Second Order Non linear ode matlab

In summary: The Euler and Heun methods are simplified versions of the Runge-Kutta method, with lower accuracy. The 4th order Runge-Kutta method is a more accurate version of the Runge-Kutta method, and it is the most commonly used one.In summary, the problem involves a free swinging pendulum with dampening, modeled by an equation involving various parameters. The goal is to solve this equation using the Euler, Heun, and 4th order Runge-Kutta methods. However, it is possible to use only the 4th order Runge-Kutta method for a more accurate solution. The equation is solved by creating a function and using the ode45 function in MATLAB.
  • #1
SteliosVas
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Homework Statement



Okay the problem is of a free swinging pendulum with dampening which is modeled using the following equation:

Damping coefficient: c=1 s−1
Mass: m=1 kg
Gravity: g=9.81 ms−1
Link length: l=0.5 m

We know
θ(0)=90° and θ′(0)=0, solve this equation from t = 0 to t = 10 with a time interval of 0.01s The equation is:

d2θ/dt2+(c/m)*(dθ/dt)+(g/l)*sin (θ)=0

So we need to use Euler,Heun and 4th order Runge-Kutta method

Homework Equations

The Attempt at a Solution



Okay so my idea was to create a function as so:

function xdot=pendemo(t,x)
% PENDEMO Pendulum ODE derivative evaluation

xdot(1,1) = x(2,1);

xdot(2,1) = -1/(1*1)*x(2,1) - 9.81/1*sin(x(1,1));

% End of pendemo.m

and than an m.file giving the above information:

xphi = [pi/2;0];

tphi = 0; 5 %start time

tfin = 10; %end time

[t,x] = ode45('pendemo',[tphi tfin],xphi);

plot(t,x(:,1))

The only thing is how do I implement a euler/heun method? What is a 4th order Runga Kata??

thanks
 
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  • #2
Did you try to google something ? I found several MATLAB RK4 integrator codes in no time at all !
 
  • #3

FAQ: Second Order Non linear ode matlab

What is a Second Order Nonlinear ODE?

A Second Order Nonlinear ODE is a type of differential equation that involves a second derivative and a nonlinear function. It is commonly used in mathematical modeling to describe complex systems.

How do you solve a Second Order Nonlinear ODE using MATLAB?

To solve a Second Order Nonlinear ODE using MATLAB, you can use the built-in function "ode45". This function uses a numerical method called Runge-Kutta to approximate the solution of the ODE. You will need to define the initial conditions and the nonlinear function in your code.

What are the key challenges in solving Second Order Nonlinear ODEs?

One of the key challenges in solving Second Order Nonlinear ODEs is the lack of analytical solutions. This means that numerical methods, like Runge-Kutta, are often used to approximate the solution. Additionally, the choice of initial conditions and the nonlinear function can greatly affect the accuracy of the solution.

Can MATLAB solve any type of Second Order Nonlinear ODE?

Yes, MATLAB can solve a wide range of Second Order Nonlinear ODEs, as long as they can be expressed in the form of a function with a second derivative. However, for highly complex or specialized equations, you may need to use other numerical methods or write your own code.

What are some applications of Second Order Nonlinear ODEs?

Second Order Nonlinear ODEs are used in many fields, such as physics, engineering, biology, and economics. They can be used to model a variety of systems, including oscillating systems, chemical reactions, population dynamics, and more. They are also useful in predicting future behavior and analyzing the stability of systems.

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