Sentence examples for using a direct variational from inspiring English sources
The authors proved the existence of entire solutions of (1.10) by using a direct variational method and the mountain pass theorem.
According to this, we proceed as follows: The image is divided into relatively small overlap-ping patches, and the energy functional is minimized on each particular patch independently by using a direct variational minimization.
The proofs make use of a direct variational approach and of integral identities.
By means of a direct variational approach, we establish conditions ensuring the existence and multiplicity of solutions for the problem.
In order to avoid this difficulty of picking the correct extremum in a direct variational treatment we instead present a constructive proof.
Using the direct variational minimizing method, we get the following theorem.
By using direct variational method, he gave sufficientconditions for the existence of a minimizing sequence for the case of coercivepotential, or some averaged coercivity conditions of the Ahmad-Lazer-Paul type addingthe nonlinearity satisfies some growth conditions, or the convex potential.
Then by using a generalized variational functional which absorbs the periodicity condition, an eigenfunction expansion–variational method based on a unit cell is developed to solve such problems.
A size-dependent functionally graded piezoelectric beam model is developed using a variational formulation.
The governing equations are derived using a variational formulation based on a thermodynamic framework.
