# Does a Unique Solution Exist for a PDE with Specific Boundary Conditions?

• t.t.h8701
In summary, the conversation discusses a PDE with given boundary conditions and the question of whether there is a unique solution at a specific point. It is suggested that this problem can be solved using characteristics lines, but a potential issue is raised regarding extending the solution beyond the given interval due to the arbitrary function involved in the solution.
t.t.h8701
If I have a PDE like Ux-Uy=0 and U(x,0)=f(x) when x in [0,1]. Then is there an uniqueness solution exist at point (5,1)?
How can I explain it using characteristics lines?

Thanks

t.t.h8701 said:
If I have a PDE like Ux-Uy=0 and U(x,0)=f(x) when x in [0,1]. Then is there an uniqueness solution exist at point (5,1)?
How can I explain it using characteristics lines?

Thanks
If you only know f(x) between 0 and 1, you are going to have a problem extending the solution to x= 5!

The "characteristic lines" are of the form x+ y= C and any solution to this equation is of the form F(x+y) where F is an arbitrary function of one variable. Since you require that U(x,0)= F(x+0)= F(x)= f(x) for x between 0 and 1, you will need to take F(x) to be f(x) between 0 and 1 but that does not define it for x+ y= 5+ 1= 6. Consider any number of functions "f(x)" which are identical between 0 and 1 but differ outside that interval.

## 1. What is the uniqueness of a solution in scientific research?

The uniqueness of a solution in scientific research refers to the idea that there can only be one correct answer or solution to a given problem or question. This means that the solution must be able to explain and predict the observed data or phenomena in a consistent and reliable manner.

## 2. How is the uniqueness of a solution determined in scientific experiments?

The uniqueness of a solution is determined through rigorous testing and analysis in scientific experiments. This involves designing experiments that can rule out alternative explanations and only support the proposed solution. Additionally, replication of the experiment by other researchers can also help to establish the uniqueness of the solution.

## 3. Can there be multiple unique solutions to a scientific problem?

It is possible for there to be multiple unique solutions to a scientific problem, especially in complex and multifaceted research areas. However, it is important to ensure that these solutions are not contradictory and can be integrated together to form a comprehensive understanding of the problem.

## 4. Why is the uniqueness of a solution important in scientific research?

The uniqueness of a solution is important in scientific research as it allows for the development of reliable and accurate theories and models. It also ensures that the scientific community can come to a consensus on a particular topic, leading to advancements and progress in the field.

## 5. How does the uniqueness of a solution impact the validity of scientific findings?

The uniqueness of a solution is a key factor in determining the validity of scientific findings. If a solution is not unique, it may not fully explain the observed data or may be based on faulty assumptions. This can lead to unreliable conclusions and may hinder the progress of scientific understanding.

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