Sentence examples for by a classical solution from inspiring English sources
By a classical solution of problem (1.2), we mean a function u such that (uin C^{1}([a,b])), (u'in AC([a,b])), and (u(x)) satisfies (1.2) a.e. on ([a,b]).
So, by a classical solution of (1.1), we mean a function u ∈ C ( [ 0, T ] ) satisfying the following conditions: For every j = 0, 1, …, m, u j = u | ( t j, t j + 1 ) ∈ H 2 ( t j, t j + 1 ) ; u satisfies the boundary condition of (1.1) and the first equation of (1.1); u ′ ( t j + ) and u ′ ( t j − ), j = 0, 1, …, m, exist and the impulsive conditions of (1.1) hold.
With the similar method, by setting, we obtain a classical solution of the following problem.
Hence, it satisfies the Euler-Lagrange equation (as a classical solution by the regularity argument) ( − 1 ) M u ( 2 M ) ( x ) = λ 1 r ˜ ( x ) u ( x ) ( − s ≤ x ≤ s ).
Consequently, the hypotheses of Theorem 4.1 are satisfied, which implies that problem (5.1 - 5.3 5.1 - 5.3assical solution given by (5.5).
This implies the existence of the resolvent operator, and the mild solution is then given by (1.2). which is actually a classical solution if.
A classical solution was derived by letting the series exactly satisfy the governing differential equation and all the boundary conditions at every field and boundary point.
Thus, a classical solution can be derived by letting the series exactly satisfy the governing differential equation at every field point and all the boundary conditions at every boundary point, respectively.
(c) If (varphi, z in D(A_{0})), and f̃ satisfies the local Lipschitz condition (5.6), then there exists a classical solution of (5.7 - 5.9) given by (5.5). .
If (varphi, z in D(A_{0})), and f̃ satisfies the local Lipschitz condition (5.6), then there exists a classical solution of (5.7 - 5.9) given by (5.5).
