The dual equations and analytical solutions of two-dimensional crack problems in piezoelectric ceramics;

New sufficient conditions for the non-existence of positive solution to a nonlinear difference equation with unbounded delay and to its dual equation are obtained, and some of the results in the literature are improved.

By making Laplace and Fourier transformation as well as sine and cosine transformation to moving differential equations and various responses,the dual equation which is constructed from boundary conditions lastly was solved.

The correspondence principle and variational method were employed to introduce a Hamiltonian system method for dealing with the bending problem of viscoelastic cantilever-beams,so that fundamental eigenvectors of dual equations,i.

The Newton-PGCG algorithm is proposed to solve the Kuhn-Tucker equations.

By using the Fourier transform,the problem can be solved with a pair of dual integral equations in which the unknown variable is the jump of displacements across the crack surfaces.

With Fourier transform,the problem is evolved as dual integral equations where the unknown variable is taken as the jump of the displacements across the cract surface.

By using the Fourier transform,theproblem can be solved with the help of a pair of dual integral equations in which the unknown variable is thejump of the displacements across the crack surfaces.