How to find a differential equation if I know its solution?

[jonjacson]

Hello folks,

Let me explain this.

Much effort, time, methods and books is devoted to the science of finding solutions of differential equations. But I cannot find anywhere the reverse problem.

Basically I am thinking about the work that Maxwell did finding the differential equations that describe Electrodynamics. Is there a mathematical branch studying this? I mean you have solutions, and you need to find a set of differential equations compatible with them.

Is this studied in a systematic manner?

Thanks.

 

[fresh_42]

The main issue of such an approach is how to define a problem, behavior, experiment, observation or whatever, i.e. how to describe it, for which differential equations might be a solution. I assume that usually the description itself already bears the equations, e.g. quantities that change in time and depend on each other.

 

[jonjacson]

The main issue of such an approach is how to define a problem, behavior, experiment, observation or whatever, i.e. how to describe it, for which differential equations might be a solution? I assume that usually the description itself already bears the equations, e.g. quantities that change in time.

Yes, maybe we should search the simplest set of partial differential equations compatible with the solution. Isn't it strange that nobody talks about this?

Basically it is what Physics does since its origins.

 

[fresh_42]

Not only in physics. Many other systems in biology, economics, society research, chemistry, ... and so on also use differential equations. To find the solutions is usually far more complicated than the equation system itself.
How many solutions to Einsteins field equations exist?

Those solutions heavily depend on initial conditions and that is an essential part in investigations of differential equation systems. They can decide by tiny changes whether you get an attractor, a repeller or a chaotic system.
And even then, which kind of them?

Building up the equations from observations is IMO the simple part.
Sometimes even a picture of a vector field is sufficient.

 

[jonjacson]

Not only in physics. Many other systems in biology, economics, society research, chemistry, ... and so on also use differential equations. To find the solutions is usually far more complicated than the equation system itself.
How many solutions to Einsteins field equations exist?

Those solutions heavily depend on initial conditions and that is an essential part in investigations of differential equation systems. They can decide by tiny changes whether you get an attractor, a repeller or a chaotic system.
And even then, which kind of them?

Building up the equations from observations is IMO the simple part.
Sometimes even a picture of a vector field is sufficient.

But why is that there are unexplained phenomena?

Rotation velocity in galaxies is observed but there is no theory explaining them for example.

 

[fresh_42]

Rotation velocity in galaxies is observed but there is no theory explaining them for example.

Oh, there are. We don't lack on theories, e.g. Dark Matter, MOND and probably some more. It's their falsifications or ideally verification that we lack of. One can certainly build up an equation system that describes the observations, but that doesn't provide us a reason why these equations should apply. The art is to derive them from other known and accepted systems or the (very) tough way: derive old and well-known systems from proposed new ones that explain galaxies' rotations. We did this by explaining Newton's gravity as a special case in GR. Next time a new model will have to derive GR! The existence of dark matter seems to be more likely. And in this case we already have our equations.

 

[jonjacson]

But Dark Matter and Mond are just a fit. I can vive you any galaxy distribution... You always will be able to rearrange an invented new material that fits the data. The same with Mond

 

[fresh_42]

The latest I read about it was, that some scientists modeled the universe (including dark matter) from scratch on a highly advanced computer and found an astonishing agreement of their galaxy distributions with reality. Unfortunately I forgot the source, but it was "pop science" anyway. Nevertheless, this is pretty good evidence that the equations are correct. This shows that the difficulty is not to find the correct differential equations. Therefore distribution and behavior of (visible) matter in the universe isn't an example of the difficulty on how to find a correct equation system which is all I wanted to say.

 

[jonjacson]

The latest I read about it was, that some scientists modeled the universe (including dark matter) from scratch on a highly advanced computer and found an astonishing agreement of their galaxy distributions with reality. Unfortunately I forgot the source, but it was "pop science" anyway. Nevertheless, this is pretty good evidence that the equations are correct. This shows that the difficulty is not to find the correct differential equations. Therefore distribution and behavior of (visible) matter in the universe isn't an example of the difficulty on how to find a correct equation system which is all I wanted to say.

http://www.deus-consortium.org/

"... for different dark energy models". I am sure with trial and error you can find an imaginary distribution that meets real data.

http://wwwmpa.mpa-garching.mpg.de/galform/virgo/millennium/

It says they show what the simulation says, but I don't find their comparison with real data.

 

[wrobel]

nobody talks because the question is senseless. Any smooth function satisfies a differential equation. For example the function $y(x)=x^4+3\cos x$ is a solution to the following ODE $y'=4x^3-3\sin x$

 

[jonjacson]

I know that, but that was not the point.

When Maxwell studied electromagnetism he didn't use that method to find his equations, that was the example I have in mind. His introduction of the displacement current to make everything consistent is what I try to learn if there is any theory explaining that method.

 

[Ssnow]

This makes sense in physics (or in science in general) when you know the behavior of a phenomena (usually the analytic expression or a large number of analytic expressions) and you want know if there is an equation that permit you to describe the phenomena in major generality ... usually this involve the use of derivatives in order to construct a differential equation as in the case presented by @wrobel but I think there isn't a general algorithm. It is a question that concern the kind of language used in order to describe the phenomena, there is not a mechanical procedure ...

