Sentence examples for by numerical methods such from inspiring English sources

Analytical solutions form an important basis for comparison and verification of results obtained by numerical methods such as the finite element method.

Considering that (15) is a nonlinear equation, its solution (hat {gamma }) can be obtained efficiently by numerical methods such as a Newton's iterative method.

Solving partial differential equations by numerical methods such as the finite element method requires the solution of large sparse linear systems.

Dynamic response of the wind turbine tower subjected to wind forces is usually estimated by numerical methods such as finite element method (FEM).

The numerical results can also be employed to verify the accuracy of the solution by numerical methods, such as the finite element and difference methods, when analyzing laminated composites made of quasicrystals.

However, for laterally or three-dimensionally heterogeneous Earth models the validation of the use of Earth-flattening approximation should be eventually checked by comparing the solutions with those computed by numerical methods such as the FDTD without the approximation for each of the same models.

By employing (15), and in (10) can be effectively obtained by standard numerical methods such as the function "fsolve" in Matlab.

The exact expression of P u * is not easily obtained and can be found by the numerical methods, such as the bisection method.

We need point out that the fourth-order compact difference method, which is employed to discretize the differential equation, can be replaced by other numerical methods, such as the FE method and the finite volume method.

The optimal solution to the subproblem (P2-2) is achieved at ( widehat{beta}={beta}^ ), where β* is the point at which η d,EE reaches its maximum for β ∈ [0, + ∞) and is obtained by solving the equation ( frac{d{eta}_{d,mathrm{E}mathrm{E}}}{dbeta }=0 ), as shown in Appendix 4. The exact expression of β* is not easily obtained and can be found by the numerical methods, such as the bisection method.

Solving FDEs by the classical mesh-dependent numerical methods such as the finite difference methods (FDM), the finite volume methods (FVM), and the finite element methods (FEM) requires the mesh generation, which appears to be computationally costly.