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> And then instead of attempting (and failing) to solve the DE analytically through the algebraic methods, apply modern numerical methods (Runge-Kutta, etc). Y
by barumi 6y ago
> And then instead of attempting (and failing) to solve the DE analytically through the algebraic methods, apply modern numerical methods (Runge-Kutta, etc).
Your assertion doesn't make sense. Runge-Kutta does not solve differential equations. That's not what they are used for. A solution to a differential equation is an equation that satisfies both the differential equations and the boundary conditions. Runge-Kutta, or any other iterative numerical method, does nothing if the sort. The best they do is provide estimates on what values the solution might exhibit at a given point within their domain. These estimates may be of use for some application, but they are not solutions.
If you care about solutions to differential equations but for some reason don't want to learn methods that allow you to determine their solutions then the next best thing is getting up to speed with approximate methods such as the Galerkin method, which for some classes of problems and choices of trial functions can actually output exact analytical solutions.
- d_silin 6y agoSemantics. I want some numbers out of the equations that will match the real-life engineering problem. Approximations/estimates are fine as long as I know the limits of their applicability.
- barumi 6y ago> Semantics. Nope, don't confuse semantics with being just plain wrong. The definition of a solution to a differential equation is quite straight-forward, and a value within the problem domain is not it, even if you use the solution to determine that value. > I want some numbers out of the equations that will match the real-life engineering problem. If that's what you want then what you want is not the solution to the differential equation. You seek something else, and in a scenario where you tolerate a significant margin of error. Meanwhile, you can be taught to determine the exact solution. > Approximations/estimates are fine as long as I know the limits of their applicability. That's not what Runge-Kutta gives you. You might get it with Galerkin methods, but not with numerical approximations.
- deleted 6y ago[deleted]