Sentence examples for are provided to validate from inspiring English sources

Various numerical results are provided to validate the proposed formulation.

Accurate numerical results are provided to validate the proposed algorithm.

Various numerical experiments are provided to validate the new schemes.

Finally, numerical simulations are provided to validate the theoretical results.

Numerical simulations are provided to validate the results.

Simulation results are provided to validate the proposed algorithm.

Finally, examples are provided to validate the theoretical results.

The test results are provided to validate the effectiveness.

Three simulation examples are provided to validate the proposed approaches.

Numerical simulation results are provided to validate the approach.

Numerical studies in the context of several statistical models are provided to validate the theoretical results.