Please check if this works out for your study:
A = 1.0;
B = 1.0;
fun = @(t, x)[-A*x(1)*x(2)^4; % 1st ODE
-B*x(1)^2*x(2)^3]; % 2nd ODE
tspan = [0 1e2];
y0 = 1.0; % initial value of y(t)
z0 = 1.125*y0; % initial value of z(t)
x0 = [y0;
z0];
[T, Z] = ode45(fun, tspan, x0); % Solver assigns solutions to Z array
y = Z(:,1); % numerical solution for y(t)
z = Z(:,2); % numerical solution for z(t)
% Fit well if A = 1; B = 1, and A = 1; B = -1
f1 = fit(T, y, 'rat23') % fit Rat23 model into y(t) data
f2 = fit(T, z, 'rat33') % fit Rat33 model into z(t) data
% Fit well if A = -1; B = 1
% f1 = fit(T, y, 'rat22') % fit Rat22 model into y(t) data
% f2 = fit(T, z, 'rat22') % fit Rat22 model into z(t) data
subplot(2, 1, 1)
plot(f1, T, y)
subplot(2, 1, 2)
plot(f2, T, z)
The Approximate Analytical Solution is a Rational Function:
f1 =
General model Rat23:
f1(x) = (p1*x^2 + p2*x + p3) /
(x^3 + q1*x^2 + q2*x + q3)
Coefficients (with 95% confidence bounds):
p1 = 1.925 (1.768, 2.083)
p2 = -0.4781 (-0.9759, 0.01979)
p3 = -0.246 (-0.5008, 0.008754)
q1 = 2.363 (1.718, 3.009)
q2 = -0.9528 (-1.554, -0.3518)
q3 = -0.2443 (-0.5014, 0.01292)
f2 =
General model Rat33:
f2(x) = (p1*x^3 + p2*x^2 + p3*x + p4) /
(x^3 + q1*x^2 + q2*x + q3)
Coefficients (with 95% confidence bounds):
p1 = 0.5208 (0.5195, 0.5221)
p2 = 12.22 (10.04, 14.4)
p3 = 50.82 (43.67, 57.96)
p4 = 21.52 (18.88, 24.16)
q1 = 24.87 (20.4, 29.34)
q2 = 68.33 (59.2, 77.46)
q3 = 19.15 (16.8, 21.5)
Results:
