Week 2 M1Lesson 1.pptx

Module 1 Numerical Methods: Introduction, Discrete Algebra, Accuracy, Errors  M1: Numerical Methods: Introduction, Discrete Algebra, Accuracy, Errors L1: ANALYSIS VS NUMERICAL ANALYSIS Learning Outcome Explain the importance of Numerical Analysis in Solving Engineering P...

Module 1 Numerical Methods: Introduction, Discrete Algebra, Accuracy, Errors  M1: Numerical Methods: Introduction, Discrete Algebra, Accuracy, Errors L1: ANALYSIS VS NUMERICAL ANALYSIS Learning Outcome Explain the importance of Numerical Analysis in Solving Engineering Problems. What is Analysis in Mathematics? to solve an engineering problem through equations which can be simplified to an answer through the procedures of Algebra, Calculus, Differential Equations, Partial Differential Equations, and etc. the OLD and COMMON way of solving engineering problems analytical method is the term referring to the way of solving problem using analysis. Algebra Problem using Analytical Method I am three times as old as you were when i was as old as you are now, When you got to be my age together our ages will be 84. How old are we now? What are we looking for? Tabular method as tool for analyzing age problem Past (-a Person years) Present Future (+b years) X – my age Y – your age X (I) X–a X X+b Y (you) Y–a Y Y+b Algebra Problem using Analytical Person Past (-a Method years) Present Future (+b years) X (I) X–a X X+b Y (you) Y–a Y Y+b Let’s interpret each statement in the problem 1. I am three times as old as you were… 3. When you got to be my age… X = 3(Y – a) Y+b=X 2. …when i was as old as you are now. 4. …together our ages will be 84. X-a=Y X + b + Y + b = 84 Algebra Problem using Analytical Person Past (-a Method years) Present Future (+b years) X = 3(Y – a) Y+b=X X (I) X–a X X+b X-a=Y Y (you) Y–a Y Y+b X + b + Y + b = 84 Further simplifying the equations to solve the unknowns 5. Combine equations involving the var. a. 7. Rearranging the 2 equations. X = 3(Y – (X – Y)) 4X – 6Y = 0 6. Combine equations involving var. b. 3X – Y = 84 X + X + (X – Y) = 84 Algebra Problem using Analytical Method I am three times as old as you were when i was as old as you are now, When you got to be my age together our ages will be 84. How old are we now? Final answer Check the solution from the given equations X – 36 yrs. old 4(36) – 6(24) = 0 Y – 24 yrs. old 3(36) – 24 = 84 Differential Equation Problem using Analytical Method Find the general solution of the DE What are we looking for? Possible analytical methods to be used Methods of Undetermined Coefficients General Solution of DE (MUC) Method of reduction of order Methods of Variation of Parameters (VP) Differential Equation Problem using Analytical Method Find the general solution of the DE 1. Finding the Complimentary Solution Equation has real and distinct roots Equation has real and repeated roots Equation has conjugate complex roots Differential Equation Problem using Analytical Method Find the general solution of the DE 2. Finding the Particular Solution Differential Equation Problem using Analytical Method Find the general solution of the DE 3. Generating the final answer. Differential Equation Problem using Analytical Method Table for Solving using Methods of Undetermined Coefficients (MUC) or or Some drawbacks of Analytical Method? Time consuming during a manual solution process. Impractical for nonlinear systems and complex geometry. Fluid flow through a pipe Non-uniform stress 3-dimensional Heat Transfer distribution in a body In some cases, the solutions are complicated and very lengthy. that is why engineers and scientists used numerical method… What is a Numerical Analysis in Mathematics? a method of solving engineering problems by using the fundamental arithmetic operation; addition, subtraction, multiplication, division, and compare. Since arithmetic operations are exactly what computers do, numerical analysis is strongly connected to the computer. Computer performs arithmetic operations using the binary