how to solve a set of differential equation systems like this?

Question

Daniel Niu 2022-11-16

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https://ww2.mathworks.cn/matlabcentral/answers/1852948-how-to-solve-a-set-of-differential-equation-systems-like-this

编辑： Torsten 2022-11-18

采纳的回答： Torsten

Dear friend,

How to solve a ordinary differential equation systems like above using MATLAB?

a=b=c=1

Your help would be highly appreciated.

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John D'Errico 2022-11-16

编辑：John D'Errico 2022-11-16

@Sam Chak

Actually, the point where you identify a singularity is an interesting one, since in order for a singularity to exist, one would need to have

x^2 + (c*t-y)^2 == 0

And that implies two things MUST happen, x==0, AND y = c*t. Note that both differential equations have a corresponding term in the denominator. But then look carefully. In the numerator of both equations, they have a similar term on top. As such, that fraction is actually going to have a finite limit at that point. At the exact point of interest, the result is effectively a Nan, but everywhere else, it is well defined. Not really any different from the fraction x/x.

Sam Chak 2022-11-16

@John D'Errico, Thank you for pointing out this and the explanation.

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Answer 1

Torsten 2022-11-17

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https://ww2.mathworks.cn/matlabcentral/answers/1852948-how-to-solve-a-set-of-differential-equation-systems-like-this#answer_1102933

编辑：Torsten 2022-11-17

在 MATLAB Online 中打开

ar = 0.01;

af = 0.001;

x0 = 3;

y0 = -3;

Lr = 0;

Lf = 0;

z0 = [x0; y0; Lr; Lf];

[T,Z] = ode45(@(t,z)fun(t,z,ar,af),[0 11.5],z0);

plot(Z(:,1),Z(:,2),Z(:,3),zeros(size(T)))

ylim([-3 0.5])

function dz = fun(t,z,ar,af)

x = z(1);

y = z(2);

Lr = z(3);

Lf = z(4);

sr = exp(-ar*Lr);

sf = exp(-af*Lf);

dz = zeros(4,1);

dz(1) = sf* (Lr-x)/sqrt((Lr-x)^2+(0-y)^2);

dz(2) = sf* (0-y)/sqrt((Lr-x)^2+(0-y)^2);

dz(3) = sr;

dz(4) = norm([dz(1) dz(2)]);

end

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Daniel Niu 2022-11-18

Dear Torsten,

Sorry to bother you again. I want to know if the rabbit is not moving horizontal but from (0,0) to (-100,100)

how to modify the code?

Thank you!

Torsten 2022-11-18

编辑：Torsten 2022-11-18

在 MATLAB Online 中打开

Uninteresting case since both fox and rabbit will run on the same straight line.

ar = 0.01;

af = 0.001;

x0r = 0;

y0r = 0;

x0f = 3;

y0f = -3;

Lr0 = 0;

Lf0 = 0;

z0 = [x0r; y0r; x0f; y0f; Lr0; Lf0];

[T,Z] = ode45(@(t,z)fun(t,z,ar,af),[0 11.5],z0);

plot(Z(:,1),Z(:,2),Z(:,3),Z(:,4))

function dz = fun(t,z,ar,af)

xr = z(1);

yr = z(2);

xf = z(3);

yf = z(4);

Lr = z(5);

Lf = z(6);

sr = exp(-ar*Lr);

sf = exp(-af*Lf);

dz = zeros(6,1);

dz(1) = sr* (-1/sqrt(2));

dz(2) = sr* (1/sqrt(2));

dz(3) = sf* (xr-xf)/sqrt((xr-xf)^2+(yr-yf)^2);

dz(4) = sf* (yr-yf)/sqrt((xr-xf)^2+(yr-yf)^2);

dz(5) = norm([dz(1) dz(2)]);

dz(6) = norm([dz(3) dz(4)]);

end

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Answer 2

Torsten 2022-11-17

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https://ww2.mathworks.cn/matlabcentral/answers/1852948-how-to-solve-a-set-of-differential-equation-systems-like-this#answer_1102578

在 MATLAB Online 中打开

Hint:

The length L(t) of a curve in parametric form (x(t),y(t)) where x(t) and y(t) are given by differential equations

dx/dt = f(x,y)
dy/dt = g(x,y)

can be computed by additionally solving a differential equation for L:

dL/dt = sqrt(f(x,y).^2+g(x,y).^2)

with initial condition

L(0) = 0.

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Torsten 2022-11-17

编辑：Torsten 2022-11-17

r1 is part of your ODE system:

dz1/dt = s_f* (r1-z1)/sqrt((r1-z1)^2+(r2-z2)^2)

dz2/dt = s_f* (r2-z2)/sqrt((r1-z1)^2+(r2-z2)^2)

And I think the speed of the rabbit will also decrease with time...

Sam Chak 2022-11-17

编辑：Sam Chak 2022-11-17

Hi @Daniel Niu

Your original "Fox-chasing-Rabbit" example assumes that the Rabbit is moving at a constant velocity

. That's why the solution for the Rabbit's position is

.

If the Rabbit's velocity is time-varying, or is reactively dependent on the position of the Fox {

,

}, then naturally there exists

, and I think that is what @Torsten tried to tell you.

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