Sentence examples for boundary conditions we get from inspiring English sources

Namely, for a two-dimensional system of ordinary equations and a particular case of boundary conditions we get the following.

Applying the proper boundary conditions, we get the solution of the deformations (i.e., vertical displacements and rotations) in the strip footing.

We see that for both local and non-local boundary conditions we get (U t)) in matrix form as U t)=mathbf{H_{N}^{T}}mathbf{E}_{Ntimes N} boldsymbol { mho}_{mathbf{N}}^{boldsymbol {omega}}(mathbf{t})+mathbf{G_{N}^{T}} boldsymbol {mho}_{mathbf{N}}^{boldsymbol {omega }}(mathbf{t}).

By using integration by parts and the periodic boundary conditions, we get ( d v m d t, A u m ) = 1 2 d d t ( ∥ u m ∥ 2 + | A u m | 2 ), ( [ G ( u m ) ] x ( 4 ), A u m ) = ∫ Ω [ G ( u m ) ] x ( 4 ) A u m d x = ∫ Ω G u m ′ ( u m ) u m x u m x x x x x d x.

It also matters whether Dirichlet or Neumann boundary conditions are considered: for the Dirichlet boundary condition we get a better remainder estimate.

For the concavity of u and the boundary condition, we get 1 - ∑ i = 1 m - 2 α i u ( 0 ) ≤ ∑ i = 1 m - 2 α i max u Δ ( t ).

For the concavity of u and the boundary condition, we get 1 - ∑ i = 1 m - 2 β i u ( T ) ≤ ∑ i = 1 m - 2 β i ( T - ξ i ) max u Δ ( t ).

According to Green's formula, Lemma 1 and the Dirichlet boundary condition, we get ∫ Ω ∑ l = 1 m e i ( t ) ∂ ∂ x l ( D i l ∂ e i ( t ) ∂ x l ) d x ≤ − ∫ Ω ∑ l = 1 m D i l d l 2 ( e i ( t ) ) 2 d x. (11).

end{aligned} (23) According to Green's formula and the Dirichlet boundary condition, we get int_{Omega} sum_{l = 1}^{m} z_{i} ( t )frac{partial}{ partial x_{l}} biggl( D_{il} frac{partial z_{i} ( t )}{partial x_{l}} biggr),dx = - sum_{l = 1}^{m} int_{Omega} D_{il} biggl( frac{partial z_{i} ( t )}{partial x_{l}} biggr)^{2},dx.

According to Green's formula and the Dirichlet boundary condition, we get ∫ Ω ∑ k = 1 l ( u i − u i ∗ ) ∂ ∂ x k ( D i k ∂ ( u i − u i ∗ ) ∂ x k ) d x = − ∑ k = 1 l ∫ Ω D i k ( ∂ ( u i − u i ∗ ) ∂ x k ) 2 d x. (14).

For homogeneous Dirichlet boundary condition we obtain (varphi_{j}, lambda_{j} )= biggl( biggl[ frac {(j+1)pi}{b-a} biggr]^{2}, sqrt{frac{2}{b-a}}sin biggl[ frac {(x-a pi(j+1)}{b-a} biggr] biggr), (3.13) and for homogeneous Neumann boundary condition we get (varphi_{j}, lambda_{j} )= biggl( biggl[ frac{pi j}{b-a} biggr]^{2}, sqrt{frac{2}{b-a}}sin biggl[ frac{(x-a pi j+1b-a} biggr] biggr).