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Here is the second order differential equation:

$\frac{d^2x}{dt^2} + \sin(x) = 0$

with initial conditions $x(0)=1$ and $x'(0)=0$.

The written code in matlab that numerically solves and plots the results is shown below:

function second_order
t=0:0.001:14;
initial_x=1;
initial_dxdt = 0;
[t,x]=ode45(@rhs,t,[initial_x initial_dxdt]);
plot(t,x(:,1));
xlabel('t');ylabel('x');

function dxdt=rhs(t,x)
dxdt_1=x(2);
dxdt_2=-sin(x(1));
dxdt = [dxdt_1; dxdt_2);
end
end

The plot shows the sinusoidal curve with maximum amplitude plus and minus one. This plot is correct. However, when I turn the second order differential equation into first one as follows:

$ \frac{d^2x}{dt^2}\frac{dx}{dt} + \sin(x)\frac{dx}{dt} = 0$

$\frac{1}{2}(\frac{dx}{dt})^2 = \cos(x)-\cos(1)$ and that leads to first order ODE,

$\frac{dx}{dt}=\sqrt{2(\cos(x)-\cos(1))}$

The matlab to plot first ODE is shown below:

function first_order
t=0:0.001:14;
initial_x=1;
[t,x]=ode45(@rhs, t, initial_x);
plot(t,x);
xlabel('t'); ylabel('x');

function dxdt=rhs(t,x)
dxdt = sqrt(2)*sqrt(cos(x)-cos(1));
end
end

The problem is that I am not getting the same plot as in second ODE code. Instead, I am getting a straight line at $x=1$. I know that the code is correct because I tested it with other first order differential equation. Therefore, why I am not getting the same plot even thought the first and second order differential equations are the same. First order is basically a solution of the second order. But values of $x$ should be the same. Is this approach not applicable? or am I doing something wrong?

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Apparently the preceding answers don't satisfy you because they are more mathematical answers than about software and/or about how to handle software (Matlab).

I am not matlab expert. So my answer cannot be considered as definitive and might be discussed.

You coded the equation $$x'(t)=\frac{dx}{dt}=\sqrt{2\left(\cos(x)-\cos(1)\right)} \tag 1$$ With the initial specification $x(0)=1$.

From the behaviour of software that you report, one can suppose that $x'_0=x'(0)=\sqrt{2\left(\cos(1)-\cos(1)\right)} =0$ was first computed.

Then applying a time increment $\delta t$ at time $t=0$ the nest time $t_1=0+\delta t= \delta t$ and the next point $(x_1,t_1)$ was computed : $x_1=x_0+x'_0\delta t=1+0(\delta t)=1$.

And so on for successive increments, $x'$ remains nul and $x$ remains equal to $1$. So the software give a correct solution $$x(t)=1.$$ As already pointed out, this is not the unique solution of the ODE Eq.$(1)$ with condition $x(0)=1$.

Apparently the software doesn't give spontaneously another solution. As a consequence one have to code the software in order to get another solution since we know that several solutions exist.

Consider the first step of the numerical calculus : $\begin{cases} t_1=\delta t \\ x_1=1+\delta x_1 \end{cases}$

In Eq.$(1)$ necessarily $\quad \cos(x_1)-\cos(1)> 0$ thus $x_1<1$ and $\delta x_1<0$.

$\frac{\delta x_1}{\delta t}<0 \quad$ is not consistent with Eq.$(1)$ where the square root is positive.

Thus Eq.$(1)$ is not correct. The correct equation according to the initial condition $x(0)=1$ is $$x'(t)=\frac{dx}{dt}=-\sqrt{2\left(\cos(x)-\cos(1)\right)} \tag 2$$ This is a correct derivation of $\frac12(\frac{dx}{dt})^2=\cos(x)-\cos(1)$.

$\underline{\text{So, first you have to correct the code}}$ to apply Eq.$(2)$ instead of $(1)$.

But how to make the algorithm start a non-constant incremental process ? Preliminary inspection shows that :

$\delta x=-\sqrt{2\left(\cos(1+\delta x)-\cos(1)\right)}\:\delta t=-\sqrt{2\left(\cos(1)\sin(\delta x))-\sin(1)\sin(\delta x)-\cos(1)\right)}\:\delta t\simeq -\sqrt{-2\sin(1)\delta x}\:\delta t$

After simplification :$\quad \delta x=-2\sin(1)(\delta t)^2$

The first increment $\delta t$ gives : $$\begin{cases} t_1=\delta t \\ x_1=1-2\sin(1)(\delta t)^2 \end{cases}$$

$\underline{\text{Second, you have to correct the code }}$ for the above starting point instead of $t=0\:,\:x=1$.

You mentioned in comment, citation : "I already did that with 0.9 and 0.8. Instead on line, I am getting something else but not sinusoidal curve". But this was not exactly the right way.

