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Numerical Solutions of Ordinary Differential Equations

Dr. S. P. Harsha

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EULERS METHOD:

First order system

with an initial condition x(t0) = x0

The method proposed by Euler was based on a

finite-difference approximation of a continuous first

derivative dx/dt.

Taylors approximation of a first-order continuous

derivative defined as

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Cont

The Eulers solution procedure is marching from the initial

time t0 to the final time tf = t0 + twith a constant time step t.

The following equations are solved successively in the

computational process:

Use Eulers method to obtain a numerical solution of thedifferential equation

Over a period of time from 0 to 12 s. The initial condition is

x(0) = 10, and the system time constant is 4 s.

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Cont

First rewrite the differential equation in the same form

The first step in implementing the numerical solution is to

choose a value of the integration interval t.

For this system, the time constant is 4 s and to illustrate thisrelationship, the Euler method is implemented for ts of 0.5,

1.0, 2.0, and 4.0 s.

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Table 5.1. MATLAB script for the solution of Example use of the Euler

method of integration

% MATLAB script to integrate a first order

% ODE using the Euler method.

%

dt = 0.5; % time step

t(1) = 0.0; % initial time

tf = 12.0; % final time

%

x(1) = 10.0; % Set initial condition

%for i = 2: tf/dt + 1

k1 = 1/4*x(i 1)+1;

x(i) = x(i 1) + k1*dt;

t(i) = t(i 1)+ dt; % increment time end

endDr. S. P. Harsha

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Cont

Table 5.2. Comparison of Euler method solution for various step sizes

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RungeKutta Method

In the Runge Kutta method is that the higher derivatives are

approximated by finite-difference expressions and thus do not

have to be calculated from the original differential equation.

The approximating expressions are calculated by use of data

obtained from tentative steps taken from t0 toward t0 + t. The

number of steps used to estimate x(t0 + t) determines the

order of the RungeKutta method.

In the most common version of the method, four tentative

steps are made within each time step, and the successive value

of the dependent variable is calculated as

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RungeKutta Method

Table 5.4. MATLAB script for implementation of the fourth-order Runge

Kutta numerical integration scheme

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RungeKutta Method Algorithm

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RungeKutta Method Algorithm

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Step Size

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Step 1. Starting at some time t0, integrate to t0 + t, using a default time

step, t

Step 2. Return to the original time, and integrate to t0 + t by using two

time steps of t/2.Step 3. Compare the difference between the results of Steps 1 and 2 to a

user defined tolerance.

Step 4. If the tolerance is exceeded, decrease the step size and return to

Step 1; if not, nominally increase the step size and move to the next time

step.

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Example:

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Consider the following nonlinear differential equation:

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Dr. S. P. Harsha