number system comprising of ones and zeroes. What is a Numerical Analysis in Mathematics? Supposed we are to find the value of cube root of 2 without using a calculator and you forgot everything about math except for the fundamental arithmetic operation… One way would be trial and error… but this is not the exact value… What is a Numerical Analysis in Mathematics? From the previous situation, it is noted that Numerical Analysis gives only approximate result. And this result would posed some errors. That is why a good Numerical Method ensures that the approximate results are stable and much closer to the true value and are less likely to catch errors. To achieve higher accuracy and stability, computer software are being utilized such as MATLAB, Mathcad, and others. Analysis vs Numerical Analysis? Differ only NUMERICAL ANALYSIS on the ANALYSIS SOLUTION Scope of Numerical Analysis? 1. Given raw data, solve 2. Linear algebraic the roots of its equation equations given the f(x) = 0 for x. coefficients, solve values of x. Scope of Numerical Analysis? 3. Data Optimization. 4. Curve Fitting Determine x that gives optimum f(x). Scope of Numerical Analysis? 5. Integration. Find the 6. Ordinary differential are under the curve. equations Scope of Numerical Analysis? 7. Partial differential equations 1. Algebra Problem Using Numerical Method Find the value of x in the equation 2x – 3 = x + 4 on the interval [2,9]. One way would be trial and error and perform the arithmetic operation… 1. at x = 2 3. at x = 4 5. at x = 6 7. at x = 8 𝟏≠𝟔 𝟓 ≠𝟖 𝟗 ≠𝟏𝟎 𝟏𝟑≠𝟏𝟐 2. at x = 3 4. at x = 5 6. at x = 7 8. at x = 9 𝟑 ≠𝟕 𝟕 ≠𝟗 𝟏𝟏=𝟏𝟏 𝟏𝟓≠𝟏𝟑 1. Algebra Problem Using Numerical Method Find the value of x in the equation 2x – 3 = x + 4 on the interval [2,9]. but why does the solution seem long??? The computer perform this long and iterative arithmetic operation in just a second or less. 2. Algebra Problem Using Numerical What is the closestMethod range of integer values of x that would satisfy the equation on the interval [5,18]? One way would be trial and error and perform the arithmetic operation… 1. at x = 12 3. at x = 14 True Value 𝟏𝟒≠𝟏𝟓.𝟓 𝟏𝟔≠𝟏𝟓.𝟕𝟓 2. at x = 13 4. at x = 15 Approximate Value 𝟏𝟓≠𝟏𝟓. 𝟔𝟐𝟓 𝟏𝟕≠𝟏𝟓.𝟖𝟕𝟓 𝒙 ≈[𝟏𝟑 ,𝟏𝟒] DE Problem Using Analytical Method A parachutist of mass 68.1 kg jumps out of a stationary hot air balloon. It is subject to air resistance directly proportional to the velocity with drag coefficient equal to 12.5 kg/s. Tabulate the values of velocity prior to opening the chute until achieving terminal speed. DE Problem Using Analytical Method Create Free Body Diagram and Formulate the differential equation DE Problem Using Analytical Method Solve the Differential Equation 𝒎𝒈 −𝒄𝒗 =𝒎 𝒅𝒗 𝒅𝒕 DE Problem Using Analytical Method Use the general solution to solve 𝒄 𝒈𝒎 −( 𝒎 𝒕) 𝒗 (𝒕 )= [𝟏 −𝒆 ] values of v up to terminal speed 𝒄 DE Problem Using Numerical 𝒄 𝒅𝒗 Method 1. Use Euler Method/Finite divided Difference Method 𝒈− 𝒎 𝒗= 𝒅𝒕 DE Problem Using Numerical 𝒄 𝒅𝒗 Method 2. Solve for 𝒈− 𝒎 𝒗= 𝒅𝒕 DE Problem Using Numerical Method 3. Solve for velocity at any time interval 𝒄 𝒗 ( 𝒕 𝒊+𝟏 ) =𝒗 ( 𝒕 𝒊 ) +[ 𝒈 − 𝒗 ( 𝒕 𝒊 ) ] (𝒕 𝒊 +𝟏 − 𝒕 𝒊) 𝒎 1. at t = 0 to t = 2 2. at t = 2 to t = 4 let’s compare the two methods… Analytical Method vs Numerical Method ANALYTICAL NUMERICAL Analysis vs Numerical Analysis Analysis uses equations derived from algebra, calculus and DE to come up an exact answer while Numerical Analysis utilizes arithmetic operations to solve complex problems with approximate answers. Analysis does not perform well for non-linear systems or complex geometry while Numerical Analysis, with the aid of computer software, can perform it in a matter of second or less. Assignment!!! Review problems in Algebra, Trigonometry, Differential Calculus, Integral Calculus and Differential Equations "So do not fear, for I am with you; do not be dismayed, for I am your God. I will strengthen you and help you; I will uphold you with my righteous right hand." Isaiah 41:10 God Bless!