In this paper,the generalised two-dimensional differential transform method (DTM) of solving the time-fractional coupled KdV equations is proposed.The fractional derivative is described in the Caputo sense.The presented method is a numerical method based on the generalised Taylor series expansion which constructs an analytical solution in the form of a polynomial.An illustrative example shows that the generalised two-dimensional DTM is effective for the coupled equations.
The eigenvalue problem of the Hamiltonian operator associated with plane elasticity problems is investigated.The eigenfunctions of the operator are directly solved with mixed boundary conditions for the displacement and stress in a rectangular region.The completeness of the eigenfunctions is then proved,providing the feasibility of using separation of variables to solve the problems.A general solution is obtained with the symplectic eigenfunction expansion theorem.
The completeness theorem of the eigenfunction systems for the product of two 2 × 2 symmetric operator matrices is proved.The result is applied to 4 × 4 infinite-dimensional Hamiltonian operators.A modified method of separation of variables is proposed for a separable Hamiltonian system.As an application of the theorem,the general solutions for the plate bending equation and the free vibration of rectangular thin plates are obtained.Finally,a numerical test is analysed to show the correctness of the results.
The eigenvalue problem for the Hamiltonian operator associated with the mathematical model for the deflection of a thin elastic plate is investigated.First,the problem for a rectangular plate with simply supported edges is solved directly.Then,the completeness of the eigenfunctions is proved,thereby demonstrating the feasibility of using separation of variables to solve the problem. Finally,the general solution is obtained by using the proved expansion theorem.