 

[fresh_42]

I think there isn't a general algorithm. It is a question that concern the kind of language used in order to describe the phenomena, there is not a mechanical procedure ...

That's what I think, too. The key lies in

when you know the behavior of a phenomena

where the question is: What does it mean to know a behavior? It usually means to know some quantities that change in time or place. And the mathematical description of this knowledge are differential equations. There cannot be a standard bridge between the abstract knowledge of behavior and the specific equations. As soon one starts to get hold on the knowledge of behavior in mathematical terms, differential equations will often be the tool of choice.

 

[Ssnow]

To @fresh_42 , in the previous thread I had implicitly assumed the description of the phenomena with an analytic espression, so I chosen also the language (the analysis). But as you observe the question is more general, and with possible links in the philosophy, the behavior can be also an inequality or a sequence of numbers ... Thank you for the clarification!

 

[jonjacson]

And isn't anybody doing research to "mecanize" that process? Mathematicians are curious people its rare that nobody tried to give general rules or theorems, isn't it?

 

[fresh_42]

And isn't anybody doing research to "mecanize" that process? Mathematicians are curious people its rare that nobody tried to give general rules or theorems, isn't it?

Oh, they do! E.g. https://en.wikipedia.org/wiki/Deduction_theorem
Unfortunately you will have to stay either quite general and prove logical statements, or you will have a special situation which requires special methods. As I've said: sometimes even a drawn vector field can be sufficient to assemble the equations. But you need to have something. One section of research is the study of the dependencies on initial conditions. Or already known systems are adapted to new situations.

It would be interesting to know whether there is a sort of dictionary of differential equation systems, like:

predator-prey → Lotka-Volterra
epidemiology → SIR models
electromagnetism → Maxwell equations
quantum mechanics → Schrödinger equation
hydro dynamics → continuity equation, Navier-Stokes, Euler
transformation groups → Lie theory
national economy → Solow-Swan model
etc.

Plus there will be many and often not named ones which apply to different problems, like those for radioactivity, growth of populations, chemical kinetics and so on and so on. It would be a pretty thick lexicon.

 

[pasmith]

I know that, but that was not the point.

When Maxwell studied electromagnetism he didn't use that method to find his equations, that was the example I have in mind. His introduction of the displacement current to make everything consistent is what I try to learn if there is any theory explaining that method.

Without the displacement current, electric charge would not be conserved. Most physicists would regard that as a problem.

Substituting the then known relationships between charge density and electric field and current density and magnetic field into the equation for conservation of charge tells you that making it balance requires that a quantity proportional to the time rate of change of the electric field must be added to the current density, or that a quantity proportional to the divergence of the electric field must be added to the charge density, or both. But the charge density is already proportional to the divergence of the electric field, so the only sensible option is that a quantity proportional to the time rate of change of the electric field be added to the current density.

All of this is a consequence of the specific model, not an instance of a general theory or method.

 

[jonjacson]

Without the displacement current, electric charge would not be conserved. Most physicists would regard that as a problem.

Substituting the then known relationships between charge density and electric field and current density and magnetic field into the equation for conservation of charge tells you that making it balance requires that a quantity proportional to the time rate of change of the electric field must be added to the current density, or that a quantity proportional to the divergence of the electric field must be added to the charge density, or both. But the charge density is already proportional to the divergence of the electric field, so the only sensible option is that a quantity proportional to the time rate of change of the electric field be added to the current density.

All of this is a consequence of the specific model, not an instance of a general theory or method.

Yes, I agree that electric charge should be conserved. :)

Thanks for your post is very interesting.

Oh, they do! E.g. https://en.wikipedia.org/wiki/Deduction_theorem
Unfortunately you will have to stay either quite general and prove logical statements, or you will have a special situation which requires special methods. As I've said: sometimes even a drawn vector field can be sufficient to assemble the equations. But you need to have something. One section of research is the study of the dependencies on initial conditions. Or already known systems are adapted to new situations.

It would be interesting to know whether there is a sort of dictionary of differential equation systems, like:

predator-prey → Lotka-Volterra
epidemiology → SIR models
electromagnetism → Maxwell equations
quantum mechanics → Schrödinger equation
hydro dynamics → continuity equation, Navier-Stokes, Euler
transformation groups → Lie theory
national economy → Solow-Swan model
etc.

Plus there will be many and often not named ones which apply to different problems, like those for radioactivity, growth of populations, chemical kinetics and so on and so on. It would be a pretty thick lexicon.

I didn't know that theorem thanks.

I thought about a similar dictionary too, what real systems could be described with a differential equation would be a very useful tool for science.

 

[Stephen Tashi]

Yes, maybe we should search the simplest set of partial differential equations compatible with the solution. Isn't it strange that nobody talks about this?

Try talking about it by stating the question in mathematical form. Perhaps nobody talks about it because it hasn't been properly formulated as a specific question.

If you can't formulate the general question in mathematical form, try giving a simple example. For example what makes something a "solution". What is a "solution" - the solution set to an equation? How do we determine whether one set of differential equations is "simpler" than another?