Hadamard and well-posed problems.

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In summary, a well-posed problem, as defined by Hadamard, has three properties: the solution is unique, the solution depends continuously on the data, and the solution satisfies the initial conditions. The solution for a problem may also depend on the data and initial conditions, but as long as it satisfies these three properties, it can be considered a well-posed problem.
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Kizaru
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Hadamard and "well-posed" problems.

I can't really find much clarification on Hadamard's definition of a well-posed problem.
My confusion comes from knowing exactly what is meant by the second and third properties:
2) The solution is unique
3) The solution depends continuously on the data, in some reasonable topology. http://en.wikipedia.org/wiki/Well-posed_problem" [Broken]

For example, a solution would exist for a PDE y * uxx = x * uyy with u(0,y) = 2 and u(x,0) = 2 that is simply a constant. But is the solution, u=2, considered unique? Does the solution depend on the data?

Does the solution for u(x,y) fail property 3 because it no longer satisfies the PDE if the initial conditions are changed?

Edit: Maybe this belongs in the Differential Equations forum. Sorry. Can this be moved? Thanks.
 
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  • #2


Kizaru said:
I can't really find much clarification on Hadamard's definition of a well-posed problem.
My confusion comes from knowing exactly what is meant by the second and third properties:
2) The solution is unique
3) The solution depends continuously on the data, in some reasonable topology. http://en.wikipedia.org/wiki/Well-posed_problem" [Broken]

For example, a solution would exist for a PDE y * uxx = x * uyy with u(0,y) = 2 and u(x,0) = 2 that is simply a constant. But is the solution, u=2, considered unique?
Is there any other function of x and y that will satisfy those conditions? If not, then it is unique.

Does the solution depend on the data?
Well suppose the problem were exactly the same differential equation but u(0,y)= 3 and u(x,0)= 3. Then u(x,y)= 3 is a solution. Since changing the "data" changes the solution, yes, the solution depends on the data.

Does the solution for u(x,y) fail property 3 because it no longer satisfies the PDE if the initial conditions are changed?
No, property 3 does not say that the solution must be independent of the initial conditions, it say it must depend on them continuously. suppose the intial conditions were [itex]u(x, 0)= 2+\delta[/itex], [itex]u(0, y)= 2+ \delta[/itex]. Then [/itex]u(x,y)= 2+ \delta[/ itex] is obviously a solution. And, as [itex]\delta[/itex] goes to 0, that solution goes to u(x,y)= 2. In that situation, the solution depends continuously on the initial values.

Edit: Maybe this belongs in the Differential Equations forum. Sorry. Can this be moved? Thanks.
Okay, I will move it.
 
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  • #3


Thank you so much! It's much clearer than my professor's explanation :)
 

What is a Hadamard problem?

A Hadamard problem is a type of mathematical problem that involves finding a solution or set of solutions to an equation or set of equations. These problems are named after French mathematician Jacques Hadamard, who first studied them in the early 20th century.

What makes a problem "well-posed"?

A well-posed problem is one that has a unique solution, is solvable, and the solution is sensitive to changes in the initial conditions of the problem. This means that small changes in the initial conditions will result in small changes in the solution, making the problem stable and reliable.

How are Hadamard problems related to well-posed problems?

Hadamard problems are a subset of well-posed problems. This means that all Hadamard problems are well-posed, but not all well-posed problems are Hadamard problems. Hadamard problems have the additional requirement of linear independence between the equations, which ensures a unique solution.

What techniques are used to solve Hadamard and well-posed problems?

There are various techniques used to solve Hadamard and well-posed problems, such as numerical methods, analytical methods, and optimization techniques. The choice of technique depends on the specific problem and its complexity.

What are some real-world applications of Hadamard and well-posed problems?

Hadamard and well-posed problems have various applications in physics, engineering, economics, and other fields. For example, they can be used to model and solve problems in fluid mechanics, electrical circuits, and economic systems. They are also important in data analysis and machine learning algorithms.

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