Jawaharlal Nehru Technological University Hyderabad B.Tech I Year MATHEMATICAL METHODS Previous Question Paper (July - 2021 - R13)

Using Newton's forward difference formula, we construct the difference table for x = {0, 1, 2, 3} and y = {2, 5, 13, 35}:

Here, x₀ = 0, h = 1, and the value to interpolate is x = 2.5. Thus, u = (x - x₀) / h = (2.5 - 0) / 1 = 2.5. Newton's forward difference formula is:

Substituting the values y₀ = 2, Δy₀ = 3, Δ²y₀ = 5, and Δ³y₀ = 7:

To prove Δ∇ = Δ - ∇, we apply the operators to a function f(x):

Now consider (Δ - ∇)f(x):

Since both expressions yield the same result, Δ∇ = Δ - ∇. For the second identity, the forward difference operator is defined as Δf(x) = f(x+h) - f(x). The shifting operator E is defined as Ef(x) = f(x+h). Substituting this into the definition of Δ:

Therefore, Δ = E - 1, which implies E = 1 + Δ.