MATLAB解决常微分方程

Question

How can I use matlab to solve the following Ordinary Differential Equations?

x''/y = y''/x = -( x''y + 2x'y' + xy'')

with two known points, such as t=0: x(0)= x0, y(0) = y0; t=1: x(1) = x1, y(1) = y1 ? It doesn't need to be a complete formula if it is difficult. A numerical solution is ok, which means, given a specific t, I can get the value of x(t) and y(t).

If matlab is hard to do this, mathematica is also OK. But as I am not familiar with mathematica, so I would prefer matlab if possible.

Looking forward to help, thanks!

I asked the same question on stackexchange, but haven't get good answer yet. https://math.stackexchange.com/questions/812985/matlab-or-mathematica-solve-ordinary-differential-equations

Hope I can get problem solved here!

What I have tried is:

---------MATLAB

syms t

>> [x, y] = dsolve('(D2x)/y = -(y*D2x + 2Dx*Dy + x*D2y)', '(D2y)/x = -(y*D2x + 2Dx*Dy + x*D2y)','t')


Error using sym>convertExpression (line 2246)
Conversion to 'sym' returned the MuPAD error: Error: Unexpected 'identifier'.
[line 1, col 31]

Error in sym>convertChar (line 2157)
s = convertExpression(x);

Error in sym>convertCharWithOption (line 2140)
    s = convertChar(x);

Error in sym>tomupad (line 1871)
    S = convertCharWithOption(x,a);

Error in sym (line 104)
        S.s = tomupad(x,'');

Error in dsolve>mupadDsolve (line 324)
sys = [sys_sym sym(sys_str)];

Error in dsolve (line 186)
sol = mupadDsolve(args, options);

--------MATLAB

Also, I tried to add conditions, such as x(0) = 2, y(0)=8, x(1) = 7, y(1) = 18, and the errors are still similar. So what I think is that this cannot be solve by dsolve function.

So, again, the key problem is, given two known points, such as when t=0: x(0)= x0, y(0) = y0; t=1: x(1) = x1, y(1) = y1 , how I get the value of x(t) and y(t)?

Update: I tried ode45 functions. First, in order to turn the 2-order equations into 1-order, I set x1 = x, x2=y, x3=x', x4=y'. After some calculation, the equation becomes:

x(1)' = x(3)                                                (1)

x(2)' = x(4)                                                (2)

x(3)' = x(2)/x(1)*(-2*x(1)*x(3)*x(4)/(1+x(1)^2+x(2)^2))     (3)

x(4)' = -2*x(1)*x(3)*x(4)/(1+x(1)^2+x(2)^2)                 (4)

So the matlab code I wrote is:

myOdes.m

function xdot = myOdes(t,x)

xdot = [x(3); x(4); x(2)/x(1)*(-2*x(1)*x(3)*x(4)/(1+x(1)^2+x(2)^2)); -2*x(1)*x(3)*x(4)/(1+x(1)^2+x(2)^2)]

end

main.m
t0 = 0;
tf = 1;
x0 = [2 3 5 7]';
[t,x] = ode45('myOdes',[t0,tf],x0);
plot(t,x)

It can work. However, actually this is not right. Because, what I know is that when t=0, the value of x and y, which is x(1) and x(2); and when t=1, the value of x and y. But the ode functions need the initial value: x0, I just wrote the condition x0 = [2 3 5 7]' randomly to help this code work. So how to solve this problem?

UPDATE: I tried to use the function bvp4c after I realized that it is a boundary value problem and the following is my code (Suppose the two boundry value conditions are: when t=0: x=1, y=3; when t=1, x=6, y=9. x is x(1), y is x(2) ):

1. bc.m
function res = bc(ya,yb)
res = [ ya(1)-1; ya(2)-3; yb(1) - 6; yb(2)-9];
end
2. ode.m
function dydx = ode(t,x)
dydx = [x(3); x(4); x(2)/x(1)*(-2*x(1)*x(3)*x(4)/(1+x(1)^2+x(2)^2)); -2*x(1)*x(3)*x(4)/(1+x(1)^2+x(2)^2)];
end
3. mainBVP.m
solinit = bvpinit(linspace(0,6,10),[1 0 -1 0]);
sol = bvp4c(@ode,@bc,solinit);
t = linspace(0,6);
x = deval(sol,t);
plot(t,x(1,:));
hold on
plot(t,x(2,:));

plot(t,x(3,:));

plot(t,x(4,:));
x(1,:)
x(2,:)

It can work, but I don't know whether it is right. I will check it again to make sure it is the right code.

