525602:Advanced Numerical Methods for ME School of Mechanical Engineering

Prescribed text : Numerical Method for Engineering, Seventh Edition, Steven C.Chapra, Raymond P. Canale.,McGraw Hill 2014

Recommended reading : Numerical Methods for Engineers and Scientists, Amos Gilat and Vish Subramaniam, Wiley,2008 Numerical Methods Using MATLAB, John.H.Methews., Kurtis D.Fink.,Prentice Hall , Fourth Edition,2004 ระเบียบวิธีเชิงตัวเลขในงานวิศวกรรม, ปราโมทย์ เดชะอำไพ., สำนักพิมพ์แห่งจุฬาลงกรณ์มหาวิทยาลัย พ.ศ.2556

Course Description Truncation errors and the Taylor series Numerical Solutions for ODEs Numerical Solutions for PDEs Finite difference method Optimization

Errors in Numerical Solutions Truncation Errors Round-Off Errors Total Error Local Error Global Error

Truncation Errors :The Taylor series Zero-order approximation First-order approximation

The Taylor series approximation

The Remainder for the Taylor Series Expansion

Truncation Errors: Example Taylor’s series expansion: The exact value: The Zero-order approximation; The truncation error:

Truncation Errors: Example Taylor’s series expansion: The exact value: The first-order approximation; The truncation error:

Round-Off Errors

Total Error The true error: The true relative error:

Total Error The Global Error is the total discrepancy due to past as well as present steps The Local Error refers to the error incurred over a single step. It is calculated with a Taylor series expansion.

The Global Error The relative global error:

The Total Error: Example

The Total Error: Example The percent relative global error (Step 1); The percent relative global error (Step 2);

The Total Error: Example Exact estimates of the errors in Euler’s method Problem statement:

The Total Error: Example The percent relative local error (Step 1);

The Total Error: Example The percent relative local error (Step 2);

Numerical Solutions for ODEs Introduction to Ordinary Differential Equations (ODE)

Study Objectives Solve Ordinary differential equation (ODE) problems. Appreciate the importance of numerical method in solving ODE. Assess the reliability of the different techniques. Select the appropriate method for any particular problem.

Computer Objectives Develop programs to solve ODE. Use software packages to find the solution of ODE

Learning Objectives of Lesson 1 Recall basic definitions of ODE, order, linearity initial conditions, solution, Classify ODE based on( order, linearity, conditions) Classify the solution methods

Derivatives Derivatives Partial Derivatives Ordinary Derivatives v is a function of one independent variable Partial Derivatives u is a function of more than one independent variable

Differential Equations Ordinary Differential Equations involve one or more Ordinary derivatives of unknown functions Partial Differential Equations involve one or more partial derivatives of unknown functions

Ordinary Differential Equations Ordinary Differential Equations (ODE) involve one or more ordinary derivatives of unknown functions with respect to one independent variable x(t): unknown function t: independent variable

Order of a differential equation The order of an ordinary differential equations is the order of the highest order derivative First order ODE Second order ODE Second order ODE

Solution of a differential equation A solution to a differential equation is a function that satisfies the equation.

Linear ODE Linear ODE Non-linear ODE An ODE is linear if The unknown function and its derivatives appear to power one No product of the unknown function and/or its derivatives Linear ODE Non-linear ODE

Nonlinear ODE An ODE is linear if The unknown function and its derivatives appear to power one No product of the unknown function and/or its derivatives

Uniqueness of a solution In order to uniquely specify a solution to an n th order differential equation we need n conditions Second order ODE Two conditions are needed to uniquely specify the solution

Auxiliary conditions auxiliary conditions Boundary Conditions The conditions are not at one point of the independent variable Initial Conditions all conditions are at one point of the independent variable

Boundary-Value and Initial value Problems Boundary-Value Problems The auxiliary conditions are not at one point of the independent variable More difficult to solve than initial value problem Initial-Value Problems The auxiliary conditions are at one point of the independent variable same different

Classification of ODE ODE can be classified in different ways Order First order ODE Second order ODE Nth order ODE Linearity Linear ODE Nonlinear ODE Auxiliary conditions Initial value problems Boundary value problems

Analytical Solutions Analytical Solutions to ODE are available for linear ODE and special classes of nonlinear differential equations.

Numerical Solutions Numerical method are used to obtain a graph or a table of the unknown function Most of the Numerical methods used to solve ODE are based directly (or indirectly) on truncated Taylor series expansion

Classification of the Methods Numerical Methods for solving ODE Single-Step Methods Estimates of the solution at a particular step are entirely based on information on the previous step Multiple-Step Methods Estimates of the solution at a particular step are based on information on more than one step

More Lessons in this unit Taylor series methods Midpoint and Heun’s method Runge-Kutta methods Multiple step Methods Solving systems of ODE Boundary value Problems

The Taylor series Zero-order approximation First-order approximation

The Taylor series approximation

The Remainder for the Taylor Series Expansion

The sequence of events in the application of ODEs for engineering problem solving

Differential equations Analytical methods Exact solution Numerical methods Approximate solution