In this note, we shall discuss the weak form of several ODEs via integration by parts. To be more precise, we are interested in the following two simple ODEs
in
and
in
where the given function is assumed to be continuous.
1. THE CASE OF
Initially, if solves in the classical sense, then must be of class . We call a strong solution. In the weak sense, we can only assume that is continuous. But we need more conditions. To be exact, we require the following
holds for any test function with . If the above is satisfied, we call a weak solution. We shall prove the following
(more…)Theorem 1. Assume that . Then any weak solution to is also a strong solution.