First, change the sign of the square root. Second, try with a smaller $\delta t$, for example : $$\begin{cases} t_1=0.001 \\ x_1=1-2\sin(1)(0.001)^2 \end{cases}$$

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  • $\begingroup$ I have given you a check mark for an answer because in general I've got idea from your about where is the problem source. What I did is that I decided to represent second equation using different function so that there is no square root. But I did not have time to test it with matlab because I was somewhere else then. Now I see that you attempted to explain the issue again and developed a different solution to second equation even though its different than mine. I haven't check your or mine yet, important thing is that I understand my problem. Thanks. $\endgroup$ – Aschoolar May 27 at 19:51
  • $\begingroup$ The key point is not the square root (which anyways must have the correct sign). The key point is out to boost the software to start to not always the same solution when the solution is not unique. If no resource is provided for that in the software, one have to set a very slightly different starting point. Generally that is sufficient to make the algorithm going on a different trajectory which is a solution of the equation. $\endgroup$ – JJacquelin May 28 at 10:58
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$$ \frac{d^2x}{dt^2} + \sin(x) = 0 \tag 1$$ $$ \frac{d^2x}{dt^2}\frac{dx}{dt} + \sin(x)\frac{dx}{dt} = 0\tag 2$$ When you multiply Eq.$(1)$ by $\frac{dx}{dt}$ in order to get Eq.$(2)$, you introduces additional solutions which are solutions of $(2)$ but not solutions of $(1)$. The additional solutions are the solutions of $$\frac{dx}{dt}=0 \quad\implies\quad x(t)=c\tag 3$$ The boundary conditions $x(0)=1$ and $x'(0)=0$ give $c=1$. So an additional solution was introduced : $$x(t)=1$$ Thus is is correct that, according to the boundary conditions, the drawing of the solution of $(2)$ is the same as the drawing of the solution of $(1)$ plus the straight line $x=1$.

$$ $$

IN ADDITION after the comments :

The analytic solution of $\quad\frac{dx}{dt}=\sqrt{2(\cos(x)-\cos(1))}\quad$ is $$\begin{cases} x(t)=1\\ x(t)=2\text{ am}\left(\sqrt{\frac{1-\cos(1)}{2}}(c-t) \:\bigg|\: \csc^2(\frac12) \right) \end{cases}$$ $\text{am}$ denotes the Jacobi amplitude function.

With the condition $x(0)=1\quad\implies\quad c=\csc(\frac12)F\left(\frac12 \:\big|\:\csc^2(\frac12) \right)\simeq 1.67499$

$F$ denotes the elliptic integral of the first kind.

enter image description here

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  • $\begingroup$ You mentioned at the end, " that the drawing of the solution of (2) is the same as the drawing of the solution (1) plus the straight line x=1.". However, my drawing for the solution of (2) is straight line x=1 only. There is no sinusoidal curve. $\endgroup$ – Aschoolar May 21 at 14:53
  • $\begingroup$ Then your problem is not a mathematical problem. It is a software problem (of the software itself or of the manner to use the software). May be in the case of non-unique solution (for equation $2$ of course, not for equation $1$ ) the software or the way to use it is not convenient. Better raise the question on a software forum instead of a mathematical forum. $\endgroup$ – JJacquelin May 21 at 16:03
  • $\begingroup$ I am convinced that the trouble comes form the software. Probably the algorithm used is built to give only one solution, not all the solutions even if the mathematical problem has several solutions. I suggest to test the software in setting $x'(0)=$ a very small value instead exactly $0$. Try with smaller values tending to $0$ and see what happen. $\endgroup$ – JJacquelin May 22 at 6:37
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    $\begingroup$ @Aschoolar : You start at $x=1$. The derivative there is $\sqrt{2(\cos(x)-\cos(1))}=\sqrt{2(\cos(1)-\cos(1))}=0$. The numerical method "sees no reason" to deviate from the constant function, all derivative values it computes are always zero, as the intermediate points are computed from $x=1$ with offsets proportional to $0$. Close to $x=1$ the solutions are close to the solutions of $\dot x=\sqrt{1-x^2}$ which has the same non-uniqueness issues as the standard example $\dot y=\sqrt{y}$ at $y=0$. $\endgroup$ – LutzL May 22 at 9:52
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    $\begingroup$ Of course. That is what happens if the algorithm is as Lutz describes. But if the algorithm compute $\frac{d^2x}{dt^2}$ this second derivative is not nul. So the algorithm will take a small variation of $\frac{dx}{dt}$ which will no longer be exactly nul as Lutz said. The iterative process will start and a solution $x(t)$ not continuously equal to $1$ will appear. That is why I said that probably the algorithm doesn't includes the convenient routines to obtain a second solution. Or may be this is implemented in the software but the user have to know how to make it work. $\endgroup$ – JJacquelin May 22 at 10:38
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You can get the "curved" solution from a first order equation by further parametrizing the integrated equation $$ \frac12\dot x^2+\frac12\left(2\sin\frac x2\right)^2=\frac12\left(2\sin\frac 12\right)^2 $$ by recognizing this as a circle equation for $(\dot x,2\sin\frac x2)$ and thus setting $$ 2\sin\frac{x(t)}2 = 2\sin\frac 12\cdot \sin(u(t)),\\ \dot x(t)= 2\sin\frac 12\cdot \cos(u(t)), $$ $u(0)=\frac\pi2$, and then comparing the derivative of the first equation with the second equation, $$ \cos\frac{x(t)}2\cdot\dot x(t) = 2\sin\frac 12\cdot\cos(u(t))\dot u(t) $$ so that for non-constant solutions (so that $\cos(u(t))\ne 0$) $$ \dot u(t) = \cos(\frac{x(t)}2)=\sqrt{1-\sin^2\frac 12\cdot \sin^2(u(t))} $$ For the plot you have then to reverse the parametrization to $$ x(t)=2\arcsin\left(\sin\frac 12\cdot \sin(u(t))\right) $$

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