Sentence examples for modes of the solution from inspiring English sources

A Galerkin projection is used to derive a system of deterministic equations for the stochastic modes of the solution.

The main idea of the method is to use a unique set of particles to transport the stochastic modes of the solution.

An elegant solution developed in the context of spectral methods by Eitan Tadmor and coworkers is to add diffusion only to the high frequency modes of the solution and can lead to stabilization without sacrificing accuracy.

The adaptation relies on an estimate of the eigenvalues and eigenvectors of the Galerkin Jacobian matrix of the deterministic system of the stochastic modes of the solution and on a correspondence between these approximate eigenvalues and eigenvectors for the intermediate states considered at the interface.

The stable states Xss1 and Xss2 were computed as the steady states of the deterministic reaction-rate model (1) and confirmed by identifying the modes of the solution of the CME after a sufficiently long stochastic simulation time.

The influence of the self-stressing modes on the solution is firt demonstrated theoretically and then shown practically on the results obtained via a family of equilibrium elements in which the number of self-stressing modes relative to the number of vibration modes may be varied as a parameter.

In this case setting pthreshold to a large value might lead to discovering only one mode of the solution.

To ensure asymptotic matching condition Eq. (15) is satisfied, one must set (16) 𝓟 [ Φ 0 − ℛ Φ 0 + Φ 1 − ℛ Φ 1 − ϕ 0 ] = 0. Here, 𝓟 is the operator that projects boundary sources for the one-dimensional half space problem onto the mode of the solution that does not decay as ζ → 0. We give more details about 𝓟 in Section 3.

The physical relevance of each type of mode for the solution is clarified through two numerical test cases, a homogeneous medium and a circular bar waveguide example, excited by a point source.

In this context, the complete solution in a series form to the Timoshenko beam is investigated, and it is shown for the first time that a particular mode shape of the solution is naturally expressed by an ordered pair of characteristic values, rather than a single characteristic value.

This method provides a more suitable mode of controlling the solution deposition than the existing deposition methods such as spray painting.