Sentence examples for by numerical solutions using from inspiring English sources

These theoretical results are verified by numerical solutions using the fourth order Runge-Kutta algorithm.

Our theoretical findings are confirmed by numerical solutions using three two-dimension benchmark problems, i.e. the channel flow, the cavity flow, and the decaying Taylor Green vortex flow.

We resort to the numerical solutions using MCMC and specifically, the Metropolis-Hasting (M-H) algorithm.

The accuracy of analytical solutions is verified by numerical solutions obtained using a shooting technique that uses a Runge-Kutta-Felhberg integration scheme and a Newton-Raphson correction scheme.

The numerical solution uses finite difference method.

Energy 34, 910 921] and also independent results obtained by numerical solutions of the related integral equation via use of the singularity subtraction technique and quadratures.

These predictions were confirmed by numerical solutions (Figs. 4,5).

Information about the stability and oscillatory behavior is obtained by numerical solutions.

The catalyst effectiveness factor is given by a simple expression that depends solely on the Thiele modulus and first series coefficient which is found by numerical solution of the dual-series using the method of weighted residuals.

The values obtained were compared with those obtained by numerical solution of the exact model and by using approximate methods reported in the literature.

The predictor is designed using Lyapunov-based methods, by numerical solution of a linear matrix inequality.