CSE 551
Computational Methods
2019/2020 Fall
Chapter 9-A
Ordinary Differential
Equations
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Numerical Methods for Solving Ordinary Differential Equations, Lecture notes of Numerical Methods in Engineering
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CSE 551
Computational Methods 2019/2020 Fall Chapter 9-A Ordinary Differential Equations
Outline Taylor Series Methods Runge-Kutta Methods Stability and Adaptive Runge-Kutta and Multistep Methods
Taylor Series Methods
- (^) Initial-Value Problem: Analytical versus Numerical Solution
- (^) Solving Differential Equations and Integration
- (^) Taylor Series Methods
- (^) Taylor Series Method of Higher Order
- (^) Types of Errors
Initial-Value Problem: Analytical versus Numerical Solution
- (^) An ordinary differential equation (ODE)
- (^) equation - one or more derivatives of an unknown function
- (^) A solution - differential equation
- (^) specific function - satisfies the equation
- (^) a DE not, in general,
- (^) not determine a unique solution function
- (^) accompanied by auxiliary conditions
- (^) together with the DE
- (^) specify the unknown function precisely.
Inıtial-Value Problem
- (^) initial-value problem for a first-order DE.
- (^) x - function of t ,
- (^) (1): initial-value problem
- (^) t - time and t = a - initial instant in time.
- (^) determine the value of x at any time t before or after a.
- (^) function f – depend on t and x
- (^) If f not involve x , - second example –
- (^) DE solved - indefinite integration.
- (^) illustrate,
- (^) C - x( 5 ) = 17. C = 4 ln ( 5 ) − arctan ( 5 ) − 108.
- (^) numerical solution DE:
- (^) (a) the closed form solution may be very complicated and difficult to evaluate or
- (^) (b) there is no other choice; that is, no closed-form solution can be found
- (^) e.g., for the DE
- (^) solution - taking the integral of the right-hand side.
- (^) can be done in principle but not in practice.
- (^) a function x exists
- (^) dx/dt - right-hand member (3)
- (^) but it is not possible to write x(t) in terms of familiar functions..
Solving Differential Equations and Integration
- (^) close connection between
- (^) solving DEs and integration
- (^) e.g.:
- (^) Integrating from t to t + h ,
- (^) Replacing the integral - numerical integration rules formula for solving the differential equation.
- (^) Euler’s method - obtained from the left rectangle approximation
- (^) The trapezoid rule
- (^) Fundamental Theorem of Calculus,
- (^) approximate numerical value for the integral
- (^) can be computed by solving the following initial- value problem for x(b) :
Taylor Series Methods
- (^) represent the solution of a DE locally by a few terms of its Taylor series.
- (^) assume that
- (^) solution function x - represented Taylor series:
Euler’s Method Pseudocode
- (^) Taylor series method of order 1 - Euler’s method
- (^) approximate values of the solutions to the initial- value problem:
- (^) over the interval [ a, b ],
- (^) first two terms - Taylor series (5) :
- (^) the formula:
- (^) can be used to step from t = a to t = b with n steps of size h = (b − a)/n.
- (^) The pseudocode can be written as follows, where some prescribed values for n , a , b , and
- (^) xa are used: