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In summary, we have a problem of estimating a 3rd order differential equation with 4 variables and 4 equations derived from the geodesics equation and the Schwarzschild metric. The variables are r, theta, phi, and t, and the equations involve derivatives with respect to proper time (tau). The equations have analytical solutions, but if a numerical solution is desired, they can be rewritten as a system of first order differential equations. The independent

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Probably if you can write down your equations it will be much clearer what your problem actually is.

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Equation 1:

[tex]\frac{d}{d\tau}(\frac{2r}{r-2m}(\frac{dr}{d\tau})) + \frac{2m}{(r-2m)^{2}}(\frac{dr}{d\tau})^{2} - r(\frac{d\theta}{d\tau})^{2} - rsin^{2}\theta (\frac{d\phi}{d\tau})^{2} + \frac{mc^{2}}{r^{2}}(\frac{dt}{d\tau})^{2} = 0[/tex]

Equation 2:

[tex]\frac{d}{d\tau}(r^{2}\frac{d\theta}{d\tau}) - r^{2}sin\theta cos\theta (\frac{d\phi}{d\tau}) = 0[/tex]

Equation 3:

[tex]\frac{d}{d\tau}(r^{2}sin^{2}\theta (\frac{d\phi}{d\tau})) = 0[/tex]

Equation 4:

[tex]\frac{d}{d\tau}(\frac{r-2m}{r}(\frac{dt}{d\tau})) = 0[/tex]

where m and c are constants

[tex]\frac{d}{d\tau}(\frac{2r}{r-2m}(\frac{dr}{d\tau})) + \frac{2m}{(r-2m)^{2}}(\frac{dr}{d\tau})^{2} - r(\frac{d\theta}{d\tau})^{2} - rsin^{2}\theta (\frac{d\phi}{d\tau})^{2} + \frac{mc^{2}}{r^{2}}(\frac{dt}{d\tau})^{2} = 0[/tex]

Equation 2:

[tex]\frac{d}{d\tau}(r^{2}\frac{d\theta}{d\tau}) - r^{2}sin\theta cos\theta (\frac{d\phi}{d\tau}) = 0[/tex]

Equation 3:

[tex]\frac{d}{d\tau}(r^{2}sin^{2}\theta (\frac{d\phi}{d\tau})) = 0[/tex]

Equation 4:

[tex]\frac{d}{d\tau}(\frac{r-2m}{r}(\frac{dt}{d\tau})) = 0[/tex]

where m and c are constants

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If I'm not mistaken, for the Schwarzschild metric, there is analytic solution.

But if you still insist on numerical solution you can try Rungge-Kutta fourth order. One of the easiest to program. But you need to rewrite your equations as system of first order DE of the form.

[tex]X'(\tau) = A(\tau) X(\tau)[/tex]

where X is a column vector and A is a matrix.

By the way why do you say your equation is 3rd order?

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Yes the equation is 2nd order not 3rd.

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[tex]\frac{d}{d\tau}(r^{2}\frac{d\theta}{d\tau}) = 2r\frac{dr}{d\tau}\frac{d\theta}{d\tau} + r^{2}(\frac{d\theta}{d\tau})^{2}[/tex]

Is this correct?

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[tex]\frac{d}{d\tau}(r^{2}\frac{d\theta}{d\tau}) = A \cot (\theta) [/tex]

where A is the arbitrary constant from eq. 2

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Any suggestions on how to solve for r and [tex]\theta[/tex] from the first two equations if the solution from equation 3 is :

[tex]\frac{d\phi}{d\tau} = \frac{constant3}{r^{2}sin^{2}\theta}[/tex]

and equation 4 is:

[tex]\frac{dt}{d\tau} = \frac{(constant4)r}{r-2m}[/tex]

[tex]\frac{d\phi}{d\tau} = \frac{constant3}{r^{2}sin^{2}\theta}[/tex]

and equation 4 is:

[tex]\frac{dt}{d\tau} = \frac{(constant4)r}{r-2m}[/tex]

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tau takes the place of s in geodesic equations and is called proper time in a "local free-fall frame replacing s by:

[tex]ds^{2} = -c^{2}d\tau^{2}[/tex]

[tex]ds^{2} = -c^{2}d\tau^{2}[/tex]

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To input a 3rd order differential equation into a computer program for estimation, you will need to use the proper syntax for the programming language you are using. This typically involves defining the variables and constants in the equation and using the appropriate operators and functions for differentiation.

The process for estimating a 3rd order differential equation with a computer program involves first inputting the equation into the program and then using numerical methods such as Euler's method or Runge-Kutta methods to solve the equation and obtain an estimated solution. This solution can then be compared to known solutions or real-world data to assess its accuracy.

Yes, a well-written computer program using appropriate numerical methods can accurately estimate a 3rd order differential equation. However, the accuracy of the estimation will also depend on the quality of the input equation and the chosen numerical methods. It is important to validate the results by comparing them to known solutions or real-world data.

While computer programs can accurately estimate 3rd order differential equations, they may have limitations when it comes to solving complex or highly nonlinear equations. In these cases, the estimation may be less accurate and may require more advanced numerical methods or specialized programming techniques.

To improve the accuracy of your 3rd order differential equation estimation, you can try using more sophisticated numerical methods, such as higher order Runge-Kutta methods, or adjusting the step size of the estimation. It is also important to carefully review and validate the input equation and consider any potential sources of error in the estimation process.

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