Different operators

 Different operators

We will study the following operators:

1) The Shifting Operator (E):

Ef(x) = f(x+h) i.e Ey₀= y₁

E²f(x) = f(x+2h) i.e E²y₀ = y₂

Eⁿf(x) = f(x+nh) i.e Eⁿy₀ = yₙ

Here n takes integral or fractional, positive or negative values.

For example:

E⁻¹f(x) = f(x-h) i.e E⁻¹ y₂ = y₁

E¹/²f(x) = f[x+(1/2)h] i.e E¹/² y₁ = y₃ₗ₂

Properties of Operator E

1) Operator E is distributive.

2) Operator E is commutative with respect to constant.

3) Operator E obeys laws of indices.

2) Forward difference operator (∆):

If x₀, x₁,x₂,......., xₙ are equally spaced with interval of differencing h and if y = f(x), then 

∆f(x) = f(x+h) - f(x)

i.e ∆yᵢ = yᵢ₊₁ - yᵢ for i = 0,1,2,3,..... n-1,

The symbol ∆ is called forward difference operator and  ∆yᵢ is called first forward difference.

Similarly, the second forward differences are ∆²yᵢ =∆yᵢ₊₁ - ∆yᵢ

For example: ∆²y₀ =∆y₁ - ∆y₀ = (y₂ - y₁)-(y₁-y₀)

                                  = y₂ -2y₁+y₀

Clearly any higher order differences can easily be expressed in terms of ordinates.

Forward difference table.

Here the first entry y₀ is called leading term and ∆y₀, ∆²y₀,.... are called leading differences.

Properties of ∆:

1) ∆[f(x)±𝝓(x)] = ∆f(x)± ∆𝝓(x)

2) ∆[cf(x)] = c∆f(x); c is constant.

3) ∆ᵐ∆ⁿf(x) = ∆ᵐ⁺ⁿf(x), m,n being positive integers.

4) ∆c = 0.

5) If given n observation (xᵢ ,yᵢ), then ∆ⁿy=0.

Example : Evaluate ∆²x³, when h= 1.

Solution. ∆²x³ = ∆[(x+1)³-x³] = ∆[x³+3x²+3x+1-x³] = ∆(3x²+3x+1) = 3∆x²+3∆x+∆1 = [3(x+1)²-3x²+3(x+1)-3x+0]

= 3x²+6x+3-3x²+3x+3-3x = 6x+6 = 6(x+1)Ans.

3) Backward difference operator (∇):

If x₀, x₁,x₂,......., xₙ are equally spaced with interval of differencing h and if y = f(x), then 

∇f(x) = f(x) - f(x-h)

i.e ∇yᵢ = yᵢ - yᵢ₋₁, for i = 1,2,3,.....,n

The symbol  ∇ is called the backward difference operator and ∇yᵢ is called first backward difference.

Also the second backward differences are 

∇²yᵢ = ∇yᵢ - ∇yᵢ₋₁, i = 2,3,4,.....,n

For example

 ∇²y₂ = ∇(∇y₂) =∇( y₂- y₁) =  ∇y₂- ∇ y₁ = (y₂- y₁) -( y₁-y₀) = y₂ -2y₁+y₀

Similarly, the higher order backward differences are defined as follows:

∇ᵏyᵢ = ∇ᵏ⁻¹yᵢ - ∇ᵏ⁻¹yᵢ₋₁

Backward difference table.

Example: Find the first backward difference of x²+2x,

Solution. We know that

 ∇f(x) = f(x) - f(x-h)

∇(x²+2x) = [x²+2x] - [(x-h)²+2(x-h)]

                =  [x²+2x] - [(x²+h²-2xh+2x-2h]

                = 2hx +2h-h² Ans.

4) Central difference operator (𝜹):

If x₀, x₁,x₂,......., xₙ are equally spaced with interval of differencing h and if y = f(x), then the first order central difference of y is defined as:

𝜹f(x+h/2) = f(x+h) - f(x) or

𝜹f(x) = f(x+h/2) - f (x-h/2)

The symbol 𝜹 is called the central difference operator.

Central difference table.

5) Averaging operator (𝝁):

𝝁f(x) = [f(x+h/2) + f(x-h/2)]/2

6) Differential operator (D):

Df(x) = f'(x)

D²f(x) = f"(x) and so on.