Solving ordinary differential equations

How to solve systems of linear differential equations? Here we describe three ways: Runge - Kutta method, and two similar ways to each other using the method of state space.

Let our process is described by the following differential equation:

This equation can be written as:

(*)

where:

We have to find: and

Initial conditions:  and

   1 Method: solution by the method of numerical integration (Runge - Kutta method)

x0 = [1
      1];
tspan = 0:0.1:10;
[T,x1] = ode45(@odefun,tspan,x0);

figure;
subplot(2,1,1);plot(T,x(:,1));title('x1')
subplot(2,1,2);plot(T,x(:,2));title('x2')

Differ. equation is written in a function and stored in a separate file called odefun.m:

function [dx] = odefun(t,x)
dx = zeros(2,1);
A = [-1 0
      0 0.5];
dx = A*x;

   

The solution of equation (x1 and x2)

  2 Method: solving equation using the state space method 

1. Differentiating  consistently equation (*) :

2. Then if :

3. Expansion of  in a Taylor series around the point :

4. Series before  is:

then

The are two ways to implement the algorithm:

1 way:

x0 = [1
      1];
tspan = 0:0.1:10;
A = [-1 0
      0 0.5];
x = zeros(2,length(tspan));
for i = 1:length(tspan)
    t = tspan(i);
    F = expm(A*t);
    x(:,i) = F*x0;
    T(i) = tspan(i);
end

x = x';
figure;
subplot(2,1,1);plot(T,x(:,1));title('x1')
subplot(2,1,2);plot(T,x(:,2));title('x2')

The solution of equation (x1 and x2)

2 way:

x0 = [1
      1];
tspan = 0:0.1:10;

dt = tspan(2) - tspan(1);
A = [-1 0
      0 0.5];
x(:,1) = x0';
F = expm(A*dt);
for i = 2:length(tspan)
    x(:,i) = F*x(:,i-1);
    T(i) = tspan(i);
end
x = x';
figure;
subplot(2,1,1);plot(T,x(:,1));title('x1')
subplot(2,1,2);plot(T,x(:,2));title('x2')

The solution of equation (x1 and x2)