Sentence examples for using the variational approaches from inspiring English sources

In 2005, Guo and Yu [6] took the lead in using the variational approaches to study the existence of multiple periodic solutions for (1.1), and a multiplicity result was given.

Recently, using the variational approaches, the multiplicity of the periodic solutions for the following system: { u ′ ( t ) = − Λ u ( t + r ) − f ( t, u ( t − r ) ), u ( 0 ) = − u ( 2 r ), u ( 0 ) = u ( 4 r ). was studied by Wu and Wu in [7].

Calculations have been performed using the variational approach, developed in [27].

Therefore, the capability of analyzing cracked symmetric laminates using the variational approach has been enhanced significantly.

Governing equations and both classical and non-classical boundary conditions of motion are obtained using the variational approach.

Under the proper assumptions, we obtain the existence of nontrivial solutions to perturbed p-Laplacian system by using the variational approach.

By using the variational approach, under different conditions, the author obtained that system (1.3) has at least one or (N+1) geometrically distinct T-periodic solutions.

Phase-field evolution equations, considered in this work, are derived using the variational approach, and correspond to the conservative Allen Cahn-type equAllen Cahn-type

In the present paper, static deformation of an eccentrically stiffened plates with partial composite action was analyzed by using the variational approach.

Recently, in [1] and [14], by using the variational approach, Mawhin investigated the following second order nonlinear difference systems with ϕ-Laplacian: Δ ϕ [ Δ u ( n − 1 ) ] = ∇ u F [ n, u ( n ) ] + h ( n ) ( n ∈ Z ), (1.3).

Based on the principle of the minimum potential energy, the governing differential equations were derived by using the variational approach with taking into consideration of strain energy of connectors between plate and stiffeners and associated boundary conditions as well.