runge-kutta

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rana 2012-4-22

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hello i have this equation y''+3y'+5y=1 how can i solve it by programming a runge kutta 4'th order method ? i know how to solve it by using a pen and paper but i can not understand how to programe it please any one can solve to me this problem ? i dont have any idea about how to use ODE and i read the help in matlab but did not understand how to solve this equation please any one can solve this and help me with it ? thanx

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Answer 1

Richard Brown 2012-4-23

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在 MATLAB Online 中打开

Writing a new answer so that I can use markup (MathWorks, please fix this!!). You need a couple of things - first, the time values

t = 0:h:100

You can now work out how many steps you need:

n = numel(t)

Now, we need an array in which to store your results: each column should correspond to a value from your t vector

x = zeros(2, n)

We know the initial condition, so we'll pop that in

x(:, 1) = [0; 0]

Now we need a loop to store the subsequent x values. The basic structure will look like this

for i = 2:n % We've already filled in the first column
  k1 = h*f(x(:, i-1));
  k2 = ...
  k3 = ...
  k4 = ...
  % Now define the new x vector based on k1, ... , k4
  x(:, i) = x(:, i-1) + ...
end
% Plot your results
plot(t, x)

And that should pretty much be all you need

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rana 2012-4-23

i really soooooooo thankfull for you Richard

thanx for your help

Richard Brown 2012-4-24

No problem, you might want to accept the answer so that others can benefit from it

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Answer 2

Richard Brown 2012-4-22

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I'll give you the high level overview, you'll need to write the code though

Turn it into a system of two first order equations (define a new variable z = y')
Find a decent pseudocode representation of the algorithm, either from your lecture notes or from e.g. section 8.3 of Kincaid and Chaney, "Numerical Analysis", or Algorithm 5.2 of Burden and Faires, "Numerical Analysis", both of which are readily available introductory books that should be in your library.
Try and code it up in MATLAB
Report back if you have any problems

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Richard Brown 2012-4-23

You've defined your system correctly, however I suggest you use a function, because you're going to need to evaluate it at many different x values, which can get confusing. Something like

f = @(x) [x(2); -3*x(2) - 5*x(1) + u];

You can then call f(x) (where x is a 2-element vector) to get your derivatives (also as a vector).

Wikipedia also has the basic RK4 here <http://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods#Common_fourth-order_Runge.E2.80.93Kutta_method>, you should be able to use that pretty much the way it is written ...

rana 2012-4-23