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Computer Science/Numerical Methods/ Finite difference method( backward difference)

Ali Seron Jason
 Ali Seron Jason

Computer Science/Numerical Methods/ Finite difference method( backward difference)

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Finite Difference Method: Backward Difference

The Finite Difference Method (FDM) is a numerical technique used for approximating derivatives. The backward difference method is a commonly used approach when calculating derivatives at a given point using previous data points.

 Definition of Backward Difference

The backward difference of a function f(x)f(x) at a point xix_i with step size hh is given by:

Δbf(xi)=f(xi)−f(xi−1)\Delta_b f(x_i) = f(x_i) – f(x_{i-1})

For higher-order differences, we define:

Δb2f(xi)=Δbf(xi)−Δbf(xi−1)\Delta^2_b f(x_i) = \Delta_b f(x_i) – \Delta_b f(x_{i-1}) Δb3f(xi)=Δb2f(xi)−Δb2f(xi−1)\Delta^3_b f(x_i) = \Delta^2_b f(x_i) – \Delta^2_b f(x_{i-1})

 Backward Difference Approximation for Derivatives

The first derivative using backward difference is approximated as:

f′(xi)≈f(xi)−f(xi−1)hf'(x_i) \approx \frac{f(x_i) – f(x_{i-1})}{h}

For the second derivative:

f′′(xi)≈f(xi)−2f(xi−1)+f(xi−2)h2f”(x_i) \approx \frac{f(x_i) – 2f(x_{i-1}) + f(x_{i-2})}{h^2}

 Example Calculation

Given Data Points:

xx 1.0 1.2 1.4
f(x)f(x) 2.718 3.320 4.055

Step Size: h=0.2h = 0.2

Using the backward difference formula:

f′(1.4)≈f(1.4)−f(1.2)hf'(1.4) \approx \frac{f(1.4) – f(1.2)}{h} f′(1.4)≈4.055−3.3200.2=3.675f'(1.4) \approx \frac{4.055 – 3.320}{0.2} = 3.675

Final Answer: f′(1.4)≈3.675f'(1.4) \approx 3.675

 When to Use Backward Difference?

 When you have data points and want to approximate derivatives using previous values.
 When solving numerical PDEs like heat equations and wave equations.
 In finite difference schemes for boundary conditions in numerical simulations.

Would you like a Python code to implement this?

Computer Science/Numerical Methods/ Finite difference method( backward difference)

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