How to solve a non-linear equation where each element of the array is a parameterized function?

2015 6 18
2 个回答
更新时间：2015 6 22
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Dear all,
I need to solve an equation with the following form:
[f(a,b,c); g(a,b,c); p(a,b,c)] = [0; 0; q(t)]
Ok, it's more like
[f(a,b,c,da,db,dc,d2a,d2b,d2c); g(a,b,c,da, db,dc,d2a,d2b,d2c); p(a,b,c,da,db,dc,d2a,d2b,d2c);] = [0; 0; q(t)]
(where "da" means "da/dt" and "d2a" means "d2a/dt2", got using fulldiff())
Or, exactly:
A=
12*d2Gama*cos(Alpha)*cos(Gama)^2 - (9*d2Beta*cos(Gama))/10 - (9*d2Alpha*cos(Beta)*sin(Gama))/10 - 12*d2Gama*cos(Alpha) + 12*d2Gama*cos(Beta)*cos(Gama)^2 - (9*dAlpha^2*cos(Beta)*cos(Gama)*sin(Beta))/10 - 12*dGama^2*cos(Alpha)*cos(Gama)*sin(Gama) - 12*dGama^2*cos(Beta)*cos(Gama)*sin(Gama) + 12*dGama^2*cos(Gama)^2*sin(Alpha)*sin(Beta) + (9*dAlpha*dBeta*sin(Beta)*sin(Gama))/5 + 12*d2Gama*cos(Gama)*sin(Alpha)*sin(Beta)*sin(Gama); (9*d2Alpha*cos(Beta)*cos(Gama))/10 - 12*dGama^2*cos(Beta) - (9*d2Beta*sin(Gama))/10 + 12*d2Gama*sin(Alpha)*sin(Beta) + 12*dGama^2*cos(Alpha)*cos(Gama)^2 + 12*dGama^2*cos(Beta)*cos(Gama)^2 - 12*d2Gama*cos(Gama)^2*sin(Alpha)*sin(Beta) - (9*dAlpha^2*cos(Beta)*sin(Beta)*sin(Gama))/10 - (9*dAlpha*dBeta*cos(Gama)*sin(Beta))/5 + 12*d2Gama*cos(Alpha)*cos(Gama)*sin(Gama) + 12*d2Gama*cos(Beta)*cos(Gama)*sin(Gama) + 12*dGama^2*cos(Gama)*sin(Alpha)*sin(Beta)*sin(Gama); 12*dGama^2*sin(Beta)*sin(Gama) - (9*dBeta^2)/10 - 12*d2Gama*cos(Gama)*sin(Beta) - (9*dAlpha^2*cos(Beta)^2)/10 + 12*dGama^2*cos(Beta)*cos(Gama)*sin(Alpha) + 12*d2Gama*cos(Beta)*sin(Alpha)*sin(Gama)];
B= [0 0 log(t+1)+sin(t)./(t+1)].';
fsolve(A-B)
but let's try to learn the simplest first.
I need to get expressions for a(t), b(t) and c(t).
I believe there's no exact solution, so, it could be an approximated solution, since it returns expressions Alpha(t), Beta(t) and Gama(t).
How do we do Matlab solve it?
Thanks, very much

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回答（2 个）

Torsten
Torsten 2015-6-19

0 个投票

Add the equations
dalpha/dt=alphadot
dbeta/dt=betadot
dgamma/dt=gammadot,
to your system, substitute
dalpha by alphadot
dbeta by betadot
dGama by gammadot
d2alpha by dalphadot/dt
d2beta by dbetadot/dt
d2Gama by dgammadot/dt
in your equations and use ODE15i to solve.
Best wishes
Torsten.

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Thanks very much Torsten.
Using what you suggest, I made the code below to learn about ODE15i considering a f(t,y,y')=0 problem. I'll let it here if One someday has the same question.
However, could you help me how to include an f(t,x,x',x",y,y',y",z,z',z")=0 inside an ODE15i()? Is it possible?
function res = myfunc(t,y,yp)
%myfunc Function to test ODE
%
% See also ODE15I.
% Marlon
% Got from Matlab Help.
res = t*y^2 * yp^3 - y^3 * yp^2 + t*(t^2 + 1)*yp - t^2 * y;
---
%%Test ODE15i
clear
clc
syms t y
t0 = 1; %Initial guess to t0
y0 = sqrt(3/2); %Initial guess to y0
yp0 = 0; %Initial guess to yp0
[y0,yp0] = decic(@myfunc,t0,y0,1,yp0,0); %"calculated guess" to y0 and yp0
odeset('refine',[10]);
[t,y] = ode15i(@myfunc,[1 10],y0,yp0); %Numerical solution y(t)
ytrue = sqrt(t.^2 + 0.5); %Real solution (to compare)
n=4; %Polynomial degree
p=polyfit(t,y, n); %Polynomial coefficients
f = polyval(p,t); %Values of the polynomial
syms tsim ytrue
yt=0;
for i=1:n+1
yt=yt+p(i)*tsim^(n+1-i); %Polynomial wrote as symbolic
end
%Prepare to plot
numb=1:1:t(end);
yt_plot=subs(yt,tsim,numb);
ytp=diff(yt,tsim);
ytp_plot=subs(ytp,tsim,numb);
%plot and compare
plot(t,y,'o',t,f,'r', t, ytrue,'k', numb, yt_plot, 'm',numb,ytp_plot, 'cyan'); hold;
legend('NumericSolution','Polynomial (numeric)','ExactSol','Polynomial (function)', 'yp');

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For example. I tried this, but it's not running:
function res = myfunc2(t,y,yp)
%myfunc2 Function to test ODE with 2 variables and their diffs
%
% See also ODE15I.
% Marlon
% Got from Matlab Help.
res = [t*y(1)^2 * yp(1)^3 - y(1)^3 * yp(1)^2 + t*(t^2 + 1)*yp(1) - t^2 * y(1); t*y(2)^2 * yp(2)^3 - y(2)^3 * yp(2)^2 + t*(t^2 + 1)*yp(2) - t^2 * y(2)];
---
clear
clc
t0 = 1; %Initial guess to t0
y1_0 = sqrt(3/2); %Initial guess to y1_0
yp1_0 = 0; %Initial guess to yp1_0
y2_0 = sqrt(3/2); %Initial guess to y2_0
yp2_0 = 0; %Initial guess to yp2_0
[t,y] = ode45(@myfunc2,[1 10],[y1_0 y2_0],[yp1_0 yp2_0]); %Numerical solution y(t)

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Torsten
Torsten 2015-6-22
I already included the way how to deal with 2nd order derivatives in my reply .
If you have the equation
f(t,y,y',y'')=0,
replace it by the two equations
y'-y1=0;
f(t,y,y1,y1')=0
Best wishes
Torsten.

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