Question

How is the finite difference formulation for the first derivative related to the Taylor series expansion of the solution function?

Short answer

Answer: The finite difference formulation for the first derivative is derived from the Taylor series expansion of the solution function by isolating the first derivative term. In particular, it is derived by expanding the function at the point x + h and canceling out the function terms, leading to the approximation f'(x) ≈ (f(x + h) - f(x)) / h.

Step-by-step solution

Define finite difference formulation for the first derivative

The finite difference formulation for the first derivative approximates the derivative of a function using the difference between function values at adjacent points. For a function f(x), the forward finite difference approximation of f'(x) can be expressed as: f'(x) ≈ (f(x + h) - f(x)) / h where h is a small increment in the x-value.

Express Taylor series expansion for a function

The Taylor series is an infinite series representation of a function in the form of a sum of its polynomial derivatives evaluated at a specific point. For a function f(x) expanded around the point x = a, the Taylor series can be expressed as: f(x) = f(a) + f'(a)(x - a) + (f''(a)/2!)(x - a)^2 + (f'''(a)/3!)(x - a)^3 + ...

Apply Taylor series expansion to f(x + h)

To relate the forward finite difference approximation to the Taylor series expansion, we can expand the function f(x) at the point x + h using Taylor series: f(x + h) = f(x) + f'(x)h + (f''(x)/2!)h^2 + (f'''(x)/3!)h^3 + ...

Derive the finite difference from Taylor series expansion

Now, we want to isolate the first derivative f'(x) by rearranging the expansion of f(x + h): f'(x) ≈ (f(x + h) - f(x)) / h From the Taylor expansion, the left-hand side can be approximated as: (f(x) + f'(x)h + (f''(x)/2!)h^2 - f(x)) / h ≈ f'(x) Simplifying and canceling out the f(x) terms, we get: (f'(x)h + (f''(x)/2!)h^2) / h ≈ f'(x) Finally, dividing both sides by h: f'(x) + (f''(x)/2!)h ≈ f'(x) Since h is a small increment, (f''(x)/2!)h is much smaller in magnitude, the approximation holds: f'(x) ≈ (f(x + h) - f(x)) / h Hence, we have shown that the finite difference formulation for the first derivative is related to the Taylor series expansion of the solution function.