12Ordinary DifferentialEquationsInitial-Value Problems.ppt

12Ordinary DifferentialEquations:Initial-Value Problems 2005. 06. 01 ????? ??? ??? ??? ??? Introduction Techniques for solving first-order ordinary differential equations are the subject of this chapter the differential equation is written in the form with the value of function given at ,i.e., . Basic idea is to divide the internal of interest into discrete steps and find approximations to the function y at those values of . Taylor Method Runge-Kutta Method Multistep Method 12.1 TAYLOR METHODS Consider two methods that are based on the Taylor series representation of unknown function . Euler’s method Uses only the first term in the Taylor series Higher order Taylor method 12.1 TAYLOR METHODS 12.1.1 Euler’s Method The simplest method of approximating the solution of the differential equation Start by treating the function as a constant, , and replace the derivative by the forward difference quotient. This gives or ( where ) Geometrically, this corresponds to using the line tangent to the true solution curve to find the value of . 12.1 TAYLOR METHODS 12.1.1 Euler’s Method Euler Method gives , for For example, Example 12.1 Solving a Simple ODE with Euler’s Method(1/2) , (i.e., a=0, b=1) , . First, find the approximate solution for h =0.5 (n =2) Approximation at is Second, find the approximate solution at : Example 12.1 Solving a Simple ODE with Euler’s Method(2/2) To find a better approximate solution(use n=10 intervals) MATLAB Functi