existance of solution of a (mathematical) problem
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From: fisico32 on 16 Nov 2009 18:11 Hello Forum, a question about existence of solution. Given a problem, we decide for the best differential equation that would model the situation, apply initial conditions and boundary conditions. What would cause the problem to not have a solution? For sure, there is an actual physical solution occurring for an observed phenomenon.Why is there not a mathematical solution for it? Is is the difficulty of the problem or some inherent contradiction/flow in how the problem was defined mathematically? Any simple example? thanks fisico32
From: Jerry Avins on 16 Nov 2009 18:24 fisico32 wrote: > Hello Forum, > > a question about existence of solution. > > Given a problem, we decide for the best differential equation that would > model the situation, apply initial conditions and boundary conditions. > What would cause the problem to not have a solution? > > For sure, there is an actual physical solution occurring for an observed > phenomenon.Why is there not a mathematical solution for it? > Is is the difficulty of the problem or some inherent contradiction/flow in > how the problem was defined mathematically? > Any simple example? Not all solutions can be expressed in closed form. Is that related to what you mean? Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
From: Michael Plante on 16 Nov 2009 22:13 >fisico32 wrote: >> Hello Forum, >> >> a question about existence of solution. >> >> Given a problem, we decide for the best differential equation that would >> model the situation, apply initial conditions and boundary conditions. >> What would cause the problem to not have a solution? >> >> For sure, there is an actual physical solution occurring for an observed >> phenomenon.Why is there not a mathematical solution for it? >> Is is the difficulty of the problem or some inherent contradiction/flow in >> how the problem was defined mathematically? >> Any simple example? > >Not all solutions can be expressed in closed form. Is that related to >what you mean? > I don't have an example, but there's at least one further difficulty. http://en.wikipedia.org/wiki/Initial_value_problem See the "Existence and uniqueness" section. I checked two books that discuss Lipschitz (they happen to be books on numerical approaches, but have a small amount of theory up front), but neither give an example of a differential equation that lacks a solution, nor does the link above. I think this might be what the OP is getting at, though. If I had to guess, a differential equation that properly describes a real-world situation should have a solution (though there are frequently OTHER difficulties if you go the numerical route, but I don't think that's what's being asked).
From: Tim Wescott on 17 Nov 2009 00:36 On Mon, 16 Nov 2009 21:13:04 -0600, Michael Plante wrote: >>fisico32 wrote: >>> Hello Forum, >>> >>> a question about existence of solution. >>> >>> Given a problem, we decide for the best differential equation that > would >>> model the situation, apply initial conditions and boundary conditions. >>> What would cause the problem to not have a solution? >>> >>> For sure, there is an actual physical solution occurring for an > observed >>> phenomenon.Why is there not a mathematical solution for it? Is is the >>> difficulty of the problem or some inherent contradiction/flow > in >>> how the problem was defined mathematically? Any simple example? >> >>Not all solutions can be expressed in closed form. Is that related to >>what you mean? >> >> > I don't have an example, but there's at least one further difficulty. > > http://en.wikipedia.org/wiki/Initial_value_problem > > See the "Existence and uniqueness" section. I checked two books that > discuss Lipschitz (they happen to be books on numerical approaches, but > have a small amount of theory up front), but neither give an example of > a differential equation that lacks a solution, nor does the link above. > I think this might be what the OP is getting at, though. > > If I had to guess, a differential equation that properly describes a > real-world situation should have a solution (though there are frequently > OTHER difficulties if you go the numerical route, but I don't think > that's what's being asked). OTOH, if the real world situation involves Newtonian dynamics with friction, then it may be practically impossible to adequately (never mind 'properly', whatever that may mean) describe it with differential equations. -- www.wescottdesign.com
From: Rune Allnor on 17 Nov 2009 05:55
On 17 Nov, 00:11, "fisico32" <marcoscipio...(a)gmail.com> wrote: > Hello Forum, > > a question about existence of solution. You have asked a number of question over the past several months that seem to share one common denominator: You confuse mathemathics with physics. And vice versa. Try to start with the following: 1) Mathemathics is a philosophical excercise, based on axioms and logic. Mathemathicians prove theorems based on axioms and logic. 2) Physics is the study of th ereal world. Physicicsts attempt to understand and describe the real world. It truns out that mathemathics is a convenient foundation for understanding the real world (as opposed to various historical attempts that were based on deities and spirits and the likes) and understanding and using physics. However, as interlinked mathemathics and physics are, one should be aware of that: 1) There are aspects of mathemathics that have no direct use in physics, and thus have no 'physical' interpretation. 2) For physics to be expressable and understandable in terms of maths, problem statements have to be simplified (some time over-simplified) 3) For computations to be practical or possible, mathemathical simplifications are made in the algebra. If you start with an (over)simplified problem statement and continue to make algebraic approximations (e.g. sin(x)~x for small x), it is perfectly possible to end up with a mathemathical expression that has no solution. But again, you seem to be stuck in the double trap that "all maths must have a physical interpretation" and "physics is perfectly applied mathemathics." You might be better off if you manage to break out of those constraints. Rune |