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
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Chapter2: Second-order Linear Odes
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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]
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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