Part (a)
Forward Difference Operator (
Δ
𝑓
(
𝑥
)
Δf(x)):
The forward difference operator is defined as:
Δ
𝑓
(
𝑥
)
=
𝑓
(
𝑥
+
ℎ
)
−
𝑓
(
𝑥
)
,
Δf(x)=f(x+h)−f(x),
where
ℎ
h is the step size.
Backward Difference Operator (
∇
𝑓
(
𝑥
)
∇f(x)):
The backward difference operator is defined as:
∇
𝑓
(
𝑥
)
=
𝑓
(
𝑥
)
−
𝑓
(
𝑥
−
ℎ
)
.
∇f(x)=f(x)−f(x−h).
Sum of Forward and Backward Differences (
Δ
+
∇
Δ+∇):
Adding
Δ
Δ and
∇
∇:
Δ
𝑓
(
𝑥
)
+
∇
𝑓
(
𝑥
)
=
[
𝑓
(
𝑥
+
ℎ
)
−
𝑓
(
𝑥
)
]
+
[
𝑓
(
𝑥
)
−
𝑓
(
𝑥
−
ℎ
)
]
.
Δf(x)+∇f(x)=[f(x+h)−f(x)]+[f(x)−f(x−h)].
Simplify:
Δ
𝑓
(
𝑥
)
+
∇
𝑓
(
𝑥
)
=
𝑓
(
𝑥
+
ℎ
)
−
𝑓
(
𝑥
−
ℎ
)
.
Δf(x)+∇f(x)=f(x+h)−f(x−h).
Central Difference Operator (
𝛿
𝜇
δμ):
The central difference operator is defined as:
𝛿
𝜇
=
𝑓
(
𝑥
+
ℎ
)
−
𝑓
(
𝑥
−
ℎ
)
2
.
δμ=
2
f(x+h)−f(x−h)
.
Proof:
From the above, divide the sum of
Δ
Δ and
∇
∇ by 2:
𝛿
𝜇
=
1
2
(
Δ
+
∇
)
.
δμ=
2
1
(Δ+∇).
Thus, we have shown that:
𝛿
𝜇
=
1
2
(
Δ
+
∇
)
.
δμ=
2
1
(Δ+∇).
Mathbot Says...
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