3.0 Numerical Integration and Differentiation

Question 1

Higher-order approximations of the first derivative can be obtained by

Answer 1

involving more points on both sides of the point where the derivative is needed

Question 2

You can calculate the finite-difference approximation of the first derivative in three different ways

Answer 2

1. Backward Difference
   
   2. Central Difference
   3. Forward Difference

Question 3

Finite Difference Method

Answer 3

• Consider two points on a curve
  
  • The approximate slope is given by the first derivative
  • The close the two points to one another, the better the approximation
  • Delta x and delta y are called the finite differences
  • The slope is a finite-difference approximation of the first derivative of y with respect to x

Question 4

The finite-difference approximations can be derived for the second derivative. The first-order approximations of the second derivative are provided

Answer 4

1. Backward difference
   
   2. Central difference
   3. Forward difference

Question 5

In the higher order approximations of the second derivative

Answer 5

you can see more than 3 points are involved

Question 6

How to use Finite Difference

Answer 6

Case 1: Mathematical function is available

* Plot the function using suitable intervals and then compute the finite-difference approximations using the plot data

Case 2: Only numerical data is available

* Many relationships between variables in engineering problems are derived from experiments or field observations. In such cases, the numerical data are gathered and plotted, and then a mathematical relationship is fitted into the plot
* For example, load is applied in increments and deformation measured for each load increment in determining load-deformation relationship of building materials
* We can apply the finite-difference technique directly to the data, however we need to make sure that the noise in the data is not significant

Question 7

Noise in the Data

Answer 7

• Noise can arise from various sources instrumental or observational errors are the most common in experiments)
  
  • If you apply finite differences directly to noisy data, you might and probably will get unacceptable results
  • You can apply filtering to clean up the data as much as possible)

Question 8

If Noise is significant

Answer 8

1. You can either get a best-fit function (usually a polynomial) by curve fitting, and then differentiate numerically by the finite difference technique
   
   2. OR filter the noise out, then apply the finite-difference technique directly to the filtered numerical data

Question 9

Rules for improving accuracy of numerical differentiation

Answer 9

1. filter out the noise
2. make interval (delta x) smaller
3. central differentiation approximation

Question 10

Many engineering problems involve:

Answer 10

1. Determination of the Area under a curve
   
   2. Integration of a differential equation
      Numerical integration can be applied in both cases often enabling a relatively easy solution

Question 11

Methods of Numerical integration

Answer 11

1. Rectangular Method
   
   • Left-sided rectangle
   • Right-sided rectangle
     2. Trapezoidal method
     3. Simpson’s rule
   • It requires an odd number of data points
   • Evenly spaced data (uniform delta x)

Question 12

Generally the ___ difference gives a slightly better approximation and is most commonly used

Answer 12

central