NUMERICAL METHOD CHAPTE 1 numerical solution part 1.pptx

NUMERICAL METHOD CHAPTER 1 numerical solution part 2 .pptx

1. Addis Ababa Science and Technology University College of Engineering Environmental Engineering (4 th Year) Course: Numerical Analysis for Environmental Engineering (EnEg-4103) Prerequisite: Math and Programming BY SHELEMA. A 2025

2. Course contents Chapter 1: Basic concepts in error estimation Chapter 2: Numerical solution of non-linear and Transcendental equations Chapter 3: Numerical solution of linear equations , eigen values & eigen vectors Chapter4: Interpolations, finite differences and Curve fitting Chapter 5: Numerical Differentiation & Integration chapter 6: Numerical Solutions of Ordinary Differential Equations 7 Application of Software’s

3. ASSESSMENTS Continuous Assessment (50%) Quiz /attendance : 5 % Tests: 15% Assignments: 20% Project: 10% Final Exam (50)

4. Chapter 1: Basic concepts in error estimation Introduction to Numerical analysis Sources of errors Absolute and relative errors Approximations of errors Truncation errors and the Maclaurin series Propagation of errors

5. Introduction Numerical analysis Numerical methods in environmental engineering are computational techniques that use algorithms to approximate solutions to complex mathematical problems. which can be used in simulating environmental systems and analyzing data. These methods, are crucial for modeling pollution dispersal, water resources, and ecosystem behavior It Enable engineers to design and manage systems more effectively using tools like MATLAB or Python.

6. Introduction Numerical analysis Numerical analysis is the branch of mathematics that is used to find approximations to difficult problems such as: Finding the roots of non−linear equations Integration involving complex expressions solving systems of equations solving differential equations for which analytical solutions do not exist

7. Introduction Numerical analysis Numerical analysis is the study of algorithms that use numerical approximation or development of algorithms for solving all kinds of problems of continuous mathematics. It used for solving mathematical problems that cannot be solved or are difficultto solve analytically. An analytical solution is an exact answer in the form of a mathematical expression in terms of the variables associated with the problem that is being solved. A numerical solution is an approximate numerical value (a number) for the solution.

8. Why numerical methods are essential? Complexity of Environmental Systems: Environmental systems are often too complex to be solved with analytical or classical mathematical methods. Data Analysis: They provide ways to analyze and interpret large environmental datasets, helping to extract meaningful information and answer research questions. Modeling and Prediction : Numerical methods allow for the creation of models that simulate environmental processes, predict future conditions, and assess the impact of various interventions. Computer-Aided Solutions : They harness the power of computers to perform complex calculations and solve problems involving large systems of equations, nonlinearities, and intricate geometries.

9. Key applications in environmental engineering: Root-finding methods: Used to find solutions for equations where the output is zero. Numerical integration : Essential for calculating areas under curves or volumes, which are common in environmental calculations. Solving Ordinary and Partial Differential Equations: Used to model dynamic environmental processes like fluid flow, heat transfer, and chemical reactions. Optimization : Employed to find the best solutions for environmental problems, such as designing the most efficient wastewater treatment system.

10. 1.2 Errors and Sources of errors In numerical methods, error estimation involves   quantifying the discrepancy between an approximate numerical solution and the true, exact solution.   The two main types of errors are   Round-off error,   which arises from computer limitations in storing numbers with finite precision, and   Truncation error ,   which results from approximating an exact mathematical process (like an infinite series) with a finite one.   Additionally, Modeling errors stem from simplifications in the mathematical model itself, and Data errors are inaccuracies present in the initial input data.  

11. Round off Error: which result when numbers having limited significant figures are used to represent exact numbers and Caused by representing a number approximately. p =3.14159265358 p =3.141592

12. TRUNCATION ERROR TRUNCATION ERROR : Error caused by truncating or approximating a mathematical procedure.

13. Maclaurin series

14. 2 derivating fuction using approxmation

15. Types of error , Absolute and relative errors True Error : Defined as the difference between the true value in a calculation and the approximate value found using a numerical method etc. True Error = True Value – Approximate Value Relative True Error: Defined as the ratio between the true error, and the true value. Approximate Error : error is defined as the difference between the present approximation and the previous approximation. Relative Approximate Error: Defined as the ratio between the approximate error and the present approximation. ) = Present Approximation – Previous Approximation Approximate Error (

16. Absolute and relative errors

17. Example 1 on types error d) find the relative true error for

18. (d) relative true error . From part b, Relative True Error is defined as as a percentage,

19. Exercise 1 a) Aproximate value b) True error c) relative true error d) read it for

20. SOLUTION

21. Example 2 on types of error

22. Exercis 2 Find Relative Approximate Error for example 2

23. Uncertainity and Significant Figuree Only rarely given data are exact, since they originate from measurements. Therefore there is probably error in the input information. The question is “how much error is present in our calculation and is it tolerable?” Accuracy : How close is a computed or measured value to the true value Precision (or reproducibility): How close is a computed or measured value to previously computed or measured values. Inaccuracy (or bias): A systematic deviation from the actual value. Imprecision (or uncertainty): Magnitude of scatter.

24. SIGNIFICANT FIGURES A significant figure is any digit 1 to 9 and any zero which is not a place holder . 1.350 -- 4 significant figures: since the zero is not needed to make sense of the number. 0.00320 -- 3 significant figures: the first three zeros are just place holders. Number of significant figures indicates precision. Significant digits of a number are those that can be used with confidence. e.g., the number of certain digits plus one estimated digit.  53,800 How many significant figures?

25. Brain storming How many significant figures are there in each of the following?    (1) 0.00042   (2) 0.14700   (3) 4.2 x 10 6 (4) -154.090 x 10 -27

26. Propagation of errors .A "numerical method propagation of errors" refers to techniques for tracking how uncertainties and inaccuracies from input data and calculations accumulate and influence the final result in a numerical computation. The purpose of this section is to study how errors in numbers can propagate through mathematical functions. If we multiply two numbers that have errors,we would like to estimate the error in the product. Functions of a Single Variable Functions of More than One Variable

27. . F ind the bounds for the propagation in adding two numbers. For example if one is calculating X + Y where X = 1.5 ± 0.05 Y = 3.4 ± 0.04 Solution Maximum possible value of X = 1.55 and Y = 3.44 Maximum possible value of X + Y = 1.55 + 3.44 = 4.99 Minimum possible value of X = 1.45 and Y = 3.36. Minimum possible value of X + Y = 1.45 + 3.36 = 4.81 Hence 4.81 ≤ X + Y ≤4.99.