Ngô Quốc Anh

February 25, 2023

A weak form of several ODEs via integration by parts

Filed under: Uncategorized — Ngô Quốc Anh @ 21:59

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

u'=f in [a,b]

and

u''=f in [a,b]

where the given function f is assumed to be continuous.

1. THE CASE OF u'=f

Initially, if u solves u'=f in the classical sense, then u must be of class C^1. We call u a strong solution. In the weak sense, we can only assume that u is continuous. But we need more conditions. To be exact, we require the following

\displaystyle \int_a^b u(x) \phi'(x) dx = - \int_a^b f(x) \phi(x) dx

holds for any test function \phi \in C^1([a,b]) with \phi (a)=\phi(b)=0. If the above is satisfied, we call u a weak solution. We shall prove the following

Theorem 1. Assume that f \in C([a,b]). Then any weak solution to u'=f is also a strong solution.

(more…)

Create a free website or blog at WordPress.com.