Computer Science/Numerical Methods/ Finite difference method( backward difference)
Computer Science/Numerical Methods/ Finite difference method( backward difference)
Contents
- 1 Finite Difference Method: Backward Difference
- 2 Definition of Backward Difference
- 3 Backward Difference Approximation for Derivatives
- 4 Example Calculation
- 5 When to Use Backward Difference?
- 6 Computer Science/Numerical Methods/ Finite difference method( backward difference)
- 7 18MAT21 Module 5 NUMERICAL METHOD CONTENTS: • …
- 8 B.Tech 4th Semester MATHEMATICS- …
- 9 The finite difference method
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?