Partial Differential Equations : Existence & Uniqueness

Study concepts, example questions & explanations for Partial Differential Equations

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Example Questions

Example Question #1 : Existence & Uniqueness

Determine if the statement is true or false:

The wave equation has at most one solution. 

Possible Answers:

False

True

Correct answer:

True

Explanation:

This statement is true by the Uniqueness Theorem.

 is twice differential equation in terms of  and .

Now, consider the energy integral

After performing integration by parts results in the following,

From here, the initial conditions and boundary conditions are applied.

Therefore,

which proves the wave equations has only one solution and thus is unique.

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