Answer 1

EDIT: This is how I would solve the problem. Note: I don't really like the matlabFunction creator but this is simply a personal preference for various reasons I won't go into.

% Seperate function of the first order state equations
function dz = firstOrderEqns(t,z)
    dz(4,1) = 0;
    dz(1) = -2.*z(3).*z(1).*z(4)./(1 + z(4).^2 + z(2).^2);
    dz(2) = z(1);
    dz(3) = -2.*z(2).*z(3).*z(1)./(1 + z(4).^2 + z(2).^2);
    dz(4) = z(3);
end

% runfirstOrderEqns
%% Initial conditions i.e. @ t=0
z1 = 5; % dy/dt = 5 (you didn't specify these initial conditions, 
%  these will depend on the system which you didn't really specify
z2 = 0; % y = 0
z3 = 5; % dx/dt = 5 (The same as for z1)
z4 = 0; % x = 0
IC = [z1, z2, z3, z4];
%% Run simulation
% Time vector: i.e closed interval [0,20]
t = [0,20]; % This is where you have to know about your system
           % i.e what is it's time domain.
           % Note: when a system has unstable poles at
           % certain places the solver can crash you need
           % to understand these.
% using default settings (See documentation ode45 for 'options') 
[T,Y] = ode45(@firstOrderEqns,t,IC);

%% Plot function
plot(T,Y(:,1),'-',T,Y(:,2),'-.',T,Y(:,3),':',T,Y(:,4),'.');
legend('dy/dt','y','dx/dt','x')

As in my comments I have made a lot of assumtions that you need to fix for example, you didn't specify what the initial conditions for the first derivatives of the states are ie (z1, z3) which is important for the response of the system. Also you didn't specify the time interval your interested for the simulation etc.

Note: The second m file can be used with any state function in the correct format

Answer 2

As mentioned, this isn't a math site, so try to give code or something showing some effort. However, the first step you need to do is turn the DE into normal form (ie, no 2nd derivatives). You do this by making a separate variable equal to the derivative. Then, you use

syms x y % or any variable instead of x or y

to define variables as symbolic. Use matlabfunction to create a symbolic function based on these variables. Finally, you can use the ode45 function to solve the symbolic function while passing variable values. I recommend you look up the full documentation in matlab in order to understand it better, but here is a very basic syntax:

MyFun= matlabFunction(eq,'vars',{x,y});
[xout,yout]=ode45(@(x,Y) MyFun(variables),[variable values],Options);

Hopefully this puts you in the right direction, so try messing around with it and provide code if you need more help.

Answer 3

The following is the answer we finally get @Chriso: use matlab bvp4c function to solve this boundary value problem (Suppose the two boundry value conditions are: when t=0: x=1, y=3; when t=1, x=6, y=9. x is x(1), y is x(2) ):

1. bc.m
function res = bc(ya,yb)
res = [ ya(1)-1; ya(2)-3; yb(1) - 6; yb(2)-9];
end
2. ode.m
function dydx = ode(t,x)
dydx = [x(3); x(4); x(2)/x(1)*(-2*x(1)*x(3)*x(4)/(1+x(1)^2+x(2)^2)); -2*x(1)*x(3)*x(4)/(1+x(1)^2+x(2)^2)];
end
3. mainBVP.m
solinit = bvpinit(linspace(0,6,10),[1 0 -1 0]);
sol = bvp4c(@ode,@bc,solinit);
t = linspace(0,6);
x = deval(sol,t);
plot(t,x(1,:));
hold on
plot(t,x(2,:));

plot(t,x(3,:));

plot(t,x(4,:));
x(1,:)
x(2,:)