299-3530-4f58-bc55-975d1bb0dd1a%2Fnye46a_processed.png&w=3840&q=75)
Transcribed Image Text:a) Consider the ordinary differential equation
dy
= f (t, y).
dt
(1)
By integrating equation (1) over the interval [tn, tn+1] and approximating ƒ(t, y) as a
constant on this interval, derive the forward Euler scheme
Yn+1 = Yn + hf (tn, Yn),
(2)
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 (2) and show that it is first order. [5]
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