Finite difference discretization for systems of higher ODEs

[Ojo Saheed]

How can I use finite difference to discretize a system of fourth order differential equations?
for example:
y⁽⁴⁾+5y⁽³⁾-2y^''+3y^'-y=0

 

[Chestermiller]

This doesn't look like a system of 4th order ODEs. It looks like a single 4th order ODE.

Chet

 

[Ojo Saheed]

This doesn't look like a system of 4th order ODEs. It looks like a single 4th order ODE.

Chet

Thanks for the reply Chet. Actually, I had thought that given an example of one the system of equations might suffice for a demonstration. In any case, a better representation of the system goes thus:
y₁⁽⁴⁾=a₁y₁^''+a₂y₂^''+a₃y₃^''+a₄y₁+a₅y₂+a₆y₃+a₇
y₂⁽⁴⁾=b₁y₁^''+b₂y₂^''+b₃y₃^''+b₄y₁+b₅y₂+b₆y₃+b₇
y₃⁽⁴⁾=c₁y₁^''+c₂y₂^''+c₃y₃^''+c₄y₁+c₅y₂+c₆y₃+c₇
where a, b and c terms are constant coefficients

 

[Chestermiller]

Thanks for the reply Chet. Actually, I had thought that given an example of one the system of equations might suffice for a demonstration. In any case, a better representation of the system goes thus:
y₁⁽⁴⁾=a₁y₁^''+a₂y₂^''+a₃y₃^''+a₄y₁+a₅y₂+a₆y₃+a₇
y₂⁽⁴⁾=b₁y₁^''+b₂y₂^''+b₃y₃^''+b₄y₁+b₅y₂+b₆y₃+b₇
y₃⁽⁴⁾=c₁y₁^''+c₂y₂^''+c₃y₃^''+c₄y₁+c₅y₂+c₆y₃+c₇
where a, b and c terms are constant coefficients

If you're trying to solve this numerically, one way it is often done is to convert to a set of first order ODEs. In this particular case, there would be 12 first order ODEs to integrate.

Define,
z_j=y_j (j = 1,3)
z_j+3=y'_j (j = 1,3)
z_j+6=y''_j (j=1,3)
z_j+9=y'''_j (j=1,3)
Then
z₁' = z₄
z₄' = z₇
z₇' = z₁₀
z₁₀' = a₁z₇+a₂z₈+a₃z₉+a₄z₁+a₅z₂+a₆z₃+a₇
etc.

You end up with 12 coupled first order linear ODEs explicit in the derivatives. You can use an automatic integrator to solve them or use your own coding of Runge Kutta or forward Euler, or backward Euler, or whatever.

Chet

 

[blue_raver22]

Finite difference discretization is a numerical method used to approximate the solution of a differential equation by dividing the continuous domain into a finite number of discrete points. This method is commonly used for systems of higher order ordinary differential equations (ODEs) such as the example given, a fourth order ODE.

To use finite difference discretization for a system of fourth order ODEs, we can follow these steps:

1. Discretize the domain: First, we need to discretize the continuous domain of the ODE into a finite number of discrete points. This can be done by dividing the domain into equal intervals and choosing a finite number of points within each interval.

2. Approximate derivatives: Next, we need to approximate the derivatives of the ODE at each discrete point. This can be done using finite difference formulas, which involve calculating the difference between function values at adjacent points.

3. Substitute into the ODE: Once we have approximated the derivatives, we can substitute them into the original ODE at each discrete point. This will result in a system of algebraic equations.

4. Solve the system: Finally, we can solve the resulting system of algebraic equations to obtain the numerical solution at each discrete point. This solution can then be used to approximate the solution to the original ODE over the entire domain.

In the example given, the fourth order ODE can be discretized by dividing the domain into equal intervals and choosing a finite number of points within each interval. The derivatives can then be approximated using finite difference formulas, such as the central difference formula. These approximations can be substituted into the ODE at each discrete point, resulting in a system of algebraic equations. Solving this system will give us the numerical solution to the original ODE at each discrete point.