Sentence examples for evolution of the solutions from inspiring English sources

In order to see these dynamic properties clearly, we drew the figures for the evolution of the solutions of system (1.2) by using the function dde23 in Matlab; see Figs. 1(a)–1(b).

We notice that the main idea as regards the time-integral nonlocal problems is that only when the time is large, the time-integral term dominates the evolution of the solutions.

In order to see these dynamic properties clearly, we draw the figures for the evolution of the solutions of system (3.5) by using the function ddesd in Matlab; see Figure 1. Figure 1 The phase portrait and time series of system ( 3.5 ).

Let us now consider the time evolution of the solution in Fig. 6.

The fast-slow analysis allows usto derive reduced equations that determine the evolution of the solution during eachof these phases.

In addition, a typical evolution of the solution quality found by the proposed GA is shown in Fig. 5.

Then we suggest an optimized algorithm for adapting the mesh during the evolution of the solution towards the equilibrium state.

Moreover, we obtain sharp estimations for the evolution of the solution along the characteristic curves, which enable us to derive the uniform exponential decay of the associated energies.

These clearly show the movement of the PSO particles for the SVR model in the optimization space and the evolution of the solution quality.

We describe a fast solver for the inhomogeneous heat equation in free space, following the time evolution of the solution in the Fourier domain.

A dynamical adaption strategy to advance the grid in time and to follow the time evolution of the solution directly exploits the multiresolution representation.