1. Numerical methods are techniques that allow mathematical problems to be solved using arithmetic operations. They have become increasingly important for engineering problem solving. 2. A mathematical model expresses the key features of a physical system using variables and equations. It defines dependent and independent variables as well as parameters and forcing functions. 3. Sources of error in numerical solutions include truncation from approximations and round-off from limited significant figures. Relative and absolute errors are used to quantify the accuracy of solutions versus the true or analytical values.

1. NUMERICAL METHODS
(Meng2063)
______________________
CHAPTER ONE
Mathematical Modelling and Error analysis
By: Tesfahun Meshesha

2. Definition
• Numerical methods: techniques by which mathematical problems are formulated so that they can
be solved with arithmetic operations.
• The role of numerical methods in engineering problem solving has increased dramatically in recent
years.
Reasons for studying Numerical Methods
1. Numerical methods are extremely powerful problem solving tools.
2. For occasions to use commercially available prepackaged computer programs that involve
numerical methods.
3. Many problems cannot be appreciated using canned programs.
4. They are efficient vehicles for learning to use computers.
5. Reinforce your understanding of mathematics

3. Mathematical Modelling
• Formulation or equation that expresses the essential features of a physical system
• A mathematical model usually describes a system by a set of variables and a set of
equations:
Where
dependent (state) variable: is a characteristic that usually reflects the behavior or
state of the system.
independent (decision) variables: are usually dimensions such as time and space,
along which the system’s behavior is determined.
 parameters (constants): are reflective of the system’s properties or composition.
forcing functions: are external influences acting up on the system
𝑫𝒆𝒑𝒆𝒏𝒅𝒆𝒏𝒕 𝒗𝒂𝒓𝒊𝒂𝒃𝒍𝒆 = 𝒇( 𝒊𝒏𝒅𝒆𝒑𝒆𝒏𝒅𝒆𝒏𝒕 𝒗𝒂𝒓𝒊𝒂𝒃𝒍𝒆𝒔, 𝒑𝒂𝒓𝒂𝒎𝒆𝒕𝒆𝒓𝒔, 𝒇𝒐𝒓𝒄𝒊𝒏𝒈 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏𝒔)

4. Cont . . .
Consider a heat exchanger system;
Model: 𝑄 = 𝑚𝑐𝑃∆𝑇
Where
𝑄 = 𝑅𝑎𝑡𝑒 𝑜𝑓 𝐻𝑒𝑎𝑡 𝑡𝑟𝑎𝑛𝑠𝑓𝑒𝑟
𝑚 = mass flow rate
𝑐𝑃 = 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐 ℎ𝑒𝑎𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑙𝑢𝑖𝑑
∆𝑇 = 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒
From the above equation
𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒: 𝑄
𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒: 𝑚, ∆𝑇
𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟: 𝑐𝑃
𝑓𝑜𝑟𝑐𝑖𝑛𝑔 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛:
• propertyies of tube material, 𝑘
• Flow orientation, parallel or Counter
Figure: Heat Exchanger
system

5. Error analysis
• Numerical technique yield usually estimates that are close to the exact analytical solution.
• So there is deviation between exact and analytical solutions, error.
• Analyzing the errors, we can find out how much our solution is correct.
Truncation and round off errors: Regarding the source
• Truncation: errors arise from the use of approximations to represent exact mathematical operations
• Round off: developed when numbers having limited significant figures are used to represent exact numbers.
For example: 𝜋
Methods of calculating the Errors
Relative error: to get a feeling on how significant an error is
Absolute error: to get a deviation of the numerical solution from analytical solution
𝜀𝑡 =
𝑡𝑟𝑢𝑒 𝑒𝑟𝑟𝑜𝑟
𝑡𝑟𝑢𝑒 𝑣𝑎𝑙𝑢𝑒
∗ 100% 𝑤ℎ𝑒𝑟𝑒 𝜀𝑡 𝑖𝑠 𝑡𝑟𝑢𝑒 𝑝𝑒𝑟𝑐𝑒𝑛𝑡 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑒𝑟𝑟𝑜𝑟
𝐸𝑎 = 𝑃𝑟𝑒𝑠𝑒𝑛𝑡 𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑖𝑜𝑛 − 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑖𝑜𝑛 × 100%

6. Accuracy & Precision
Used to characterize the errors associated with calculation and measurement
• Accuracy: refers to how closely are a computed or measured value agrees with
the true value.
• Precision: refers to how closely individual computed or measured values agree
with each other
Examples:
1. Consider a Dart shooting game
2. Consider the temperature measurement in a room.